array_api._2022_12 module¶

class array_api._2022_12.Array[source]¶

Bases: Protocol, Generic

property T: Self¶

Transpose of the array.

The array instance must be two-dimensional. If the array instance is not two-dimensional, an error should be raised.

Returns:: out – two-dimensional array whose first and last dimensions (axes) are permuted in reverse order relative to original array. The returned array must have the same data type as the original array.
Return type:: array

Note

Limiting the transpose to two-dimensional arrays (matrices) deviates from the NumPy et al practice of reversing all axes for arrays having more than two-dimensions. This is intentional, as reversing all axes was found to be problematic (e.g., conflicting with the mathematical definition of a transpose which is limited to matrices; not operating on batches of matrices; et cetera). In order to reverse all axes, one is recommended to use the functional permute_dims interface found in this specification.

_abc_impl = <_abc._abc_data object>¶

_is_protocol = True¶

_is_runtime_protocol = True¶

property device: TDevice¶

Hardware device the array data resides on.

Returns:: out – a device object (see device-support).
Return type:: device

property dtype: TDtype¶

Data type of the array elements.

Returns:: out – array data type.
Return type:: dtype

property mT: Self¶

Transpose of a matrix (or a stack of matrices).

If an array instance has fewer than two dimensions, an error should be raised.

Returns:: out – array whose last two dimensions (axes) are permuted in reverse order relative to original array (i.e., for an array instance having shape (..., M, N), the returned array must have shape (..., N, M)). The returned array must have the same data type as the original array.
Return type:: array

property ndim: int¶

Number of array dimensions (axes).

Returns:: out – number of array dimensions (axes).
Return type:: int

property shape: tuple[int | None, ...]¶

Array dimensions.

Returns:: out – array dimensions. An array dimension must be None if and only if a dimension is unknown.
Return type:: Tuple[Optional[int], …]

Note

For array libraries having graph-based computational models, array dimensions may be unknown due to data-dependent operations (e.g., boolean indexing; A[:, B > 0]) and thus cannot be statically resolved without knowing array contents.

Note

The returned value should be a tuple; however, where warranted, an array library may choose to return a custom shape object. If an array library returns a custom shape object, the object must be immutable, must support indexing for dimension retrieval, and must behave similarly to a tuple.

property size: int | None¶

Number of elements in an array.

Note

This must equal the product of the array’s dimensions.

Returns:: out – number of elements in an array. The returned value must be None if and only if one or more array dimensions are unknown.
Return type:: Optional[int]

Note

For array libraries having graph-based computational models, an array may have unknown dimensions due to data-dependent operations.

to_device(device: TDevice, /, *, stream: int | Any | None = None) → Self[source]¶

Copy the array from the device on which it currently resides to the specified device.

Parameters:

self – array instance.
device (device) – a device object (see device-support).
stream (Optional[Union[int, Any]]) – stream object to use during copy. In addition to the types supported in array.__dlpack__(), implementations may choose to support any library-specific stream object with the caveat that any code using such an object would not be portable.

Returns:

out – an array with the same data and data type as self and located on the specified device.

Return type:

array

Note

If stream is given, the copy operation should be enqueued on the provided stream; otherwise, the copy operation should be enqueued on the default stream/queue. Whether the copy is performed synchronously or asynchronously is implementation-dependent. Accordingly, if synchronization is required to guarantee data safety, this must be clearly explained in a conforming library’s documentation.

class array_api._2022_12.ArrayNamespace(*args, **kwargs)[source]¶

Bases: Protocol, Generic

Device: TDevice¶

_abc_impl = <_abc._abc_data object>¶

_is_protocol = True¶

_is_runtime_protocol = True¶

abs: abs¶

Calculates the absolute value for each element x_i of the input array x.

For real-valued input arrays, the element-wise result has the same magnitude as the respective element in x but has positive sign.

Note

For signed integer data types, the absolute value of the minimum representable integer is implementation-dependent.

Note

For complex floating-point operands, the complex absolute value is known as the norm, modulus, or magnitude and, for a complex number $z = a + bj$ is computed as

\[\operatorname{abs}(z) = \sqrt{a^2 + b^2}\]

Note

For complex floating-point operands, conforming implementations should take care to avoid undue overflow or underflow during intermediate stages of computation.

Parameters:: x (array) – input array. Should have a numeric data type.
Returns:: out – an array containing the absolute value of each element in x. If x has a real-valued data type, the returned array must have the same data type as x. If x has a complex floating-point data type, the returned array must have a real-valued floating-point data type whose precision matches the precision of x (e.g., if x is complex128, then the returned array must have a float64 data type).
Return type:: array

Notes

Special Cases

For real-valued floating-point operands,

If x_i is NaN, the result is NaN.
If x_i is -0, the result is +0.
If x_i is -infinity, the result is +infinity.

For complex floating-point operands, let a = real(x_i), b = imag(x_i), and

If a is either +infinity or -infinity and b is any value (including NaN), the result is +infinity.
If a is any value (including NaN) and b is either +infinity or -infinity, the result is +infinity.
If a is either +0 or -0, the result is equal to abs(b).
If b is either +0 or -0, the result is equal to abs(a).
If a is NaN and b is a finite number, the result is NaN.
If a is a finite number and b is NaN, the result is NaN.
If a is NaN and b is NaN, the result is NaN.

Changed in version 2022.12: Added complex data type support.

acos: acos¶

Calculates an implementation-dependent approximation of the principal value of the inverse cosine for each element x_i of the input array x.

Each element-wise result is expressed in radians.

Note

The principal value of the arc cosine of a complex number $z$ is

\[\operatorname{acos}(z) = \frac{1}{2}\pi + j\ \ln(zj + \sqrt{1-z^2})\]

For any $z$,

\[\operatorname{acos}(z) = \pi - \operatorname{acos}(-z)\]

Note

For complex floating-point operands, acos(conj(x)) must equal conj(acos(x)).

Note

The inverse cosine (or arc cosine) is a multi-valued function and requires a branch cut on the complex plane. By convention, a branch cut is placed at the line segments $(-\infty, -1)$ and $(1, \infty)$ of the real axis.

Accordingly, for complex arguments, the function returns the inverse cosine in the range of a strip unbounded along the imaginary axis and in the interval $[0, \pi]$ along the real axis.

Note: branch cuts follow C99 and have provisional status (see branch-cuts).

Parameters:: x (array) – input array. Should have a floating-point data type.
Returns:: out – an array containing the inverse cosine of each element in x. The returned array must have a floating-point data type determined by type-promotion.
Return type:: array

Notes

Special cases

For real-valued floating-point operands,

If x_i is NaN, the result is NaN.
If x_i is greater than 1, the result is NaN.
If x_i is less than -1, the result is NaN.
If x_i is 1, the result is +0.

For complex floating-point operands, let a = real(x_i), b = imag(x_i), and

If a is either +0 or -0 and b is +0, the result is π/2 - 0j.
If a is either +0 or -0 and b is NaN, the result is π/2 + NaN j.
If a is a finite number and b is +infinity, the result is π/2 - infinity j.
If a is a nonzero finite number and b is NaN, the result is NaN + NaN j.
If a is -infinity and b is a positive (i.e., greater than 0) finite number, the result is π - infinity j.
If a is +infinity and b is a positive (i.e., greater than 0) finite number, the result is +0 - infinity j.
If a is -infinity and b is +infinity, the result is 3π/4 - infinity j.
If a is +infinity and b is +infinity, the result is π/4 - infinity j.
If a is either +infinity or -infinity and b is NaN, the result is NaN ± infinity j (sign of the imaginary component is unspecified).
If a is NaN and b is a finite number, the result is NaN + NaN j.
If a is NaN and b is +infinity, the result is NaN - infinity j.
If a is NaN and b is NaN, the result is NaN + NaN j.

Changed in version 2022.12: Added complex data type support.

acosh: acosh¶

Calculates an implementation-dependent approximation to the inverse hyperbolic cosine for each element x_i of the input array x.

Note

The principal value of the inverse hyperbolic cosine of a complex number $z$ is

\[\operatorname{acosh}(z) = \ln(z + \sqrt{z+1}\sqrt{z-1})\]

For any $z$,

\[\operatorname{acosh}(z) = \frac{\sqrt{z-1}}{\sqrt{1-z}}\operatorname{acos}(z)\]

or simply

\[\operatorname{acosh}(z) = j\ \operatorname{acos}(z)\]

in the upper half of the complex plane.

Note

For complex floating-point operands, acosh(conj(x)) must equal conj(acosh(x)).

Note

The inverse hyperbolic cosine is a multi-valued function and requires a branch cut on the complex plane. By convention, a branch cut is placed at the line segment $(-\infty, 1)$ of the real axis.

Accordingly, for complex arguments, the function returns the inverse hyperbolic cosine in the interval $[0, \infty)$ along the real axis and in the interval $[-\pi j, +\pi j]$ along the imaginary axis.

Note: branch cuts follow C99 and have provisional status (see branch-cuts).

Parameters:: x (array) – input array whose elements each represent the area of a hyperbolic sector. Should have a floating-point data type.
Returns:: out – an array containing the inverse hyperbolic cosine of each element in x. The returned array must have a floating-point data type determined by type-promotion.
Return type:: array

Notes

Special cases

For real-valued floating-point operands,

If x_i is NaN, the result is NaN.
If x_i is less than 1, the result is NaN.
If x_i is 1, the result is +0.
If x_i is +infinity, the result is +infinity.

For complex floating-point operands, let a = real(x_i), b = imag(x_i), and

If a is either +0 or -0 and b is +0, the result is +0 + πj/2.
If a is a finite number and b is +infinity, the result is +infinity + πj/2.
If a is a nonzero finite number and b is NaN, the result is NaN + NaN j.
If a is +0 and b is NaN, the result is NaN ± πj/2 (sign of imaginary component is unspecified).
If a is -infinity and b is a positive (i.e., greater than 0) finite number, the result is +infinity + πj.
If a is +infinity and b is a positive (i.e., greater than 0) finite number, the result is +infinity + 0j.
If a is -infinity and b is +infinity, the result is +infinity + 3πj/4.
If a is +infinity and b is +infinity, the result is +infinity + πj/4.
If a is either +infinity or -infinity and b is NaN, the result is +infinity + NaN j.
If a is NaN and b is a finite number, the result is NaN + NaN j.
If a is NaN and b is +infinity, the result is +infinity + NaN j.
If a is NaN and b is NaN, the result is NaN + NaN j.

Changed in version 2022.12: Added complex data type support.

add: add¶

Calculates the sum for each element x1_i of the input array x1 with the respective element x2_i of the input array x2.

Parameters:

x1 (array) – first input array. Should have a numeric data type.
x2 (array) – second input array. Must be compatible with x1 (see broadcasting). Should have a numeric data type.

Returns:

out – an array containing the element-wise sums. The returned array must have a data type determined by type-promotion.

Return type:

array

Notes

Special cases

For real-valued floating-point operands,

If either x1_i or x2_i is NaN, the result is NaN.
If x1_i is +infinity and x2_i is -infinity, the result is NaN.
If x1_i is -infinity and x2_i is +infinity, the result is NaN.
If x1_i is +infinity and x2_i is +infinity, the result is +infinity.
If x1_i is -infinity and x2_i is -infinity, the result is -infinity.
If x1_i is +infinity and x2_i is a finite number, the result is +infinity.
If x1_i is -infinity and x2_i is a finite number, the result is -infinity.
If x1_i is a finite number and x2_i is +infinity, the result is +infinity.
If x1_i is a finite number and x2_i is -infinity, the result is -infinity.
If x1_i is -0 and x2_i is -0, the result is -0.
If x1_i is -0 and x2_i is +0, the result is +0.
If x1_i is +0 and x2_i is -0, the result is +0.
If x1_i is +0 and x2_i is +0, the result is +0.
If x1_i is either +0 or -0 and x2_i is a nonzero finite number, the result is x2_i.
If x1_i is a nonzero finite number and x2_i is either +0 or -0, the result is x1_i.
If x1_i is a nonzero finite number and x2_i is -x1_i, the result is +0.
In the remaining cases, when neither infinity, +0, -0, nor a NaN is involved, and the operands have the same mathematical sign or have different magnitudes, the sum must be computed and rounded to the nearest representable value according to IEEE 754-2019 and a supported round mode. If the magnitude is too large to represent, the operation overflows and the result is an infinity of appropriate mathematical sign.

Note

Floating-point addition is a commutative operation, but not always associative.

For complex floating-point operands, addition is defined according to the following table. For real components a and c and imaginary components b and d,

	c	dj	c + dj
a	a + c	a + dj	(a+c) + dj
bj	c + bj	(b+d)j	c + (b+d)j
a + bj	(a+c) + bj	a + (b+d)j	(a+c) + (b+d)j

For complex floating-point operands, real-valued floating-point special cases must independently apply to the real and imaginary component operations involving real numbers as described in the above table. For example, let a = real(x1_i), b = imag(x1_i), c = real(x2_i), d = imag(x2_i), and

If a is -0 and c is -0, the real component of the result is -0.
Similarly, if b is +0 and d is -0, the imaginary component of the result is +0.

Hence, if z1 = a + bj = -0 + 0j and z2 = c + dj = -0 - 0j, the result of z1 + z2 is -0 + 0j.

Changed in version 2022.12: Added complex data type support.

all: all¶

Tests whether all input array elements evaluate to True along a specified axis.

Note

Positive infinity, negative infinity, and NaN must evaluate to True.

Note

If x has a complex floating-point data type, elements having a non-zero component (real or imaginary) must evaluate to True.

Note

If x is an empty array or the size of the axis (dimension) along which to evaluate elements is zero, the test result must be True.

Parameters:

x (array) – input array.
axis (Optional[Union[int, Tuple[int, ...]]]) – axis or axes along which to perform a logical AND reduction. By default, a logical AND reduction must be performed over the entire array. If a tuple of integers, logical AND reductions must be performed over multiple axes. A valid axis must be an integer on the interval [-N, N), where N is the rank (number of dimensions) of x. If an axis is specified as a negative integer, the function must determine the axis along which to perform a reduction by counting backward from the last dimension (where -1 refers to the last dimension). If provided an invalid axis, the function must raise an exception. Default: None.
keepdims (bool) – If True, the reduced axes (dimensions) must be included in the result as singleton dimensions, and, accordingly, the result must be compatible with the input array (see broadcasting). Otherwise, if False, the reduced axes (dimensions) must not be included in the result. Default: False.

Returns:

out – if a logical AND reduction was performed over the entire array, the returned array must be a zero-dimensional array containing the test result; otherwise, the returned array must be a non-zero-dimensional array containing the test results. The returned array must have a data type of bool.

Return type:

array

Notes

Changed in version 2022.12: Added complex data type support.

any: any¶

Tests whether any input array element evaluates to True along a specified axis.

Note

Positive infinity, negative infinity, and NaN must evaluate to True.

Note

If x has a complex floating-point data type, elements having a non-zero component (real or imaginary) must evaluate to True.

Note

If x is an empty array or the size of the axis (dimension) along which to evaluate elements is zero, the test result must be False.

Parameters:

x (array) – input array.
axis (Optional[Union[int, Tuple[int, ...]]]) – axis or axes along which to perform a logical OR reduction. By default, a logical OR reduction must be performed over the entire array. If a tuple of integers, logical OR reductions must be performed over multiple axes. A valid axis must be an integer on the interval [-N, N), where N is the rank (number of dimensions) of x. If an axis is specified as a negative integer, the function must determine the axis along which to perform a reduction by counting backward from the last dimension (where -1 refers to the last dimension). If provided an invalid axis, the function must raise an exception. Default: None.
keepdims (bool) – If True, the reduced axes (dimensions) must be included in the result as singleton dimensions, and, accordingly, the result must be compatible with the input array (see broadcasting). Otherwise, if False, the reduced axes (dimensions) must not be included in the result. Default: False.

Returns:

out – if a logical OR reduction was performed over the entire array, the returned array must be a zero-dimensional array containing the test result; otherwise, the returned array must be a non-zero-dimensional array containing the test results. The returned array must have a data type of bool.

Return type:

array

Notes

Changed in version 2022.12: Added complex data type support.

arange: arange¶

Returns evenly spaced values within the half-open interval [start, stop) as a one-dimensional array.

Parameters:

start (Union[int, float]) – if stop is specified, the start of interval (inclusive); otherwise, the end of the interval (exclusive). If stop is not specified, the default starting value is 0.
stop (Optional[Union[int, float]]) – the end of the interval. Default: None.
step (Union[int, float]) – the distance between two adjacent elements (out[i+1] - out[i]). Must not be 0; may be negative, this results in an empty array if stop >= start. Default: 1.
dtype (Optional[dtype]) – output array data type. If dtype is None, the output array data type must be inferred from start, stop and step. If those are all integers, the output array dtype must be the default integer dtype; if one or more have type float, then the output array dtype must be the default real-valued floating-point data type. Default: None.
device (Optional[device]) – device on which to place the created array. Default: None.

Note

This function cannot guarantee that the interval does not include the stop value in those cases where step is not an integer and floating-point rounding errors affect the length of the output array.

Returns:: out – a one-dimensional array containing evenly spaced values. The length of the output array must be ceil((stop-start)/step) if stop - start and step have the same sign, and length 0 otherwise.
Return type:: array

argmax: argmax¶

Returns the indices of the maximum values along a specified axis.

When the maximum value occurs multiple times, only the indices corresponding to the first occurrence are returned.

Note

For backward compatibility, conforming implementations may support complex numbers; however, inequality comparison of complex numbers is unspecified and thus implementation-dependent (see complex-number-ordering).

Parameters:

x (array) – input array. Should have a real-valued data type.
axis (Optional[int]) – axis along which to search. If None, the function must return the index of the maximum value of the flattened array. Default: None.
keepdims (bool) – if True, the reduced axes (dimensions) must be included in the result as singleton dimensions, and, accordingly, the result must be compatible with the input array (see broadcasting). Otherwise, if False, the reduced axes (dimensions) must not be included in the result. Default: False.

Returns:

out – if axis is None, a zero-dimensional array containing the index of the first occurrence of the maximum value; otherwise, a non-zero-dimensional array containing the indices of the maximum values. The returned array must have be the default array index data type.

Return type:

array

argmin: argmin¶

Returns the indices of the minimum values along a specified axis.

When the minimum value occurs multiple times, only the indices corresponding to the first occurrence are returned.

Note

Parameters:

x (array) – input array. Should have a real-valued data type.
axis (Optional[int]) – axis along which to search. If None, the function must return the index of the minimum value of the flattened array. Default: None.
keepdims (bool) – if True, the reduced axes (dimensions) must be included in the result as singleton dimensions, and, accordingly, the result must be compatible with the input array (see broadcasting). Otherwise, if False, the reduced axes (dimensions) must not be included in the result. Default: False.

Returns:

out – if axis is None, a zero-dimensional array containing the index of the first occurrence of the minimum value; otherwise, a non-zero-dimensional array containing the indices of the minimum values. The returned array must have the default array index data type.

Return type:

array

argsort: argsort¶

Returns the indices that sort an array x along a specified axis.

Note

Parameters:

x (array) – input array. Should have a real-valued data type.
axis (int) – axis along which to sort. If set to -1, the function must sort along the last axis. Default: -1.
descending (bool) – sort order. If True, the returned indices sort x in descending order (by value). If False, the returned indices sort x in ascending order (by value). Default: False.
stable (bool) – sort stability. If True, the returned indices must maintain the relative order of x values which compare as equal. If False, the returned indices may or may not maintain the relative order of x values which compare as equal (i.e., the relative order of x values which compare as equal is implementation-dependent). Default: True.

Returns:

out – an array of indices. The returned array must have the same shape as x. The returned array must have the default array index data type.

Return type:

array

asarray: asarray¶

Convert the input to an array.

Parameters:

obj (Union[array, bool, int, float, complex, NestedSequence[bool | int | float | complex], SupportsBufferProtocol]) –
object to be converted to an array. May be a Python scalar, a (possibly nested) sequence of Python scalars, or an object supporting the Python buffer protocol.

Tip

An object supporting the buffer protocol can be turned into a memoryview through memoryview(obj).
dtype (Optional[dtype]) –
output array data type. If dtype is None, the output array data type must be inferred from the data type(s) in obj. If all input values are Python scalars, then, in order of precedence,
- if all values are of type bool, the output data type must be bool.
- if all values are of type int or are a mixture of bool and int, the output data type must be the default integer data type.
- if one or more values are complex numbers, the output data type must be the default complex floating-point data type.
- if one or more values are floats, the output data type must be the default real-valued floating-point data type.
Default: None.

Note

If dtype is not None, then array conversions should obey type-promotion rules. Conversions not specified according to type-promotion rules may or may not be permitted by a conforming array library. To perform an explicit cast, use array_api.astype().

Note

If an input value exceeds the precision of the resolved output array data type, behavior is left unspecified and, thus, implementation-defined.
device (Optional[device]) – device on which to place the created array. If device is None and obj is an array, the output array device must be inferred from obj. Default: None.
copy (Optional[bool]) – boolean indicating whether or not to copy the input. If True, the function must always copy. If False, the function must never copy for input which supports the buffer protocol and must raise a ValueError in case a copy would be necessary. If None, the function must reuse existing memory buffer if possible and copy otherwise. Default: None.

Returns:

out – an array containing the data from obj.

Return type:

array

Notes

Changed in version 2022.12: Added complex data type support.

asin: asin¶

Calculates an implementation-dependent approximation of the principal value of the inverse sine for each element x_i of the input array x.

Each element-wise result is expressed in radians.

Note

The principal value of the arc sine of a complex number $z$ is

\[\operatorname{asin}(z) = -j\ \ln(zj + \sqrt{1-z^2})\]

For any $z$,

\[\operatorname{asin}(z) = \operatorname{acos}(-z) - \frac{\pi}{2}\]

Note

For complex floating-point operands, asin(conj(x)) must equal conj(asin(x)).

Note

The inverse sine (or arc sine) is a multi-valued function and requires a branch cut on the complex plane. By convention, a branch cut is placed at the line segments $(-\infty, -1)$ and $(1, \infty)$ of the real axis.

Accordingly, for complex arguments, the function returns the inverse sine in the range of a strip unbounded along the imaginary axis and in the interval $[-\pi/2, +\pi/2]$ along the real axis.

Note: branch cuts follow C99 and have provisional status (see branch-cuts).

Parameters:: x (array) – input array. Should have a floating-point data type.
Returns:: out – an array containing the inverse sine of each element in x. The returned array must have a floating-point data type determined by type-promotion.
Return type:: array

Notes

Special cases

For real-valued floating-point operands,

If x_i is NaN, the result is NaN.
If x_i is greater than 1, the result is NaN.
If x_i is less than -1, the result is NaN.
If x_i is +0, the result is +0.
If x_i is -0, the result is -0.

For complex floating-point operands, special cases must be handled as if the operation is implemented as -1j * asinh(x*1j).

Changed in version 2022.12: Added complex data type support.

asinh: asinh¶

Calculates an implementation-dependent approximation to the inverse hyperbolic sine for each element x_i in the input array x.

Note

The principal value of the inverse hyperbolic sine of a complex number $z$ is

\[\operatorname{asinh}(z) = \ln(z + \sqrt{1+z^2})\]

For any $z$,

\[\operatorname{asinh}(z) = \frac{\operatorname{asin}(zj)}{j}\]

Note

For complex floating-point operands, asinh(conj(x)) must equal conj(asinh(x)) and asinh(-z) must equal -asinh(z).

Note

The inverse hyperbolic sine is a multi-valued function and requires a branch cut on the complex plane. By convention, a branch cut is placed at the line segments $(-\infty j, -j)$ and $(j, \infty j)$ of the imaginary axis.

Accordingly, for complex arguments, the function returns the inverse hyperbolic sine in the range of a strip unbounded along the real axis and in the interval $[-\pi j/2, +\pi j/2]$ along the imaginary axis.

Note: branch cuts follow C99 and have provisional status (see branch-cuts).

Parameters:: x (array) – input array whose elements each represent the area of a hyperbolic sector. Should have a floating-point data type.
Returns:: out – an array containing the inverse hyperbolic sine of each element in x. The returned array must have a floating-point data type determined by type-promotion.
Return type:: array

Notes

Special cases

For real-valued floating-point operands,

If x_i is NaN, the result is NaN.
If x_i is +0, the result is +0.
If x_i is -0, the result is -0.
If x_i is +infinity, the result is +infinity.
If x_i is -infinity, the result is -infinity.

For complex floating-point operands, let a = real(x_i), b = imag(x_i), and

If a is +0 and b is +0, the result is +0 + 0j.
If a is a positive (i.e., greater than 0) finite number and b is +infinity, the result is +infinity + πj/2.
If a is a finite number and b is NaN, the result is NaN + NaN j.
If a is +infinity and b is a positive (i.e., greater than 0) finite number, the result is +infinity + 0j.
If a is +infinity and b is +infinity, the result is +infinity + πj/4.
If a is NaN and b is +0, the result is NaN + 0j.
If a is NaN and b is a nonzero finite number, the result is NaN + NaN j.
If a is NaN and b is +infinity, the result is ±infinity + NaN j (sign of the real component is unspecified).
If a is NaN and b is NaN, the result is NaN + NaN j.

Changed in version 2022.12: Added complex data type support.

astype: astype¶

Copies an array to a specified data type irrespective of type-promotion rules.

Note

Casting floating-point NaN and infinity values to integral data types is not specified and is implementation-dependent.

Note

Casting a complex floating-point array to a real-valued data type should not be permitted.

Historically, when casting a complex floating-point array to a real-valued data type, libraries such as NumPy have discarded imaginary components such that, for a complex floating-point array x, astype(x) equals astype(real(x))). This behavior is considered problematic as the choice to discard the imaginary component is arbitrary and introduces more than one way to achieve the same outcome (i.e., for a complex floating-point array x, astype(x) and astype(real(x)) versus only astype(imag(x))). Instead, in order to avoid ambiguity and to promote clarity, this specification requires that array API consumers explicitly express which component should be cast to a specified real-valued data type.

Note

When casting a boolean input array to a real-valued data type, a value of True must cast to a real-valued number equal to 1, and a value of False must cast to a real-valued number equal to 0.

When casting a boolean input array to a complex floating-point data type, a value of True must cast to a complex number equal to 1 + 0j, and a value of False must cast to a complex number equal to 0 + 0j.

Note

When casting a real-valued input array to bool, a value of 0 must cast to False, and a non-zero value must cast to True.

When casting a complex floating-point array to bool, a value of 0 + 0j must cast to False, and all other values must cast to True.

Parameters:

x (array) – array to cast.
dtype (dtype) – desired data type.
copy (bool) – specifies whether to copy an array when the specified dtype matches the data type of the input array x. If True, a newly allocated array must always be returned. If False and the specified dtype matches the data type of the input array, the input array must be returned; otherwise, a newly allocated array must be returned. Default: True.

Returns:

out – an array having the specified data type. The returned array must have the same shape as x.

Return type:

array

Notes

Changed in version 2022.12: Added complex data type support.

atan: atan¶

Calculates an implementation-dependent approximation of the principal value of the inverse tangent for each element x_i of the input array x.

Each element-wise result is expressed in radians.

Note

The principal value of the inverse tangent of a complex number $z$ is

\[\operatorname{atan}(z) = -\frac{\ln(1 - zj) - \ln(1 + zj)}{2}j\]

Note

For complex floating-point operands, atan(conj(x)) must equal conj(atan(x)).

Note

The inverse tangent (or arc tangent) is a multi-valued function and requires a branch on the complex plane. By convention, a branch cut is placed at the line segments $(-\infty j, -j)$ and $(+j, \infty j)$ of the imaginary axis.

Accordingly, for complex arguments, the function returns the inverse tangent in the range of a strip unbounded along the imaginary axis and in the interval $[-\pi/2, +\pi/2]$ along the real axis.

Note: branch cuts follow C99 and have provisional status (see branch-cuts).

Parameters:: x (array) – input array. Should have a floating-point data type.
Returns:: out – an array containing the inverse tangent of each element in x. The returned array must have a floating-point data type determined by type-promotion.
Return type:: array

Notes

Special cases

For real-valued floating-point operands,

If x_i is NaN, the result is NaN.
If x_i is +0, the result is +0.
If x_i is -0, the result is -0.
If x_i is +infinity, the result is an implementation-dependent approximation to +π/2.
If x_i is -infinity, the result is an implementation-dependent approximation to -π/2.

For complex floating-point operands, special cases must be handled as if the operation is implemented as -1j * atanh(x*1j).

Changed in version 2022.12: Added complex data type support.

atan2: atan2¶

Calculates an implementation-dependent approximation of the inverse tangent of the quotient x1/x2, having domain [-infinity, +infinity] x [-infinity, +infinity] (where the x notation denotes the set of ordered pairs of elements (x1_i, x2_i)) and codomain [-π, +π], for each pair of elements (x1_i, x2_i) of the input arrays x1 and x2, respectively. Each element-wise result is expressed in radians.

The mathematical signs of x1_i and x2_i determine the quadrant of each element-wise result. The quadrant (i.e., branch) is chosen such that each element-wise result is the signed angle in radians between the ray ending at the origin and passing through the point (1,0) and the ray ending at the origin and passing through the point (x2_i, x1_i).

Note

Note the role reversal: the “y-coordinate” is the first function parameter; the “x-coordinate” is the second function parameter. The parameter order is intentional and traditional for the two-argument inverse tangent function where the y-coordinate argument is first and the x-coordinate argument is second.

By IEEE 754 convention, the inverse tangent of the quotient x1/x2 is defined for x2_i equal to positive or negative zero and for either or both of x1_i and x2_i equal to positive or negative infinity.

Parameters:

x1 (array) – input array corresponding to the y-coordinates. Should have a real-valued floating-point data type.
x2 (array) – input array corresponding to the x-coordinates. Must be compatible with x1 (see broadcasting). Should have a real-valued floating-point data type.

Returns:

out – an array containing the inverse tangent of the quotient x1/x2. The returned array must have a real-valued floating-point data type determined by type-promotion.

Return type:

array

Notes

Special cases

For floating-point operands,

If either x1_i or x2_i is NaN, the result is NaN.
If x1_i is greater than 0 and x2_i is +0, the result is an implementation-dependent approximation to +π/2.
If x1_i is greater than 0 and x2_i is -0, the result is an implementation-dependent approximation to +π/2.
If x1_i is +0 and x2_i is greater than 0, the result is +0.
If x1_i is +0 and x2_i is +0, the result is +0.
If x1_i is +0 and x2_i is -0, the result is an implementation-dependent approximation to +π.
If x1_i is +0 and x2_i is less than 0, the result is an implementation-dependent approximation to +π.
If x1_i is -0 and x2_i is greater than 0, the result is -0.
If x1_i is -0 and x2_i is +0, the result is -0.
If x1_i is -0 and x2_i is -0, the result is an implementation-dependent approximation to -π.
If x1_i is -0 and x2_i is less than 0, the result is an implementation-dependent approximation to -π.
If x1_i is less than 0 and x2_i is +0, the result is an implementation-dependent approximation to -π/2.
If x1_i is less than 0 and x2_i is -0, the result is an implementation-dependent approximation to -π/2.
If x1_i is greater than 0, x1_i is a finite number, and x2_i is +infinity, the result is +0.
If x1_i is greater than 0, x1_i is a finite number, and x2_i is -infinity, the result is an implementation-dependent approximation to +π.
If x1_i is less than 0, x1_i is a finite number, and x2_i is +infinity, the result is -0.
If x1_i is less than 0, x1_i is a finite number, and x2_i is -infinity, the result is an implementation-dependent approximation to -π.
If x1_i is +infinity and x2_i is a finite number, the result is an implementation-dependent approximation to +π/2.
If x1_i is -infinity and x2_i is a finite number, the result is an implementation-dependent approximation to -π/2.
If x1_i is +infinity and x2_i is +infinity, the result is an implementation-dependent approximation to +π/4.
If x1_i is +infinity and x2_i is -infinity, the result is an implementation-dependent approximation to +3π/4.
If x1_i is -infinity and x2_i is +infinity, the result is an implementation-dependent approximation to -π/4.
If x1_i is -infinity and x2_i is -infinity, the result is an implementation-dependent approximation to -3π/4.

atanh: atanh¶

Calculates an implementation-dependent approximation to the inverse hyperbolic tangent for each element x_i of the input array x.

Note

The principal value of the inverse hyperbolic tangent of a complex number $z$ is

\[\operatorname{atanh}(z) = \frac{\ln(1+z)-\ln(z-1)}{2}\]

For any $z$,

\[\operatorname{atanh}(z) = \frac{\operatorname{atan}(zj)}{j}\]

Note

For complex floating-point operands, atanh(conj(x)) must equal conj(atanh(x)) and atanh(-x) must equal -atanh(x).

Note

The inverse hyperbolic tangent is a multi-valued function and requires a branch cut on the complex plane. By convention, a branch cut is placed at the line segments $(-\infty, 1]$ and $[1, \infty)$ of the real axis.

Accordingly, for complex arguments, the function returns the inverse hyperbolic tangent in the range of a half-strip unbounded along the real axis and in the interval $[-\pi j/2, +\pi j/2]$ along the imaginary axis.

Note: branch cuts follow C99 and have provisional status (see branch-cuts).

Parameters:: x (array) – input array whose elements each represent the area of a hyperbolic sector. Should have a floating-point data type.
Returns:: out – an array containing the inverse hyperbolic tangent of each element in x. The returned array must have a floating-point data type determined by type-promotion.
Return type:: array

Notes

Special cases

For real-valued floating-point operands,

If x_i is NaN, the result is NaN.
If x_i is less than -1, the result is NaN.
If x_i is greater than 1, the result is NaN.
If x_i is -1, the result is -infinity.
If x_i is +1, the result is +infinity.
If x_i is +0, the result is +0.
If x_i is -0, the result is -0.

For complex floating-point operands, let a = real(x_i), b = imag(x_i), and

If a is +0 and b is +0, the result is +0 + 0j.
If a is +0 and b is NaN, the result is +0 + NaN j.
If a is 1 and b is +0, the result is +infinity + 0j.
If a is a positive (i.e., greater than 0) finite number and b is +infinity, the result is +0 + πj/2.
If a is a nonzero finite number and b is NaN, the result is NaN + NaN j.
If a is +infinity and b is a positive (i.e., greater than 0) finite number, the result is +0 + πj/2.
If a is +infinity and b is +infinity, the result is +0 + πj/2.
If a is +infinity and b is NaN, the result is +0 + NaN j.
If a is NaN and b is a finite number, the result is NaN + NaN j.
If a is NaN and b is +infinity, the result is ±0 + πj/2 (sign of the real component is unspecified).
If a is NaN and b is NaN, the result is NaN + NaN j.

Changed in version 2022.12: Added complex data type support.

bitwise_and: bitwise_and¶

Computes the bitwise AND of the underlying binary representation of each element x1_i of the input array x1 with the respective element x2_i of the input array x2.

Parameters:

x1 (array) – first input array. Should have an integer or boolean data type.
x2 (array) – second input array. Must be compatible with x1 (see broadcasting). Should have an integer or boolean data type.

Returns:

out – an array containing the element-wise results. The returned array must have a data type determined by type-promotion.

Return type:

array

bitwise_invert: bitwise_invert¶

Inverts (flips) each bit for each element x_i of the input array x.

Parameters:: x (array) – input array. Should have an integer or boolean data type.
Returns:: out – an array containing the element-wise results. The returned array must have the same data type as x.
Return type:: array

bitwise_left_shift: bitwise_left_shift¶

Shifts the bits of each element x1_i of the input array x1 to the left by appending x2_i (i.e., the respective element in the input array x2) zeros to the right of x1_i.

Parameters:

x1 (array) – first input array. Should have an integer data type.
x2 (array) – second input array. Must be compatible with x1 (see broadcasting). Should have an integer data type. Each element must be greater than or equal to 0.

Returns:

out – an array containing the element-wise results. The returned array must have a data type determined by type-promotion.

Return type:

array

bitwise_or: bitwise_or¶

Computes the bitwise OR of the underlying binary representation of each element x1_i of the input array x1 with the respective element x2_i of the input array x2.

Parameters:

x1 (array) – first input array. Should have an integer or boolean data type.
x2 (array) – second input array. Must be compatible with x1 (see broadcasting). Should have an integer or boolean data type.

Returns:

out – an array containing the element-wise results. The returned array must have a data type determined by type-promotion.

Return type:

array

bitwise_right_shift: bitwise_right_shift¶

Shifts the bits of each element x1_i of the input array x1 to the right according to the respective element x2_i of the input array x2.

Note

This operation must be an arithmetic shift (i.e., sign-propagating) and thus equivalent to floor division by a power of two.

Parameters:

x1 (array) – first input array. Should have an integer data type.
x2 (array) – second input array. Must be compatible with x1 (see broadcasting). Should have an integer data type. Each element must be greater than or equal to 0.

Returns:

out – an array containing the element-wise results. The returned array must have a data type determined by type-promotion.

Return type:

array

bitwise_xor: bitwise_xor¶

Computes the bitwise XOR of the underlying binary representation of each element x1_i of the input array x1 with the respective element x2_i of the input array x2.

Parameters:

x1 (array) – first input array. Should have an integer or boolean data type.
x2 (array) – second input array. Must be compatible with x1 (see broadcasting). Should have an integer or boolean data type.

Returns:

out – an array containing the element-wise results. The returned array must have a data type determined by type-promotion.

Return type:

array

bool: TDtype¶

broadcast_arrays: broadcast_arrays¶

Broadcasts one or more arrays against one another.

Parameters:: arrays (array) – an arbitrary number of to-be broadcasted arrays.
Returns:: out – a list of broadcasted arrays. Each array must have the same shape. Each array must have the same dtype as its corresponding input array.
Return type:: List[array]

broadcast_to: broadcast_to¶

Broadcasts an array to a specified shape.

Parameters:

x (array) – array to broadcast.
shape (Tuple[int, ...]) – array shape. Must be compatible with x (see broadcasting). If the array is incompatible with the specified shape, the function should raise an exception.

Returns:

out – an array having a specified shape. Must have the same data type as x.

Return type:

array

can_cast: can_cast¶

Determines if one data type can be cast to another data type according type-promotion rules.

Parameters:

from (Union[dtype, array]) – input data type or array from which to cast.
to (dtype) – desired data type.

Returns:

out – True if the cast can occur according to type-promotion rules; otherwise, False.

Return type:

bool

ceil: ceil¶

Rounds each element x_i of the input array x to the smallest (i.e., closest to -infinity) integer-valued number that is not less than x_i.

Parameters:: x (array) – input array. Should have a real-valued data type.
Returns:: out – an array containing the rounded result for each element in x. The returned array must have the same data type as x.
Return type:: array

Notes

Special cases

If x_i is already integer-valued, the result is x_i.

For floating-point operands,

If x_i is +infinity, the result is +infinity.
If x_i is -infinity, the result is -infinity.
If x_i is +0, the result is +0.
If x_i is -0, the result is -0.
If x_i is NaN, the result is NaN.

complex128: TDtype¶

complex64: TDtype¶

concat: concat¶

Joins a sequence of arrays along an existing axis.

Parameters:

arrays (Union[Tuple[array, ...], List[array]]) – input arrays to join. The arrays must have the same shape, except in the dimension specified by axis.
axis (Optional[int]) – axis along which the arrays will be joined. If axis is None, arrays must be flattened before concatenation. If axis is negative, the function must determine the axis along which to join by counting from the last dimension. Default: 0.

Returns:

out – an output array containing the concatenated values. If the input arrays have different data types, normal type-promotion must apply. If the input arrays have the same data type, the output array must have the same data type as the input arrays.

Note

This specification leaves type promotion between data type families (i.e., intxx and floatxx) unspecified.

Return type:

array

conj: conj¶

Returns the complex conjugate for each element x_i of the input array x.

For complex numbers of the form

\[a + bj\]

the complex conjugate is defined as

\[a - bj\]

Hence, the returned complex conjugates must be computed by negating the imaginary component of each element x_i.

Parameters:

x (array) – input array. Should have a complex floating-point data type.

Returns:

out (array) – an array containing the element-wise results. The returned array must have the same data type as x.
.. versionadded:: 2022.12

cos: cos¶

Calculates an implementation-dependent approximation to the cosine for each element x_i of the input array x.

Each element x_i is assumed to be expressed in radians.

Note

The cosine is an entire function on the complex plane and has no branch cuts.

Note

For complex arguments, the mathematical definition of cosine is

\[\begin{split}\begin{align} \operatorname{cos}(x) &= \sum_{n=0}^\infty \frac{(-1)^n}{(2n)!} x^{2n} \\ &= \frac{e^{jx} + e^{-jx}}{2} \\ &= \operatorname{cosh}(jx) \end{align}\end{split}\]

where $\operatorname{cosh}$ is the hyperbolic cosine.

Parameters:: x (array) – input array whose elements are each expressed in radians. Should have a floating-point data type.
Returns:: out – an array containing the cosine of each element in x. The returned array must have a floating-point data type determined by type-promotion.
Return type:: array

Notes

Special cases

For real-valued floating-point operands,

If x_i is NaN, the result is NaN.
If x_i is +0, the result is 1.
If x_i is -0, the result is 1.
If x_i is +infinity, the result is NaN.
If x_i is -infinity, the result is NaN.

For complex floating-point operands, special cases must be handled as if the operation is implemented as cosh(x*1j).

Changed in version 2022.12: Added complex data type support.

cosh: cosh¶

Calculates an implementation-dependent approximation to the hyperbolic cosine for each element x_i in the input array x.

The mathematical definition of the hyperbolic cosine is

\[\operatorname{cosh}(x) = \frac{e^x + e^{-x}}{2}\]

Note

The hyperbolic cosine is an entire function in the complex plane and has no branch cuts. The function is periodic, with period $2\pi j$, with respect to the imaginary component.

Parameters:: x (array) – input array whose elements each represent a hyperbolic angle. Should have a floating-point data type.
Returns:: out – an array containing the hyperbolic cosine of each element in x. The returned array must have a floating-point data type determined by type-promotion.
Return type:: array

Notes

Special cases

Note

For all operands, cosh(x) must equal cosh(-x).

For real-valued floating-point operands,

If x_i is NaN, the result is NaN.
If x_i is +0, the result is 1.
If x_i is -0, the result is 1.
If x_i is +infinity, the result is +infinity.
If x_i is -infinity, the result is +infinity.

For complex floating-point operands, let a = real(x_i), b = imag(x_i), and

Note

For complex floating-point operands, cosh(conj(x)) must equal conj(cosh(x)).

If a is +0 and b is +0, the result is 1 + 0j.
If a is +0 and b is +infinity, the result is NaN + 0j (sign of the imaginary component is unspecified).
If a is +0 and b is NaN, the result is NaN + 0j (sign of the imaginary component is unspecified).
If a is a nonzero finite number and b is +infinity, the result is NaN + NaN j.
If a is a nonzero finite number and b is NaN, the result is NaN + NaN j.
If a is +infinity and b is +0, the result is +infinity + 0j.
If a is +infinity and b is a nonzero finite number, the result is +infinity * cis(b).
If a is +infinity and b is +infinity, the result is +infinity + NaN j (sign of the real component is unspecified).
If a is +infinity and b is NaN, the result is +infinity + NaN j.
If a is NaN and b is either +0 or -0, the result is NaN + 0j (sign of the imaginary component is unspecified).
If a is NaN and b is a nonzero finite number, the result is NaN + NaN j.
If a is NaN and b is NaN, the result is NaN + NaN j.

where cis(v) is cos(v) + sin(v)*1j.

Changed in version 2022.12: Added complex data type support.

divide: divide¶

Calculates the division of each element x1_i of the input array x1 with the respective element x2_i of the input array x2.

Note

If one or both of the input arrays have integer data types, the result is implementation-dependent, as type promotion between data type “kinds” (e.g., integer versus floating-point) is unspecified.

Specification-compliant libraries may choose to raise an error or return an array containing the element-wise results. If an array is returned, the array must have a real-valued floating-point data type.

Parameters:

x1 (array) – dividend input array. Should have a numeric data type.
x2 (array) – divisor input array. Must be compatible with x1 (see broadcasting). Should have a numeric data type.

Returns:

out – an array containing the element-wise results. The returned array must have a floating-point data type determined by type-promotion.

Return type:

array

Notes

Special cases

For real-valued floating-point operands,

If either x1_i or x2_i is NaN, the result is NaN.
If x1_i is either +infinity or -infinity and x2_i is either +infinity or -infinity, the result is NaN.
If x1_i is either +0 or -0 and x2_i is either +0 or -0, the result is NaN.
If x1_i is +0 and x2_i is greater than 0, the result is +0.
If x1_i is -0 and x2_i is greater than 0, the result is -0.
If x1_i is +0 and x2_i is less than 0, the result is -0.
If x1_i is -0 and x2_i is less than 0, the result is +0.
If x1_i is greater than 0 and x2_i is +0, the result is +infinity.
If x1_i is greater than 0 and x2_i is -0, the result is -infinity.
If x1_i is less than 0 and x2_i is +0, the result is -infinity.
If x1_i is less than 0 and x2_i is -0, the result is +infinity.
If x1_i is +infinity and x2_i is a positive (i.e., greater than 0) finite number, the result is +infinity.
If x1_i is +infinity and x2_i is a negative (i.e., less than 0) finite number, the result is -infinity.
If x1_i is -infinity and x2_i is a positive (i.e., greater than 0) finite number, the result is -infinity.
If x1_i is -infinity and x2_i is a negative (i.e., less than 0) finite number, the result is +infinity.
If x1_i is a positive (i.e., greater than 0) finite number and x2_i is +infinity, the result is +0.
If x1_i is a positive (i.e., greater than 0) finite number and x2_i is -infinity, the result is -0.
If x1_i is a negative (i.e., less than 0) finite number and x2_i is +infinity, the result is -0.
If x1_i is a negative (i.e., less than 0) finite number and x2_i is -infinity, the result is +0.
If x1_i and x2_i have the same mathematical sign and are both nonzero finite numbers, the result has a positive mathematical sign.
If x1_i and x2_i have different mathematical signs and are both nonzero finite numbers, the result has a negative mathematical sign.
In the remaining cases, where neither -infinity, +0, -0, nor NaN is involved, the quotient must be computed and rounded to the nearest representable value according to IEEE 754-2019 and a supported rounding mode. If the magnitude is too large to represent, the operation overflows and the result is an infinity of appropriate mathematical sign. If the magnitude is too small to represent, the operation underflows and the result is a zero of appropriate mathematical sign.

For complex floating-point operands, division is defined according to the following table. For real components a and c and imaginary components b and d,

	c	dj	c + dj
a	a / c	-(a/d)j	special rules
bj	(b/c)j	b/d	special rules
a + bj	(a/c) + (b/c)j	b/d - (a/d)j	special rules

In general, for complex floating-point operands, real-valued floating-point special cases must independently apply to the real and imaginary component operations involving real numbers as described in the above table.

When a, b, c, or d are all finite numbers (i.e., a value other than NaN, +infinity, or -infinity), division of complex floating-point operands should be computed as if calculated according to the textbook formula for complex number division

\[\frac{a + bj}{c + dj} = \frac{(ac + bd) + (bc - ad)j}{c^2 + d^2}\]

When at least one of a, b, c, or d is NaN, +infinity, or -infinity,

If a, b, c, and d are all NaN, the result is NaN + NaN j.
In the remaining cases, the result is implementation dependent.

Note

For complex floating-point operands, the results of special cases may be implementation dependent depending on how an implementation chooses to model complex numbers and complex infinity (e.g., complex plane versus Riemann sphere). For those implementations following C99 and its one-infinity model, when at least one component is infinite, even if the other component is NaN, the complex value is infinite, and the usual arithmetic rules do not apply to complex-complex division. In the interest of performance, other implementations may want to avoid the complex branching logic necessary to implement the one-infinity model and choose to implement all complex-complex division according to the textbook formula. Accordingly, special case behavior is unlikely to be consistent across implementations.

Changed in version 2022.12: Added complex data type support.

e: TArray¶

IEEE 754 floating-point representation of Euler’s constant.

e = 2.71828182845904523536028747135266249775724709369995...

empty: empty¶

Returns an uninitialized array having a specified shape.

Parameters:

shape (Union[int, Tuple[int, ...]]) – output array shape.
dtype (Optional[dtype]) – output array data type. If dtype is None, the output array data type must be the default real-valued floating-point data type. Default: None.
device (Optional[device]) – device on which to place the created array. Default: None.

Returns:

out – an array containing uninitialized data.

Return type:

array

empty_like: empty_like¶

Returns an uninitialized array with the same shape as an input array x.

Parameters:

x (array) – input array from which to derive the output array shape.
dtype (Optional[dtype]) – output array data type. If dtype is None, the output array data type must be inferred from x. Default: None.
device (Optional[device]) – device on which to place the created array. If device is None, the output array device must be inferred from x. Default: None.

Returns:

out – an array having the same shape as x and containing uninitialized data.

Return type:

array

equal: equal¶

Computes the truth value of x1_i == x2_i for each element x1_i of the input array x1 with the respective element x2_i of the input array x2.

Parameters:

x1 (array) – first input array. May have any data type.
x2 (array) – second input array. Must be compatible with x1 (see broadcasting). May have any data type.

Returns:

out – an array containing the element-wise results. The returned array must have a data type of bool.

Return type:

array

Notes

Special Cases

For real-valued floating-point operands,

If x1_i is NaN or x2_i is NaN, the result is False.
If x1_i is +infinity and x2_i is +infinity, the result is True.
If x1_i is -infinity and x2_i is -infinity, the result is True.
If x1_i is -0 and x2_i is either +0 or -0, the result is True.
If x1_i is +0 and x2_i is either +0 or -0, the result is True.
If x1_i is a finite number, x2_i is a finite number, and x1_i equals x2_i, the result is True.
In the remaining cases, the result is False.

For complex floating-point operands, let a = real(x1_i), b = imag(x1_i), c = real(x2_i), d = imag(x2_i), and

If a, b, c, or d is NaN, the result is False.
In the remaining cases, the result is the logical AND of the equality comparison between the real values a and c (real components) and between the real values b and d (imaginary components), as described above for real-valued floating-point operands (i.e., a == c AND b == d).

Note

For discussion of complex number equality, see complex-numbers.

Changed in version 2022.12: Added complex data type support.

exp: exp¶

Calculates an implementation-dependent approximation to the exponential function for each element x_i of the input array x (e raised to the power of x_i, where e is the base of the natural logarithm).

Note

For complex floating-point operands, exp(conj(x)) must equal conj(exp(x)).

Note

The exponential function is an entire function in the complex plane and has no branch cuts.

Parameters:: x (array) – input array. Should have a floating-point data type.
Returns:: out – an array containing the evaluated exponential function result for each element in x. The returned array must have a floating-point data type determined by type-promotion.
Return type:: array

Notes

Special cases

For real-valued floating-point operands,

If x_i is NaN, the result is NaN.
If x_i is +0, the result is 1.
If x_i is -0, the result is 1.
If x_i is +infinity, the result is +infinity.
If x_i is -infinity, the result is +0.

For complex floating-point operands, let a = real(x_i), b = imag(x_i), and

If a is either +0 or -0 and b is +0, the result is 1 + 0j.
If a is a finite number and b is +infinity, the result is NaN + NaN j.
If a is a finite number and b is NaN, the result is NaN + NaN j.
If a is +infinity and b is +0, the result is infinity + 0j.
If a is -infinity and b is a finite number, the result is +0 * cis(b).
If a is +infinity and b is a nonzero finite number, the result is +infinity * cis(b).
If a is -infinity and b is +infinity, the result is 0 + 0j (signs of real and imaginary components are unspecified).
If a is +infinity and b is +infinity, the result is infinity + NaN j (sign of real component is unspecified).
If a is -infinity and b is NaN, the result is 0 + 0j (signs of real and imaginary components are unspecified).
If a is +infinity and b is NaN, the result is infinity + NaN j (sign of real component is unspecified).
If a is NaN and b is +0, the result is NaN + 0j.
If a is NaN and b is not equal to 0, the result is NaN + NaN j.
If a is NaN and b is NaN, the result is NaN + NaN j.

where cis(v) is cos(v) + sin(v)*1j.

Changed in version 2022.12: Added complex data type support.

expand_dims: expand_dims¶

Expands the shape of an array by inserting a new axis (dimension) of size one at the position specified by axis.

Parameters:

x (array) – input array.
axis (int) – axis position (zero-based). If x has rank (i.e, number of dimensions) N, a valid axis must reside on the closed-interval [-N-1, N]. If provided a negative axis, the axis position at which to insert a singleton dimension must be computed as N + axis + 1. Hence, if provided -1, the resolved axis position must be N (i.e., a singleton dimension must be appended to the input array x). If provided -N-1, the resolved axis position must be 0 (i.e., a singleton dimension must be prepended to the input array x). An IndexError exception must be raised if provided an invalid axis position.

Returns:

out – an expanded output array having the same data type as x.

Return type:

array

expm1: expm1¶

Calculates an implementation-dependent approximation to exp(x)-1 for each element x_i of the input array x.

Note

The purpose of this function is to calculate exp(x)-1.0 more accurately when x is close to zero. Accordingly, conforming implementations should avoid implementing this function as simply exp(x)-1.0. See FDLIBM, or some other IEEE 754-2019 compliant mathematical library, for a potential reference implementation.

Note

For complex floating-point operands, expm1(conj(x)) must equal conj(expm1(x)).

Note

The exponential function is an entire function in the complex plane and has no branch cuts.

Parameters:: x (array) – input array. Should have a floating-point data type.
Returns:: out – an array containing the evaluated result for each element in x. The returned array must have a floating-point data type determined by type-promotion.
Return type:: array

Notes

Special cases

For real-valued floating-point operands,

If x_i is NaN, the result is NaN.
If x_i is +0, the result is +0.
If x_i is -0, the result is -0.
If x_i is +infinity, the result is +infinity.
If x_i is -infinity, the result is -1.

For complex floating-point operands, let a = real(x_i), b = imag(x_i), and

If a is either +0 or -0 and b is +0, the result is 0 + 0j.
If a is a finite number and b is +infinity, the result is NaN + NaN j.
If a is a finite number and b is NaN, the result is NaN + NaN j.
If a is +infinity and b is +0, the result is +infinity + 0j.
If a is -infinity and b is a finite number, the result is +0 * cis(b) - 1.0.
If a is +infinity and b is a nonzero finite number, the result is +infinity * cis(b) - 1.0.
If a is -infinity and b is +infinity, the result is -1 + 0j (sign of imaginary component is unspecified).
If a is +infinity and b is +infinity, the result is infinity + NaN j (sign of real component is unspecified).
If a is -infinity and b is NaN, the result is -1 + 0j (sign of imaginary component is unspecified).
If a is +infinity and b is NaN, the result is infinity + NaN j (sign of real component is unspecified).
If a is NaN and b is +0, the result is NaN + 0j.
If a is NaN and b is not equal to 0, the result is NaN + NaN j.
If a is NaN and b is NaN, the result is NaN + NaN j.

where cis(v) is cos(v) + sin(v)*1j.

Changed in version 2022.12: Added complex data type support.

eye: eye¶

Returns a two-dimensional array with ones on the kth diagonal and zeros elsewhere.

Note

An output array having a complex floating-point data type must have the value 1 + 0j along the kth diagonal and 0 + 0j elsewhere.

Parameters:

n_rows (int) – number of rows in the output array.
n_cols (Optional[int]) – number of columns in the output array. If None, the default number of columns in the output array is equal to n_rows. Default: None.
k (int) – index of the diagonal. A positive value refers to an upper diagonal, a negative value to a lower diagonal, and 0 to the main diagonal. Default: 0.
dtype (Optional[dtype]) – output array data type. If dtype is None, the output array data type must be the default real-valued floating-point data type. Default: None.
device (Optional[device]) – device on which to place the created array. Default: None.

Returns:

out – an array where all elements are equal to zero, except for the kth diagonal, whose values are equal to one.

Return type:

array

Notes

Changed in version 2022.12: Added complex data type support.

finfo: finfo¶

Machine limits for floating-point data types.

Parameters:

type (Union[dtype, array]) –

the kind of floating-point data-type about which to get information. If complex, the information is about its component data type.

Note

Complex floating-point data types are specified to always use the same precision for both its real and imaginary components, so the information should be true for either component.

Returns:

out – an object having the following attributes:

bits: int

number of bits occupied by the real-valued floating-point data type.
eps: float

difference between 1.0 and the next smallest representable real-valued floating-point number larger than 1.0 according to the IEEE-754 standard.
max: float

largest representable real-valued number.
min: float

smallest representable real-valued number.
smallest_normal: float

smallest positive real-valued floating-point number with full precision.
dtype: dtype

real-valued floating-point data type.

Added in version 2022.12.

Return type:

finfo object

Notes

Changed in version 2022.12: Added complex data type support.

flip: flip¶

Reverses the order of elements in an array along the given axis. The shape of the array must be preserved.

Parameters:

x (array) – input array.
axis (Optional[Union[int, Tuple[int, ...]]]) – axis (or axes) along which to flip. If axis is None, the function must flip all input array axes. If axis is negative, the function must count from the last dimension. If provided more than one axis, the function must flip only the specified axes. Default: None.

Returns:

out – an output array having the same data type and shape as x and whose elements, relative to x, are reordered.

Return type:

array

float32: TDtype¶

float64: TDtype¶

floor: floor¶

Rounds each element x_i of the input array x to the greatest (i.e., closest to +infinity) integer-valued number that is not greater than x_i.

Parameters:: x (array) – input array. Should have a real-valued data type.
Returns:: out – an array containing the rounded result for each element in x. The returned array must have the same data type as x.
Return type:: array

Notes

Special cases

If x_i is already integer-valued, the result is x_i.

For floating-point operands,

If x_i is +infinity, the result is +infinity.
If x_i is -infinity, the result is -infinity.
If x_i is +0, the result is +0.
If x_i is -0, the result is -0.
If x_i is NaN, the result is NaN.

floor_divide: floor_divide¶

Rounds the result of dividing each element x1_i of the input array x1 by the respective element x2_i of the input array x2 to the greatest (i.e., closest to +infinity) integer-value number that is not greater than the division result.

Note

For input arrays which promote to an integer data type, the result of division by zero is unspecified and thus implementation-defined.

Parameters:

x1 (array) – dividend input array. Should have a real-valued data type.
x2 (array) – divisor input array. Must be compatible with x1 (see broadcasting). Should have a real-valued data type.

Returns:

out – an array containing the element-wise results. The returned array must have a data type determined by type-promotion.

Return type:

array

Notes

Special cases

Note

Floor division was introduced in Python via PEP 238 with the goal to disambiguate “true division” (i.e., computing an approximation to the mathematical operation of division) from “floor division” (i.e., rounding the result of division toward negative infinity). The former was computed when one of the operands was a float, while the latter was computed when both operands were ints. Overloading the / operator to support both behaviors led to subtle numerical bugs when integers are possible, but not expected.

To resolve this ambiguity, / was designated for true division, and // was designated for floor division. Semantically, floor division was defined as equivalent to a // b == floor(a/b); however, special floating-point cases were left ill-defined.

Accordingly, floor division is not implemented consistently across array libraries for some of the special cases documented below. Namely, when one of the operands is infinity, libraries may diverge with some choosing to strictly follow floor(a/b) and others choosing to pair // with % according to the relation b = a % b + b * (a // b). The special cases leading to divergent behavior are documented below.

This specification prefers floor division to match floor(divide(x1, x2)) in order to avoid surprising and unexpected results; however, array libraries may choose to more strictly follow Python behavior.

For floating-point operands,

If either x1_i or x2_i is NaN, the result is NaN.
If x1_i is either +infinity or -infinity and x2_i is either +infinity or -infinity, the result is NaN.
If x1_i is either +0 or -0 and x2_i is either +0 or -0, the result is NaN.
If x1_i is +0 and x2_i is greater than 0, the result is +0.
If x1_i is -0 and x2_i is greater than 0, the result is -0.
If x1_i is +0 and x2_i is less than 0, the result is -0.
If x1_i is -0 and x2_i is less than 0, the result is +0.
If x1_i is greater than 0 and x2_i is +0, the result is +infinity.
If x1_i is greater than 0 and x2_i is -0, the result is -infinity.
If x1_i is less than 0 and x2_i is +0, the result is -infinity.
If x1_i is less than 0 and x2_i is -0, the result is +infinity.
If x1_i is +infinity and x2_i is a positive (i.e., greater than 0) finite number, the result is +infinity. (note: libraries may return NaN to match Python behavior.)
If x1_i is +infinity and x2_i is a negative (i.e., less than 0) finite number, the result is -infinity. (note: libraries may return NaN to match Python behavior.)
If x1_i is -infinity and x2_i is a positive (i.e., greater than 0) finite number, the result is -infinity. (note: libraries may return NaN to match Python behavior.)
If x1_i is -infinity and x2_i is a negative (i.e., less than 0) finite number, the result is +infinity. (note: libraries may return NaN to match Python behavior.)
If x1_i is a positive (i.e., greater than 0) finite number and x2_i is +infinity, the result is +0.
If x1_i is a positive (i.e., greater than 0) finite number and x2_i is -infinity, the result is -0. (note: libraries may return -1.0 to match Python behavior.)
If x1_i is a negative (i.e., less than 0) finite number and x2_i is +infinity, the result is -0. (note: libraries may return -1.0 to match Python behavior.)
If x1_i is a negative (i.e., less than 0) finite number and x2_i is -infinity, the result is +0.
If x1_i and x2_i have the same mathematical sign and are both nonzero finite numbers, the result has a positive mathematical sign.
If x1_i and x2_i have different mathematical signs and are both nonzero finite numbers, the result has a negative mathematical sign.
In the remaining cases, where neither -infinity, +0, -0, nor NaN is involved, the quotient must be computed and rounded to the greatest (i.e., closest to +infinity) representable integer-value number that is not greater than the division result. If the magnitude is too large to represent, the operation overflows and the result is an infinity of appropriate mathematical sign. If the magnitude is too small to represent, the operation underflows and the result is a zero of appropriate mathematical sign.

from_dlpack: from_dlpack¶

Returns a new array containing the data from another (array) object with a __dlpack__ method.

Parameters:

x (object) – input (array) object.

Returns:

out – an array containing the data in x.

Note

The returned array may be either a copy or a view. See data-interchange for details.

Return type:

array

full: full¶

Returns a new array having a specified shape and filled with fill_value.

Parameters:

shape (Union[int, Tuple[int, ...]]) – output array shape.
fill_value (Union[bool, int, float, complex]) – fill value.
dtype (Optional[dtype]) –
output array data type. If dtype is None, the output array data type must be inferred from fill_value according to the following rules:
- If the fill value is an int, the output array data type must be the default integer data type.
- If the fill value is a float, the output array data type must be the default real-valued floating-point data type.
- If the fill value is a complex number, the output array data type must be the default complex floating-point data type.
- If the fill value is a bool, the output array must have a boolean data type. Default: None.
Note

If the fill_value exceeds the precision of the resolved default output array data type, behavior is left unspecified and, thus, implementation-defined.
device (Optional[device]) – device on which to place the created array. Default: None.

Returns:

out – an array where every element is equal to fill_value.

Return type:

array

Notes

Changed in version 2022.12: Added complex data type support.

full_like: full_like¶

Returns a new array filled with fill_value and having the same shape as an input array x.

Parameters:

x (array) – input array from which to derive the output array shape.
fill_value (Union[bool, int, float, complex]) – fill value.
dtype (Optional[dtype]) –
output array data type. If dtype is None, the output array data type must be inferred from x. Default: None.

Note

If the fill_value exceeds the precision of the resolved output array data type, behavior is unspecified and, thus, implementation-defined.

Note

If the fill_value has a data type which is not of the same data type kind (boolean, integer, or floating-point) as the resolved output array data type (see type-promotion), behavior is unspecified and, thus, implementation-defined.
device (Optional[device]) – device on which to place the created array. If device is None, the output array device must be inferred from x. Default: None.

Returns:

out – an array having the same shape as x and where every element is equal to fill_value.

Return type:

array

Notes

Changed in version 2022.12: Added complex data type support.

greater: greater¶

Computes the truth value of x1_i > x2_i for each element x1_i of the input array x1 with the respective element x2_i of the input array x2.

Note

Parameters:

x1 (array) – first input array. Should have a real-valued data type.
x2 (array) – second input array. Must be compatible with x1 (see broadcasting). Should have a real-valued data type.

Returns:

out – an array containing the element-wise results. The returned array must have a data type of bool.

Return type:

array

greater_equal: greater_equal¶

Computes the truth value of x1_i >= x2_i for each element x1_i of the input array x1 with the respective element x2_i of the input array x2.

Note

Parameters:

x1 (array) – first input array. Should have a real-valued data type.
x2 (array) – second input array. Must be compatible with x1 (see broadcasting). Should have a real-valued data type.

Returns:

out – an array containing the element-wise results. The returned array must have a data type of bool.

Return type:

array

iinfo: iinfo¶

Machine limits for integer data types.

Parameters:

type (Union[dtype, array]) – the kind of integer data-type about which to get information.

Returns:

out – an object having the following attributes:

bits: int

number of bits occupied by the type.
max: int

largest representable number.
min: int

smallest representable number.
dtype: dtype

integer data type.

Added in version 2022.12.

Return type:

iinfo object

imag: imag¶

Returns the imaginary component of a complex number for each element x_i of the input array x.

Parameters:

x (array) – input array. Should have a complex floating-point data type.

Returns:

out (array) – an array containing the element-wise results. The returned array must have a floating-point data type with the same floating-point precision as x (e.g., if x is complex64, the returned array must have the floating-point data type float32).
.. versionadded:: 2022.12

inf: TArray¶: IEEE 754 floating-point representation of (positive) infinity.

int16: TDtype¶

int32: TDtype¶

int64: TDtype¶

int8: TDtype¶

isdtype: isdtype¶

Returns a boolean indicating whether a provided dtype is of a specified data type “kind”.

Parameters:

dtype (dtype) – the input dtype.
kind (Union[str, dtype, Tuple[Union[str, dtype], ...]]) –
data type kind.
- If kind is a dtype, the function must return a boolean indicating whether the input dtype is equal to the dtype specified by kind.
- If kind is a string, the function must return a boolean indicating whether the input dtype is of a specified data type kind. The following dtype kinds must be supported:
  - 'bool': boolean data types (e.g., bool).
  - 'signed integer': signed integer data types (e.g., int8, int16, int32, int64).
  - 'unsigned integer': unsigned integer data types (e.g., uint8, uint16, uint32, uint64).
  - 'integral': integer data types. Shorthand for ('signed integer', 'unsigned integer').
  - 'real floating': real-valued floating-point data types (e.g., float32, float64).
  - 'complex floating': complex floating-point data types (e.g., complex64, complex128).
  - 'numeric': numeric data types. Shorthand for ('integral', 'real floating', 'complex floating').
- If kind is a tuple, the tuple specifies a union of dtypes and/or kinds, and the function must return a boolean indicating whether the input dtype is either equal to a specified dtype or belongs to at least one specified data type kind.
Note

A conforming implementation of the array API standard is not limited to only including the dtypes described in this specification in the required data type kinds. For example, implementations supporting float16 and bfloat16 can include float16 and bfloat16 in the real floating data type kind. Similarly, implementations supporting int128 can include int128 in the signed integer data type kind.

In short, conforming implementations may extend data type kinds; however, data type kinds must remain consistent (e.g., only integer dtypes may belong to integer data type kinds and only floating-point dtypes may belong to floating-point data type kinds), and extensions must be clearly documented as such in library documentation.

Returns:

out – boolean indicating whether a provided dtype is of a specified data type kind.

Return type:

bool

Notes

Added in version 2022.12.

isfinite: isfinite¶

Tests each element x_i of the input array x to determine if finite.

Parameters:: x (array) – input array. Should have a numeric data type.
Returns:: out – an array containing test results. The returned array must have a data type of bool.
Return type:: array

Notes

Special Cases

For real-valued floating-point operands,

If x_i is either +infinity or -infinity, the result is False.
If x_i is NaN, the result is False.
If x_i is a finite number, the result is True.

For complex floating-point operands, let a = real(x_i), b = imag(x_i), and

If a is NaN or b is NaN, the result is False.
If a is either +infinity or -infinity and b is any value, the result is False.
If a is any value and b is either +infinity or -infinity, the result is False.
If a is a finite number and b is a finite number, the result is True.

Changed in version 2022.12: Added complex data type support.

isinf: isinf¶

Tests each element x_i of the input array x to determine if equal to positive or negative infinity.

Parameters:: x (array) – input array. Should have a numeric data type.
Returns:: out – an array containing test results. The returned array must have a data type of bool.
Return type:: array

Notes

Special Cases

For real-valued floating-point operands,

If x_i is either +infinity or -infinity, the result is True.
In the remaining cases, the result is False.

For complex floating-point operands, let a = real(x_i), b = imag(x_i), and

If a is either +infinity or -infinity and b is any value (including NaN), the result is True.
If a is either a finite number or NaN and b is either +infinity or -infinity, the result is True.
In the remaining cases, the result is False.

Changed in version 2022.12: Added complex data type support.

isnan: isnan¶

Tests each element x_i of the input array x to determine whether the element is NaN.

Parameters:: x (array) – input array. Should have a numeric data type.
Returns:: out – an array containing test results. The returned array should have a data type of bool.
Return type:: array

Notes

Special Cases

For real-valued floating-point operands,

If x_i is NaN, the result is True.
In the remaining cases, the result is False.

For complex floating-point operands, let a = real(x_i), b = imag(x_i), and

If a or b is NaN, the result is True.
In the remaining cases, the result is False.

Changed in version 2022.12: Added complex data type support.

less: less¶

Computes the truth value of x1_i < x2_i for each element x1_i of the input array x1 with the respective element x2_i of the input array x2.

Note

Parameters:

x1 (array) – first input array. Should have a real-valued data type.
x2 (array) – second input array. Must be compatible with x1 (see broadcasting). Should have a real-valued data type.

Returns:

out – an array containing the element-wise results. The returned array must have a data type of bool.

Return type:

array

less_equal: less_equal¶

Computes the truth value of x1_i <= x2_i for each element x1_i of the input array x1 with the respective element x2_i of the input array x2.

Note

Parameters:

x1 (array) – first input array. Should have a real-valued data type.
x2 (array) – second input array. Must be compatible with x1 (see broadcasting). Should have a real-valued data type.

Returns:

out – an array containing the element-wise results. The returned array must have a data type of bool.

Return type:

array

linspace: linspace¶

Returns evenly spaced numbers over a specified interval.

Let $N$ be the number of generated values (which is either num or num+1 depending on whether endpoint is True or False, respectively). For real-valued output arrays, the spacing between values is given by

\[\Delta_{\textrm{real}} = \frac{\textrm{stop} - \textrm{start}}{N - 1}\]

For complex output arrays, let a = real(start), b = imag(start), c = real(stop), and d = imag(stop). The spacing between complex values is given by

\[\Delta_{\textrm{complex}} = \frac{c-a}{N-1} + \frac{d-b}{N-1} j\]

Parameters:

start (Union[int, float, complex]) – the start of the interval.
stop (Union[int, float, complex]) –
the end of the interval. If endpoint is False, the function must generate a sequence of num+1 evenly spaced numbers starting with start and ending with stop and exclude the stop from the returned array such that the returned array consists of evenly spaced numbers over the half-open interval [start, stop). If endpoint is True, the output array must consist of evenly spaced numbers over the closed interval [start, stop]. Default: True.

Note

The step size changes when endpoint is False.
num (int) – number of samples. Must be a nonnegative integer value.
dtype (Optional[dtype]) –
output array data type. Should be a floating-point data type. If dtype is None,
- if either start or stop is a complex number, the output data type must be the default complex floating-point data type.
- if both start and stop are real-valued, the output data type must be the default real-valued floating-point data type.
Default: None.

Note

If dtype is not None, conversion of start and stop should obey type-promotion rules. Conversions not specified according to type-promotion rules may or may not be permitted by a conforming array library.
device (Optional[device]) – device on which to place the created array. Default: None.
endpoint (bool) – boolean indicating whether to include stop in the interval. Default: True.

Returns:

out – a one-dimensional array containing evenly spaced values.

Return type:

array

Notes

Note

While this specification recommends that this function only return arrays having a floating-point data type, specification-compliant array libraries may choose to support output arrays having an integer data type (e.g., due to backward compatibility concerns). However, function behavior when generating integer output arrays is unspecified and, thus, is implementation-defined. Accordingly, using this function to generate integer output arrays is not portable.

Note

As mixed data type promotion is implementation-defined, behavior when start or stop exceeds the maximum safe integer of an output floating-point data type is implementation-defined. An implementation may choose to overflow or raise an exception.

Changed in version 2022.12: Added complex data type support.

log: log¶

Calculates an implementation-dependent approximation to the natural (base e) logarithm for each element x_i of the input array x.

Note

The natural logarithm of a complex number $z$ with polar coordinates $(r,\theta)$ equals $\ln r + (\theta + 2n\pi)j$ with principal value $\ln r + \theta j$.

Note

For complex floating-point operands, log(conj(x)) must equal conj(log(x)).

Note

By convention, the branch cut of the natural logarithm is the negative real axis $(-\infty, 0)$.

The natural logarithm is a continuous function from above the branch cut, taking into account the sign of the imaginary component.

Accordingly, for complex arguments, the function returns the natural logarithm in the range of a strip in the interval $[-\pi j, +\pi j]$ along the imaginary axis and mathematically unbounded along the real axis.

Note: branch cuts follow C99 and have provisional status (see branch-cuts).

Parameters:: x (array) – input array. Should have a floating-point data type.
Returns:: out – an array containing the evaluated natural logarithm for each element in x. The returned array must have a floating-point data type determined by type-promotion.
Return type:: array

Notes

Special cases

For real-valued floating-point operands,

If x_i is NaN, the result is NaN.
If x_i is less than 0, the result is NaN.
If x_i is either +0 or -0, the result is -infinity.
If x_i is 1, the result is +0.
If x_i is +infinity, the result is +infinity.

For complex floating-point operands, let a = real(x_i), b = imag(x_i), and

If a is -0 and b is +0, the result is -infinity + πj.
If a is +0 and b is +0, the result is -infinity + 0j.
If a is a finite number and b is +infinity, the result is +infinity + πj/2.
If a is a finite number and b is NaN, the result is NaN + NaN j.
If a is -infinity and b is a positive (i.e., greater than 0) finite number, the result is +infinity + πj.
If a is +infinity and b is a positive (i.e., greater than 0) finite number, the result is +infinity + 0j.
If a is -infinity and b is +infinity, the result is +infinity + 3πj/4.
If a is +infinity and b is +infinity, the result is +infinity + πj/4.
If a is either +infinity or -infinity and b is NaN, the result is +infinity + NaN j.
If a is NaN and b is a finite number, the result is NaN + NaN j.
If a is NaN and b is +infinity, the result is +infinity + NaN j.
If a is NaN and b is NaN, the result is NaN + NaN j.

Changed in version 2022.12: Added complex data type support.

log10: log10¶

Calculates an implementation-dependent approximation to the base 10 logarithm for each element x_i of the input array x.

Note

For complex floating-point operands, log10(conj(x)) must equal conj(log10(x)).

Parameters:: x (array) – input array. Should have a floating-point data type.
Returns:: out – an array containing the evaluated base 10 logarithm for each element in x. The returned array must have a floating-point data type determined by type-promotion.
Return type:: array

Notes

Special cases

For real-valued floating-point operands,

If x_i is NaN, the result is NaN.
If x_i is less than 0, the result is NaN.
If x_i is either +0 or -0, the result is -infinity.
If x_i is 1, the result is +0.
If x_i is +infinity, the result is +infinity.

For complex floating-point operands, special cases must be handled as if the operation is implemented using the standard change of base formula

\[\log_{10} x = \frac{\log_{e} x}{\log_{e} 10}\]

where $\log_{e}$ is the natural logarithm, as implemented by log().

Changed in version 2022.12: Added complex data type support.

log1p: log1p¶

Calculates an implementation-dependent approximation to log(1+x), where log refers to the natural (base e) logarithm, for each element x_i of the input array x.

Note

The purpose of this function is to calculate log(1+x) more accurately when x is close to zero. Accordingly, conforming implementations should avoid implementing this function as simply log(1+x). See FDLIBM, or some other IEEE 754-2019 compliant mathematical library, for a potential reference implementation.

Note

For complex floating-point operands, log1p(conj(x)) must equal conj(log1p(x)).

Note

By convention, the branch cut of the natural logarithm is the negative real axis $(-\infty, 0)$.

The natural logarithm is a continuous function from above the branch cut, taking into account the sign of the imaginary component.

Note: branch cuts follow C99 and have provisional status (see branch-cuts).

Parameters:: x (array) – input array. Should have a floating-point data type.
Returns:: out – an array containing the evaluated result for each element in x. The returned array must have a floating-point data type determined by type-promotion.
Return type:: array

Notes

Special cases

For real-valued floating-point operands,

If x_i is NaN, the result is NaN.
If x_i is less than -1, the result is NaN.
If x_i is -1, the result is -infinity.
If x_i is -0, the result is -0.
If x_i is +0, the result is +0.
If x_i is +infinity, the result is +infinity.

For complex floating-point operands, let a = real(x_i), b = imag(x_i), and

If a is -1 and b is +0, the result is -infinity + 0j.
If a is a finite number and b is +infinity, the result is +infinity + πj/2.
If a is a finite number and b is NaN, the result is NaN + NaN j.
If a is -infinity and b is a positive (i.e., greater than 0) finite number, the result is +infinity + πj.
If a is +infinity and b is a positive (i.e., greater than 0) finite number, the result is +infinity + 0j.
If a is -infinity and b is +infinity, the result is +infinity + 3πj/4.
If a is +infinity and b is +infinity, the result is +infinity + πj/4.
If a is either +infinity or -infinity and b is NaN, the result is +infinity + NaN j.
If a is NaN and b is a finite number, the result is NaN + NaN j.
If a is NaN and b is +infinity, the result is +infinity + NaN j.
If a is NaN and b is NaN, the result is NaN + NaN j.

Changed in version 2022.12: Added complex data type support.

log2: log2¶

Calculates an implementation-dependent approximation to the base 2 logarithm for each element x_i of the input array x.

Note

For complex floating-point operands, log2(conj(x)) must equal conj(log2(x)).

Parameters:: x (array) – input array. Should have a floating-point data type.
Returns:: out – an array containing the evaluated base 2 logarithm for each element in x. The returned array must have a floating-point data type determined by type-promotion.
Return type:: array

Notes

Special cases

For real-valued floating-point operands,

If x_i is NaN, the result is NaN.
If x_i is less than 0, the result is NaN.
If x_i is either +0 or -0, the result is -infinity.
If x_i is 1, the result is +0.
If x_i is +infinity, the result is +infinity.

For complex floating-point operands, special cases must be handled as if the operation is implemented using the standard change of base formula

\[\log_{2} x = \frac{\log_{e} x}{\log_{e} 2}\]

where $\log_{e}$ is the natural logarithm, as implemented by log().

Changed in version 2022.12: Added complex data type support.

logaddexp: logaddexp¶

Calculates the logarithm of the sum of exponentiations log(exp(x1) + exp(x2)) for each element x1_i of the input array x1 with the respective element x2_i of the input array x2.

Parameters:

x1 (array) – first input array. Should have a real-valued floating-point data type.
x2 (array) – second input array. Must be compatible with x1 (see broadcasting). Should have a real-valued floating-point data type.

Returns:

out – an array containing the element-wise results. The returned array must have a real-valued floating-point data type determined by type-promotion.

Return type:

array

Notes

Special cases

For floating-point operands,

If either x1_i or x2_i is NaN, the result is NaN.
If x1_i is +infinity and x2_i is not NaN, the result is +infinity.
If x1_i is not NaN and x2_i is +infinity, the result is +infinity.

logical_and: logical_and¶

Computes the logical AND for each element x1_i of the input array x1 with the respective element x2_i of the input array x2.

Note

While this specification recommends that this function only accept input arrays having a boolean data type, specification-compliant array libraries may choose to accept input arrays having real-valued data types. If non-boolean data types are supported, zeros must be considered the equivalent of False, while non-zeros must be considered the equivalent of True.

Parameters:

x1 (array) – first input array. Should have a boolean data type.
x2 (array) – second input array. Must be compatible with x1 (see broadcasting). Should have a boolean data type.

Returns:

out – an array containing the element-wise results. The returned array must have a data type of bool.

Return type:

array

logical_not: logical_not¶

Computes the logical NOT for each element x_i of the input array x.

Note

Parameters:: x (array) – input array. Should have a boolean data type.
Returns:: out – an array containing the element-wise results. The returned array must have a data type of bool.
Return type:: array

logical_or: logical_or¶

Computes the logical OR for each element x1_i of the input array x1 with the respective element x2_i of the input array x2.

Note

Parameters:

x1 (array) – first input array. Should have a boolean data type.
x2 (array) – second input array. Must be compatible with x1 (see broadcasting). Should have a boolean data type.

Returns:

out – an array containing the element-wise results. The returned array must have a data type of bool.

Return type:

array

logical_xor: logical_xor¶

Computes the logical XOR for each element x1_i of the input array x1 with the respective element x2_i of the input array x2.

Note

Parameters:

x1 (array) – first input array. Should have a boolean data type.
x2 (array) – second input array. Must be compatible with x1 (see broadcasting). Should have a boolean data type.

Returns:

out – an array containing the element-wise results. The returned array must have a data type of bool.

Return type:

array

matmul: matmul¶

Computes the matrix product.

Note

The matmul function must implement the same semantics as the built-in @ operator (see PEP 465).

Parameters:

x1 (array) – first input array. Should have a numeric data type. Must have at least one dimension. If x1 is one-dimensional having shape (M,) and x2 has more than one dimension, x1 must be promoted to a two-dimensional array by prepending 1 to its dimensions (i.e., must have shape (1, M)). After matrix multiplication, the prepended dimensions in the returned array must be removed. If x1 has more than one dimension (including after vector-to-matrix promotion), shape(x1)[:-2] must be compatible with shape(x2)[:-2] (after vector-to-matrix promotion) (see broadcasting). If x1 has shape (..., M, K), the innermost two dimensions form matrices on which to perform matrix multiplication.
x2 (array) – second input array. Should have a numeric data type. Must have at least one dimension. If x2 is one-dimensional having shape (N,) and x1 has more than one dimension, x2 must be promoted to a two-dimensional array by appending 1 to its dimensions (i.e., must have shape (N, 1)). After matrix multiplication, the appended dimensions in the returned array must be removed. If x2 has more than one dimension (including after vector-to-matrix promotion), shape(x2)[:-2] must be compatible with shape(x1)[:-2] (after vector-to-matrix promotion) (see broadcasting). If x2 has shape (..., K, N), the innermost two dimensions form matrices on which to perform matrix multiplication.

Note

If either x1 or x2 has a complex floating-point data type, neither argument must be complex-conjugated or transposed. If conjugation and/or transposition is desired, these operations should be explicitly performed prior to computing the matrix product.

Returns:

out –

if both x1 and x2 are one-dimensional arrays having shape (N,), a zero-dimensional array containing the inner product as its only element.
if x1 is a two-dimensional array having shape (M, K) and x2 is a two-dimensional array having shape (K, N), a two-dimensional array containing the conventional matrix product and having shape (M, N).
if x1 is a one-dimensional array having shape (K,) and x2 is an array having shape (..., K, N), an array having shape (..., N) (i.e., prepended dimensions during vector-to-matrix promotion must be removed) and containing the conventional matrix product.
if x1 is an array having shape (..., M, K) and x2 is a one-dimensional array having shape (K,), an array having shape (..., M) (i.e., appended dimensions during vector-to-matrix promotion must be removed) and containing the conventional matrix product.
if x1 is a two-dimensional array having shape (M, K) and x2 is an array having shape (..., K, N), an array having shape (..., M, N) and containing the conventional matrix product for each stacked matrix.
if x1 is an array having shape (..., M, K) and x2 is a two-dimensional array having shape (K, N), an array having shape (..., M, N) and containing the conventional matrix product for each stacked matrix.
if either x1 or x2 has more than two dimensions, an array having a shape determined by broadcasting shape(x1)[:-2] against shape(x2)[:-2] and containing the conventional matrix product for each stacked matrix.

The returned array must have a data type determined by type-promotion.

Return type:

array

Notes

Changed in version 2022.12: Added complex data type support.

Raises

if either x1 or x2 is a zero-dimensional array.
if x1 is a one-dimensional array having shape (K,), x2 is a one-dimensional array having shape (L,), and K != L.
if x1 is a one-dimensional array having shape (K,), x2 is an array having shape (..., L, N), and K != L.
if x1 is an array having shape (..., M, K), x2 is a one-dimensional array having shape (L,), and K != L.
if x1 is an array having shape (..., M, K), x2 is an array having shape (..., L, N), and K != L.

matrix_transpose: matrix_transpose¶

Transposes a matrix (or a stack of matrices) x.

Parameters:: x (array) – input array having shape (..., M, N) and whose innermost two dimensions form MxN matrices.
Returns:: out – an array containing the transpose for each matrix and having shape (..., N, M). The returned array must have the same data type as x.
Return type:: array

max: max¶

Calculates the maximum value of the input array x.

Note

When the number of elements over which to compute the maximum value is zero, the maximum value is implementation-defined. Specification-compliant libraries may choose to raise an error, return a sentinel value (e.g., if x is a floating-point input array, return NaN), or return the minimum possible value for the input array x data type (e.g., if x is a floating-point array, return -infinity).

Note

Parameters:

x (array) – input array. Should have a real-valued data type.
axis (Optional[Union[int, Tuple[int, ...]]]) – axis or axes along which maximum values must be computed. By default, the maximum value must be computed over the entire array. If a tuple of integers, maximum values must be computed over multiple axes. Default: None.
keepdims (bool) – if True, the reduced axes (dimensions) must be included in the result as singleton dimensions, and, accordingly, the result must be compatible with the input array (see broadcasting). Otherwise, if False, the reduced axes (dimensions) must not be included in the result. Default: False.

Returns:

out – if the maximum value was computed over the entire array, a zero-dimensional array containing the maximum value; otherwise, a non-zero-dimensional array containing the maximum values. The returned array must have the same data type as x.

Return type:

array

Notes

Special Cases

For floating-point operands,

If x_i is NaN, the maximum value is NaN (i.e., NaN values propagate).

mean: mean¶

Calculates the arithmetic mean of the input array x.

Parameters:

x (array) – input array. Should have a real-valued floating-point data type.
axis (Optional[Union[int, Tuple[int, ...]]]) – axis or axes along which arithmetic means must be computed. By default, the mean must be computed over the entire array. If a tuple of integers, arithmetic means must be computed over multiple axes. Default: None.
keepdims (bool) – if True, the reduced axes (dimensions) must be included in the result as singleton dimensions, and, accordingly, the result must be compatible with the input array (see broadcasting). Otherwise, if False, the reduced axes (dimensions) must not be included in the result. Default: False.

Returns:

out – if the arithmetic mean was computed over the entire array, a zero-dimensional array containing the arithmetic mean; otherwise, a non-zero-dimensional array containing the arithmetic means. The returned array must have the same data type as x.

Note

While this specification recommends that this function only accept input arrays having a real-valued floating-point data type, specification-compliant array libraries may choose to accept input arrays having an integer data type. While mixed data type promotion is implementation-defined, if the input array x has an integer data type, the returned array must have the default real-valued floating-point data type.

Return type:

array

Notes

Special Cases

Let N equal the number of elements over which to compute the arithmetic mean.

If N is 0, the arithmetic mean is NaN.
If x_i is NaN, the arithmetic mean is NaN (i.e., NaN values propagate).

meshgrid: meshgrid¶

Returns coordinate matrices from coordinate vectors.

Parameters:

arrays (array) – an arbitrary number of one-dimensional arrays representing grid coordinates. Each array should have the same numeric data type.
indexing (Literal["xy", "ij"]) – Cartesian 'xy' or matrix 'ij' indexing of output. If provided zero or one one-dimensional vector(s) (i.e., the zero- and one-dimensional cases, respectively), the indexing keyword has no effect and should be ignored. Default: 'xy'.

Returns:

out – list of N arrays, where N is the number of provided one-dimensional input arrays. Each returned array must have rank N. For N one-dimensional arrays having lengths Ni = len(xi),

if matrix indexing ij, then each returned array must have the shape (N1, N2, N3, ..., Nn).
if Cartesian indexing xy, then each returned array must have shape (N2, N1, N3, ..., Nn).

Accordingly, for the two-dimensional case with input one-dimensional arrays of length M and N, if matrix indexing ij, then each returned array must have shape (M, N), and, if Cartesian indexing xy, then each returned array must have shape (N, M).

Similarly, for the three-dimensional case with input one-dimensional arrays of length M, N, and P, if matrix indexing ij, then each returned array must have shape (M, N, P), and, if Cartesian indexing xy, then each returned array must have shape (N, M, P).

Each returned array should have the same data type as the input arrays.

Return type:

List[array]

Notes

Changed in version 2022.12: Added complex data type support.

min: min¶

Calculates the minimum value of the input array x.

Note

When the number of elements over which to compute the minimum value is zero, the minimum value is implementation-defined. Specification-compliant libraries may choose to raise an error, return a sentinel value (e.g., if x is a floating-point input array, return NaN), or return the maximum possible value for the input array x data type (e.g., if x is a floating-point array, return +infinity).

Note

Parameters:

x (array) – input array. Should have a real-valued data type.
axis (Optional[Union[int, Tuple[int, ...]]]) – axis or axes along which minimum values must be computed. By default, the minimum value must be computed over the entire array. If a tuple of integers, minimum values must be computed over multiple axes. Default: None.
keepdims (bool) – if True, the reduced axes (dimensions) must be included in the result as singleton dimensions, and, accordingly, the result must be compatible with the input array (see broadcasting). Otherwise, if False, the reduced axes (dimensions) must not be included in the result. Default: False.

Returns:

out – if the minimum value was computed over the entire array, a zero-dimensional array containing the minimum value; otherwise, a non-zero-dimensional array containing the minimum values. The returned array must have the same data type as x.

Return type:

array

Notes

Special Cases

For floating-point operands,

If x_i is NaN, the minimum value is NaN (i.e., NaN values propagate).

multiply: multiply¶

Calculates the product for each element x1_i of the input array x1 with the respective element x2_i of the input array x2.

Note

Floating-point multiplication is not always associative due to finite precision.

Parameters:

x1 (array) – first input array. Should have a numeric data type.
x2 (array) – second input array. Must be compatible with x1 (see broadcasting). Should have a numeric data type.

Returns:

out – an array containing the element-wise products. The returned array must have a data type determined by type-promotion.

Return type:

array

Notes

Special cases

For real-valued floating-point operands,

If either x1_i or x2_i is NaN, the result is NaN.
If x1_i is either +infinity or -infinity and x2_i is either +0 or -0, the result is NaN.
If x1_i is either +0 or -0 and x2_i is either +infinity or -infinity, the result is NaN.
If x1_i and x2_i have the same mathematical sign, the result has a positive mathematical sign, unless the result is NaN. If the result is NaN, the “sign” of NaN is implementation-defined.
If x1_i and x2_i have different mathematical signs, the result has a negative mathematical sign, unless the result is NaN. If the result is NaN, the “sign” of NaN is implementation-defined.
If x1_i is either +infinity or -infinity and x2_i is either +infinity or -infinity, the result is a signed infinity with the mathematical sign determined by the rule already stated above.
If x1_i is either +infinity or -infinity and x2_i is a nonzero finite number, the result is a signed infinity with the mathematical sign determined by the rule already stated above.
If x1_i is a nonzero finite number and x2_i is either +infinity or -infinity, the result is a signed infinity with the mathematical sign determined by the rule already stated above.
In the remaining cases, where neither infinity nor NaN is involved, the product must be computed and rounded to the nearest representable value according to IEEE 754-2019 and a supported rounding mode. If the magnitude is too large to represent, the result is an infinity of appropriate mathematical sign. If the magnitude is too small to represent, the result is a zero of appropriate mathematical sign.

For complex floating-point operands, multiplication is defined according to the following table. For real components a and c and imaginary components b and d,

	c	dj	c + dj
a	a * c	(a*d)j	(ac) + (ad)j
bj	(b*c)j	-(b*d)	-(bd) + (bc)j
a + bj	(ac) + (bc)j	-(bd) + (ad)j	special rules

When a, b, c, or d are all finite numbers (i.e., a value other than NaN, +infinity, or -infinity), multiplication of complex floating-point operands should be computed as if calculated according to the textbook formula for complex number multiplication

\[(a + bj) \cdot (c + dj) = (ac - bd) + (bc + ad)j\]

When at least one of a, b, c, or d is NaN, +infinity, or -infinity,

If a, b, c, and d are all NaN, the result is NaN + NaN j.
In the remaining cases, the result is implementation dependent.

Note

For complex floating-point operands, the results of special cases may be implementation dependent depending on how an implementation chooses to model complex numbers and complex infinity (e.g., complex plane versus Riemann sphere). For those implementations following C99 and its one-infinity model, when at least one component is infinite, even if the other component is NaN, the complex value is infinite, and the usual arithmetic rules do not apply to complex-complex multiplication. In the interest of performance, other implementations may want to avoid the complex branching logic necessary to implement the one-infinity model and choose to implement all complex-complex multiplication according to the textbook formula. Accordingly, special case behavior is unlikely to be consistent across implementations.

Changed in version 2022.12: Added complex data type support.

nan: TArray¶: IEEE 754 floating-point representation of Not a Number (NaN).

negative: negative¶

Computes the numerical negative of each element x_i (i.e., y_i = -x_i) of the input array x.

Note

For signed integer data types, the numerical negative of the minimum representable integer is implementation-dependent.

Note

If x has a complex floating-point data type, both the real and imaginary components for each x_i must be negated (a result which follows from the rules of complex number multiplication).

Parameters:: x (array) – input array. Should have a numeric data type.
Returns:: out – an array containing the evaluated result for each element in x. The returned array must have a data type determined by type-promotion.
Return type:: array

Notes

Changed in version 2022.12: Added complex data type support.

newaxis: TArray¶: An alias for None which is useful for indexing arrays.

nonzero: nonzero¶

Returns the indices of the array elements which are non-zero.

Note

If x has a complex floating-point data type, non-zero elements are those elements having at least one component (real or imaginary) which is non-zero.

Note

If x has a boolean data type, non-zero elements are those elements which are equal to True.

Data-dependent output shape

The shape of the output array for this function depends on the data values in the input array; hence, array libraries which build computation graphs (e.g., JAX, Dask, etc.) may find this function difficult to implement without knowing array values. Accordingly, such libraries may choose to omit this function. See data-dependent-output-shapes section for more details.

Parameters:: x (array) – input array. Must have a positive rank. If x is zero-dimensional, the function must raise an exception.
Returns:: out – a tuple of k arrays, one for each dimension of x and each of size n (where n is the total number of non-zero elements), containing the indices of the non-zero elements in that dimension. The indices must be returned in row-major, C-style order. The returned array must have the default array index data type.
Return type:: Tuple[array, …]

Notes

Changed in version 2022.12: Added complex data type support.

not_equal: not_equal¶

Computes the truth value of x1_i != x2_i for each element x1_i of the input array x1 with the respective element x2_i of the input array x2.

Parameters:

x1 (array) – first input array. May have any data type.
x2 (array) – second input array. Must be compatible with x1 (see broadcasting).

Returns:

out – an array containing the element-wise results. The returned array must have a data type of bool.

Return type:

array

Notes

Special Cases

For real-valued floating-point operands,

If x1_i is NaN or x2_i is NaN, the result is True.
If x1_i is +infinity and x2_i is -infinity, the result is True.
If x1_i is -infinity and x2_i is +infinity, the result is True.
If x1_i is a finite number, x2_i is a finite number, and x1_i does not equal x2_i, the result is True.
In the remaining cases, the result is False.

For complex floating-point operands, let a = real(x1_i), b = imag(x1_i), c = real(x2_i), d = imag(x2_i), and

If a, b, c, or d is NaN, the result is True.
In the remaining cases, the result is the logical OR of the equality comparison between the real values a and c (real components) and between the real values b and d (imaginary components), as described above for real-valued floating-point operands (i.e., a != c OR b != d).

Note

For discussion of complex number equality, see complex-numbers.

Changed in version 2022.12: Added complex data type support.

ones: ones¶

Returns a new array having a specified shape and filled with ones.

Note

An output array having a complex floating-point data type must contain complex numbers having a real component equal to one and an imaginary component equal to zero (i.e., 1 + 0j).

Parameters:

shape (Union[int, Tuple[int, ...]]) – output array shape.
dtype (Optional[dtype]) – output array data type. If dtype is None, the output array data type must be the default real-valued floating-point data type. Default: None.
device (Optional[device]) – device on which to place the created array. Default: None.

Returns:

out – an array containing ones.

Return type:

array

Notes

Changed in version 2022.12: Added complex data type support.

ones_like: ones_like¶

Returns a new array filled with ones and having the same shape as an input array x.

Note

An output array having a complex floating-point data type must contain complex numbers having a real component equal to one and an imaginary component equal to zero (i.e., 1 + 0j).

Parameters:

x (array) – input array from which to derive the output array shape.
dtype (Optional[dtype]) – output array data type. If dtype is None, the output array data type must be inferred from x. Default: None.
device (Optional[device]) – device on which to place the created array. If device is None, the output array device must be inferred from x. Default: None.

Returns:

out – an array having the same shape as x and filled with ones.

Return type:

array

Notes

Changed in version 2022.12: Added complex data type support.

permute_dims: permute_dims¶

Permutes the axes (dimensions) of an array x.

Parameters:

x (array) – input array.
axes (Tuple[int, ...]) – tuple containing a permutation of (0, 1, ..., N-1) where N is the number of axes (dimensions) of x.

Returns:

out – an array containing the axes permutation. The returned array must have the same data type as x.

Return type:

array

pi: TArray¶

IEEE 754 floating-point representation of the mathematical constant π.

pi = 3.1415926535897932384626433...

positive: positive¶

Computes the numerical positive of each element x_i (i.e., y_i = +x_i) of the input array x.

Parameters:: x (array) – input array. Should have a numeric data type.
Returns:: out – an array containing the evaluated result for each element in x. The returned array must have the same data type as x.
Return type:: array

Notes

Changed in version 2022.12: Added complex data type support.

pow: pow¶

Calculates an implementation-dependent approximation of exponentiation by raising each element x1_i (the base) of the input array x1 to the power of x2_i (the exponent), where x2_i is the corresponding element of the input array x2.

Note

If both x1 and x2 have integer data types, the result of pow when x2_i is negative (i.e., less than zero) is unspecified and thus implementation-dependent.

If x1 has an integer data type and x2 has a floating-point data type, behavior is implementation-dependent (type promotion between data type “kinds” (integer versus floating-point) is unspecified).

Note

By convention, the branch cut of the natural logarithm is the negative real axis $(-\infty, 0)$.

The natural logarithm is a continuous function from above the branch cut, taking into account the sign of the imaginary component. As special cases involving complex floating-point operands should be handled according to exp(x2*log(x1)), exponentiation has the same branch cut for x1 as the natural logarithm (see log()).

Note: branch cuts follow C99 and have provisional status (see branch-cuts).

Parameters:

x1 (array) – first input array whose elements correspond to the exponentiation base. Should have a numeric data type.
x2 (array) – second input array whose elements correspond to the exponentiation exponent. Must be compatible with x1 (see broadcasting). Should have a numeric data type.

Returns:

out – an array containing the element-wise results. The returned array must have a data type determined by type-promotion.

Return type:

array

Notes

Special cases

For real-valued floating-point operands,

If x1_i is not equal to 1 and x2_i is NaN, the result is NaN.
If x2_i is +0, the result is 1, even if x1_i is NaN.
If x2_i is -0, the result is 1, even if x1_i is NaN.
If x1_i is NaN and x2_i is not equal to 0, the result is NaN.
If abs(x1_i) is greater than 1 and x2_i is +infinity, the result is +infinity.
If abs(x1_i) is greater than 1 and x2_i is -infinity, the result is +0.
If abs(x1_i) is 1 and x2_i is +infinity, the result is 1.
If abs(x1_i) is 1 and x2_i is -infinity, the result is 1.
If x1_i is 1 and x2_i is not NaN, the result is 1.
If abs(x1_i) is less than 1 and x2_i is +infinity, the result is +0.
If abs(x1_i) is less than 1 and x2_i is -infinity, the result is +infinity.
If x1_i is +infinity and x2_i is greater than 0, the result is +infinity.
If x1_i is +infinity and x2_i is less than 0, the result is +0.
If x1_i is -infinity, x2_i is greater than 0, and x2_i is an odd integer value, the result is -infinity.
If x1_i is -infinity, x2_i is greater than 0, and x2_i is not an odd integer value, the result is +infinity.
If x1_i is -infinity, x2_i is less than 0, and x2_i is an odd integer value, the result is -0.
If x1_i is -infinity, x2_i is less than 0, and x2_i is not an odd integer value, the result is +0.
If x1_i is +0 and x2_i is greater than 0, the result is +0.
If x1_i is +0 and x2_i is less than 0, the result is +infinity.
If x1_i is -0, x2_i is greater than 0, and x2_i is an odd integer value, the result is -0.
If x1_i is -0, x2_i is greater than 0, and x2_i is not an odd integer value, the result is +0.
If x1_i is -0, x2_i is less than 0, and x2_i is an odd integer value, the result is -infinity.
If x1_i is -0, x2_i is less than 0, and x2_i is not an odd integer value, the result is +infinity.
If x1_i is less than 0, x1_i is a finite number, x2_i is a finite number, and x2_i is not an integer value, the result is NaN.

For complex floating-point operands, special cases should be handled as if the operation is implemented as exp(x2*log(x1)).

Note

Conforming implementations are allowed to treat special cases involving complex floating-point operands more carefully than as described in this specification.

Changed in version 2022.12: Added complex data type support.

prod: prod¶

Calculates the product of input array x elements.

Parameters:

x (array) – input array. Should have a numeric data type.
axis (Optional[Union[int, Tuple[int, ...]]]) – axis or axes along which products must be computed. By default, the product must be computed over the entire array. If a tuple of integers, products must be computed over multiple axes. Default: None.
dtype (Optional[dtype]) –
data type of the returned array. If None,
- if the default data type corresponding to the data type “kind” (integer, real-valued floating-point, or complex floating-point) of x has a smaller range of values than the data type of x (e.g., x has data type int64 and the default data type is int32, or x has data type uint64 and the default data type is int64), the returned array must have the same data type as x.
- if x has a real-valued floating-point data type, the returned array must have the default real-valued floating-point data type.
- if x has a complex floating-point data type, the returned array must have the default complex floating-point data type.
- if x has a signed integer data type (e.g., int16), the returned array must have the default integer data type.
- if x has an unsigned integer data type (e.g., uint16), the returned array must have an unsigned integer data type having the same number of bits as the default integer data type (e.g., if the default integer data type is int32, the returned array must have a uint32 data type).
If the data type (either specified or resolved) differs from the data type of x, the input array should be cast to the specified data type before computing the product. Default: None.

Note

This keyword argument is intended to help prevent data type overflows.
keepdims (bool) – if True, the reduced axes (dimensions) must be included in the result as singleton dimensions, and, accordingly, the result must be compatible with the input array (see broadcasting). Otherwise, if False, the reduced axes (dimensions) must not be included in the result. Default: False.

Returns:

out – if the product was computed over the entire array, a zero-dimensional array containing the product; otherwise, a non-zero-dimensional array containing the products. The returned array must have a data type as described by the dtype parameter above.

Return type:

array

Notes

Special Cases

Let N equal the number of elements over which to compute the product.

If N is 0, the product is 1 (i.e., the empty product).

For both real-valued and complex floating-point operands, special cases must be handled as if the operation is implemented by successive application of multiply().

Changed in version 2022.12: Added complex data type support.

real: real¶

Returns the real component of a complex number for each element x_i of the input array x.

Parameters:: x (array) – input array. Should have a complex floating-point data type.
Returns:: out – an array containing the element-wise results. The returned array must have a floating-point data type with the same floating-point precision as x (e.g., if x is complex64, the returned array must have the floating-point data type float32).
Return type:: array

Notes

Added in version 2022.12.

remainder: remainder¶

Returns the remainder of division for each element x1_i of the input array x1 and the respective element x2_i of the input array x2.

Note

This function is equivalent to the Python modulus operator x1_i % x2_i.

Note

For input arrays which promote to an integer data type, the result of division by zero is unspecified and thus implementation-defined.

Parameters:

x1 (array) – dividend input array. Should have a real-valued data type.
x2 (array) – divisor input array. Must be compatible with x1 (see broadcasting). Should have a real-valued data type.

Returns:

out – an array containing the element-wise results. Each element-wise result must have the same sign as the respective element x2_i. The returned array must have a data type determined by type-promotion.

Return type:

array

Notes

Special cases

Note

In general, similar to Python’s % operator, this function is not recommended for floating-point operands as semantics do not follow IEEE 754. That this function is specified to accept floating-point operands is primarily for reasons of backward compatibility.

For floating-point operands,

If either x1_i or x2_i is NaN, the result is NaN.
If x1_i is either +infinity or -infinity and x2_i is either +infinity or -infinity, the result is NaN.
If x1_i is either +0 or -0 and x2_i is either +0 or -0, the result is NaN.
If x1_i is +0 and x2_i is greater than 0, the result is +0.
If x1_i is -0 and x2_i is greater than 0, the result is +0.
If x1_i is +0 and x2_i is less than 0, the result is -0.
If x1_i is -0 and x2_i is less than 0, the result is -0.
If x1_i is greater than 0 and x2_i is +0, the result is NaN.
If x1_i is greater than 0 and x2_i is -0, the result is NaN.
If x1_i is less than 0 and x2_i is +0, the result is NaN.
If x1_i is less than 0 and x2_i is -0, the result is NaN.
If x1_i is +infinity and x2_i is a positive (i.e., greater than 0) finite number, the result is NaN.
If x1_i is +infinity and x2_i is a negative (i.e., less than 0) finite number, the result is NaN.
If x1_i is -infinity and x2_i is a positive (i.e., greater than 0) finite number, the result is NaN.
If x1_i is -infinity and x2_i is a negative (i.e., less than 0) finite number, the result is NaN.
If x1_i is a positive (i.e., greater than 0) finite number and x2_i is +infinity, the result is x1_i. (note: this result matches Python behavior.)
If x1_i is a positive (i.e., greater than 0) finite number and x2_i is -infinity, the result is x2_i. (note: this result matches Python behavior.)
If x1_i is a negative (i.e., less than 0) finite number and x2_i is +infinity, the result is x2_i. (note: this results matches Python behavior.)
If x1_i is a negative (i.e., less than 0) finite number and x2_i is -infinity, the result is x1_i. (note: this result matches Python behavior.)
In the remaining cases, the result must match that of the Python % operator.

reshape: reshape¶

Reshapes an array without changing its data.

Parameters:

x (array) – input array to reshape.
shape (Tuple[int, ...]) – a new shape compatible with the original shape. One shape dimension is allowed to be -1. When a shape dimension is -1, the corresponding output array shape dimension must be inferred from the length of the array and the remaining dimensions.
copy (Optional[bool]) – boolean indicating whether or not to copy the input array. If True, the function must always copy. If False, the function must never copy and must raise a ValueError in case a copy would be necessary. If None, the function must reuse existing memory buffer if possible and copy otherwise. Default: None.

Returns:

out – an output array having the same data type and elements as x.

Return type:

array

result_type: result_type¶

Returns the dtype that results from applying the type promotion rules (see type-promotion) to the arguments.

Note

If provided mixed dtypes (e.g., integer and floating-point), the returned dtype will be implementation-specific.

Parameters:: arrays_and_dtypes (Union[array, dtype]) – an arbitrary number of input arrays and/or dtypes.
Returns:: out – the dtype resulting from an operation involving the input arrays and dtypes.
Return type:: dtype

roll: roll¶

Rolls array elements along a specified axis. Array elements that roll beyond the last position are re-introduced at the first position. Array elements that roll beyond the first position are re-introduced at the last position.

Parameters:

x (array) – input array.
shift (Union[int, Tuple[int, ...]]) – number of places by which the elements are shifted. If shift is a tuple, then axis must be a tuple of the same size, and each of the given axes must be shifted by the corresponding element in shift. If shift is an int and axis a tuple, then the same shift must be used for all specified axes. If a shift is positive, then array elements must be shifted positively (toward larger indices) along the dimension of axis. If a shift is negative, then array elements must be shifted negatively (toward smaller indices) along the dimension of axis.
axis (Optional[Union[int, Tuple[int, ...]]]) – axis (or axes) along which elements to shift. If axis is None, the array must be flattened, shifted, and then restored to its original shape. Default: None.

Returns:

out – an output array having the same data type as x and whose elements, relative to x, are shifted.

Return type:

array

round: round¶

Rounds each element x_i of the input array x to the nearest integer-valued number.

Note

For complex floating-point operands, real and imaginary components must be independently rounded to the nearest integer-valued number.

Rounded real and imaginary components must be equal to their equivalent rounded real-valued floating-point counterparts (i.e., for complex-valued x, real(round(x)) must equal round(real(x))) and imag(round(x)) must equal round(imag(x))).

Parameters:: x (array) – input array. Should have a numeric data type.
Returns:: out – an array containing the rounded result for each element in x. The returned array must have the same data type as x.
Return type:: array

Notes

Special cases

Note

For complex floating-point operands, the following special cases apply to real and imaginary components independently (e.g., if real(x_i) is NaN, the rounded real component is NaN).

If x_i is already integer-valued, the result is x_i.

For floating-point operands,

If x_i is +infinity, the result is +infinity.
If x_i is -infinity, the result is -infinity.
If x_i is +0, the result is +0.
If x_i is -0, the result is -0.
If x_i is NaN, the result is NaN.
If two integers are equally close to x_i, the result is the even integer closest to x_i.

Changed in version 2022.12: Added complex data type support.

sign: sign¶

Returns an indication of the sign of a number for each element x_i of the input array x.

The sign function (also known as the signum function) of a number $x_i$ is defined as

\[\begin{split}\operatorname{sign}(x_i) = \begin{cases} 0 & \textrm{if } x_i = 0 \\ \frac{x_i}{|x_i|} & \textrm{otherwise} \end{cases}\end{split}\]

where $|x_i|$ is the absolute value of $x_i$.

Parameters:: x (array) – input array. Should have a numeric data type.
Returns:: out – an array containing the evaluated result for each element in x. The returned array must have the same data type as x.
Return type:: array

Notes

Special cases

For real-valued operands,

If x_i is less than 0, the result is -1.
If x_i is either -0 or +0, the result is 0.
If x_i is greater than 0, the result is +1.
If x_i is NaN, the result is NaN.

For complex floating-point operands, let a = real(x_i), b = imag(x_i), and

If a is either -0 or +0 and b is either -0 or +0, the result is 0 + 0j.
If a is NaN or b is NaN, the result is NaN + NaN j.
In the remaining cases, special cases must be handled according to the rules of complex number division (see divide()).

Changed in version 2022.12: Added complex data type support.

sin: sin¶

Calculates an implementation-dependent approximation to the sine for each element x_i of the input array x.

Each element x_i is assumed to be expressed in radians.

Note

The sine is an entire function on the complex plane and has no branch cuts.

Note

For complex arguments, the mathematical definition of sine is

\[\begin{split}\begin{align} \operatorname{sin}(x) &= \frac{e^{jx} - e^{-jx}}{2j} \\ &= \frac{\operatorname{sinh}(jx)}{j} \\ &= \frac{\operatorname{sinh}(jx)}{j} \cdot \frac{j}{j} \\ &= -j \cdot \operatorname{sinh}(jx) \end{align}\end{split}\]

where $\operatorname{sinh}$ is the hyperbolic sine.

Parameters:: x (array) – input array whose elements are each expressed in radians. Should have a floating-point data type.
Returns:: out – an array containing the sine of each element in x. The returned array must have a floating-point data type determined by type-promotion.
Return type:: array

Notes

Special cases

For real-valued floating-point operands,

If x_i is NaN, the result is NaN.
If x_i is +0, the result is +0.
If x_i is -0, the result is -0.
If x_i is either +infinity or -infinity, the result is NaN.

For complex floating-point operands, special cases must be handled as if the operation is implemented as -1j * sinh(x*1j).

Changed in version 2022.12: Added complex data type support.

sinh: sinh¶

Calculates an implementation-dependent approximation to the hyperbolic sine for each element x_i of the input array x.

The mathematical definition of the hyperbolic sine is

\[\operatorname{sinh}(x) = \frac{e^x - e^{-x}}{2}\]

Note

The hyperbolic sine is an entire function in the complex plane and has no branch cuts. The function is periodic, with period $2\pi j$, with respect to the imaginary component.

Parameters:: x (array) – input array whose elements each represent a hyperbolic angle. Should have a floating-point data type.
Returns:: out – an array containing the hyperbolic sine of each element in x. The returned array must have a floating-point data type determined by type-promotion.
Return type:: array

Notes

Special cases

Note

For all operands, sinh(x) must equal -sinh(-x).

For real-valued floating-point operands,

If x_i is NaN, the result is NaN.
If x_i is +0, the result is +0.
If x_i is -0, the result is -0.
If x_i is +infinity, the result is +infinity.
If x_i is -infinity, the result is -infinity.

For complex floating-point operands, let a = real(x_i), b = imag(x_i), and

Note

For complex floating-point operands, sinh(conj(x)) must equal conj(sinh(x)).

If a is +0 and b is +0, the result is +0 + 0j.
If a is +0 and b is +infinity, the result is 0 + NaN j (sign of the real component is unspecified).
If a is +0 and b is NaN, the result is 0 + NaN j (sign of the real component is unspecified).
If a is a positive (i.e., greater than 0) finite number and b is +infinity, the result is NaN + NaN j.
If a is a positive (i.e., greater than 0) finite number and b is NaN, the result is NaN + NaN j.
If a is +infinity and b is +0, the result is +infinity + 0j.
If a is +infinity and b is a positive finite number, the result is +infinity * cis(b).
If a is +infinity and b is +infinity, the result is infinity + NaN j (sign of the real component is unspecified).
If a is +infinity and b is NaN, the result is infinity + NaN j (sign of the real component is unspecified).
If a is NaN and b is +0, the result is NaN + 0j.
If a is NaN and b is a nonzero finite number, the result is NaN + NaN j.
If a is NaN and b is NaN, the result is NaN + NaN j.

where cis(v) is cos(v) + sin(v)*1j.

Changed in version 2022.12: Added complex data type support.

sort: sort¶

Returns a sorted copy of an input array x.

Note

Parameters:

x (array) – input array. Should have a real-valued data type.
axis (int) – axis along which to sort. If set to -1, the function must sort along the last axis. Default: -1.
descending (bool) – sort order. If True, the array must be sorted in descending order (by value). If False, the array must be sorted in ascending order (by value). Default: False.
stable (bool) – sort stability. If True, the returned array must maintain the relative order of x values which compare as equal. If False, the returned array may or may not maintain the relative order of x values which compare as equal (i.e., the relative order of x values which compare as equal is implementation-dependent). Default: True.

Returns:

out – a sorted array. The returned array must have the same data type and shape as x.

Return type:

array

sqrt: sqrt¶

Calculates the principal square root for each element x_i of the input array x.

Note

After rounding, each result must be indistinguishable from the infinitely precise result (as required by IEEE 754).

Note

For complex floating-point operands, sqrt(conj(x)) must equal conj(sqrt(x)).

Note

By convention, the branch cut of the square root is the negative real axis $(-\infty, 0)$.

The square root is a continuous function from above the branch cut, taking into account the sign of the imaginary component.

Accordingly, for complex arguments, the function returns the square root in the range of the right half-plane, including the imaginary axis (i.e., the plane defined by $[0, +\infty)$ along the real axis and $(-\infty, +\infty)$ along the imaginary axis).

Note: branch cuts follow C99 and have provisional status (see branch-cuts).

Parameters:: x (array) – input array. Should have a floating-point data type.
Returns:: out – an array containing the square root of each element in x. The returned array must have a floating-point data type determined by type-promotion.
Return type:: array

Notes

Special cases

For real-valued floating-point operands,

If x_i is NaN, the result is NaN.
If x_i is less than 0, the result is NaN.
If x_i is +0, the result is +0.
If x_i is -0, the result is -0.
If x_i is +infinity, the result is +infinity.

For complex floating-point operands, let a = real(x_i), b = imag(x_i), and

If a is either +0 or -0 and b is +0, the result is +0 + 0j.
If a is any value (including NaN) and b is +infinity, the result is +infinity + infinity j.
If a is a finite number and b is NaN, the result is NaN + NaN j.
If a -infinity and b is a positive (i.e., greater than 0) finite number, the result is +0 + infinity j.
If a is +infinity and b is a positive (i.e., greater than 0) finite number, the result is +infinity + 0 j.
If a is -infinity and b is NaN, the result is NaN + infinity j (sign of the imaginary component is unspecified).
If a is +infinity and b is NaN, the result is +infinity + NaN j.
If a is NaN and b is any value, the result is NaN + NaN j.
If a is NaN and b is NaN, the result is NaN + NaN j.

Changed in version 2022.12: Added complex data type support.

square: square¶

Squares each element x_i of the input array x.

The square of a number x_i is defined as

\[x_i^2 = x_i \cdot x_i\]

Parameters:: x (array) – input array. Should have a numeric data type.
Returns:: out – an array containing the evaluated result for each element in x. The returned array must have a data type determined by type-promotion.
Return type:: array

Notes

Special cases

For floating-point operands, special cases must be handled as if the operation is implemented as x * x (see multiply()).

Changed in version 2022.12: Added complex data type support.

squeeze: squeeze¶

Removes singleton dimensions (axes) from x.

Parameters:

x (array) – input array.
axis (Union[int, Tuple[int, ...]]) – axis (or axes) to squeeze. If a specified axis has a size greater than one, a ValueError must be raised.

Returns:

out – an output array having the same data type and elements as x.

Return type:

array

stack: stack¶

Joins a sequence of arrays along a new axis.

Parameters:

arrays (Union[Tuple[array, ...], List[array]]) – input arrays to join. Each array must have the same shape.
axis (int) – axis along which the arrays will be joined. Providing an axis specifies the index of the new axis in the dimensions of the result. For example, if axis is 0, the new axis will be the first dimension and the output array will have shape (N, A, B, C); if axis is 1, the new axis will be the second dimension and the output array will have shape (A, N, B, C); and, if axis is -1, the new axis will be the last dimension and the output array will have shape (A, B, C, N). A valid axis must be on the interval [-N, N), where N is the rank (number of dimensions) of x. If provided an axis outside of the required interval, the function must raise an exception. Default: 0.

Returns:

out – an output array having rank N+1, where N is the rank (number of dimensions) of x. If the input arrays have different data types, normal type-promotion must apply. If the input arrays have the same data type, the output array must have the same data type as the input arrays.

Note

This specification leaves type promotion between data type families (i.e., intxx and floatxx) unspecified.

Return type:

array

std: std¶

Calculates the standard deviation of the input array x.

Parameters:

x (array) – input array. Should have a real-valued floating-point data type.
axis (Optional[Union[int, Tuple[int, ...]]]) – axis or axes along which standard deviations must be computed. By default, the standard deviation must be computed over the entire array. If a tuple of integers, standard deviations must be computed over multiple axes. Default: None.
correction (Union[int, float]) – degrees of freedom adjustment. Setting this parameter to a value other than 0 has the effect of adjusting the divisor during the calculation of the standard deviation according to N-c where N corresponds to the total number of elements over which the standard deviation is computed and c corresponds to the provided degrees of freedom adjustment. When computing the standard deviation of a population, setting this parameter to 0 is the standard choice (i.e., the provided array contains data constituting an entire population). When computing the corrected sample standard deviation, setting this parameter to 1 is the standard choice (i.e., the provided array contains data sampled from a larger population; this is commonly referred to as Bessel’s correction). Default: 0.
keepdims (bool) – if True, the reduced axes (dimensions) must be included in the result as singleton dimensions, and, accordingly, the result must be compatible with the input array (see broadcasting). Otherwise, if False, the reduced axes (dimensions) must not be included in the result. Default: False.

Returns:

out – if the standard deviation was computed over the entire array, a zero-dimensional array containing the standard deviation; otherwise, a non-zero-dimensional array containing the standard deviations. The returned array must have the same data type as x.

Note

Return type:

array

Notes

Special Cases

Let N equal the number of elements over which to compute the standard deviation.

If N - correction is less than or equal to 0, the standard deviation is NaN.
If x_i is NaN, the standard deviation is NaN (i.e., NaN values propagate).

subtract: subtract¶

Calculates the difference for each element x1_i of the input array x1 with the respective element x2_i of the input array x2.

The result of x1_i - x2_i must be the same as x1_i + (-x2_i) and must be governed by the same floating-point rules as addition (see add()).

Parameters:

x1 (array) – first input array. Should have a numeric data type.
x2 (array) – second input array. Must be compatible with x1 (see broadcasting). Should have a numeric data type.

Returns:

out – an array containing the element-wise differences. The returned array must have a data type determined by type-promotion.

Return type:

array

Notes

Changed in version 2022.12: Added complex data type support.

sum: sum¶

Calculates the sum of the input array x.

Parameters:

x (array) – input array. Should have a numeric data type.
axis (Optional[Union[int, Tuple[int, ...]]]) – axis or axes along which sums must be computed. By default, the sum must be computed over the entire array. If a tuple of integers, sums must be computed over multiple axes. Default: None.
dtype (Optional[dtype]) –
data type of the returned array. If None,
- if the default data type corresponding to the data type “kind” (integer, real-valued floating-point, or complex floating-point) of x has a smaller range of values than the data type of x (e.g., x has data type int64 and the default data type is int32, or x has data type uint64 and the default data type is int64), the returned array must have the same data type as x.
- if x has a real-valued floating-point data type, the returned array must have the default real-valued floating-point data type.
- if x has a complex floating-point data type, the returned array must have the default complex floating-point data type.
- if x has a signed integer data type (e.g., int16), the returned array must have the default integer data type.
- if x has an unsigned integer data type (e.g., uint16), the returned array must have an unsigned integer data type having the same number of bits as the default integer data type (e.g., if the default integer data type is int32, the returned array must have a uint32 data type).
If the data type (either specified or resolved) differs from the data type of x, the input array should be cast to the specified data type before computing the sum. Default: None.

Note

keyword argument is intended to help prevent data type overflows.
keepdims (bool) – if True, the reduced axes (dimensions) must be included in the result as singleton dimensions, and, accordingly, the result must be compatible with the input array (see broadcasting). Otherwise, if False, the reduced axes (dimensions) must not be included in the result. Default: False.

Returns:

out – if the sum was computed over the entire array, a zero-dimensional array containing the sum; otherwise, an array containing the sums. The returned array must have a data type as described by the dtype parameter above.

Return type:

array

Notes

Special Cases

Let N equal the number of elements over which to compute the sum.

If N is 0, the sum is 0 (i.e., the empty sum).

For both real-valued and complex floating-point operands, special cases must be handled as if the operation is implemented by successive application of add().

Changed in version 2022.12: Added complex data type support.

take: take¶

Returns elements of an array along an axis.

Note

Conceptually, take(x, indices, axis=3) is equivalent to x[:,:,:,indices,...]; however, explicit indexing via arrays of indices is not currently supported in this specification due to concerns regarding __setitem__ and array mutation semantics.

Parameters:

x (array) – input array.
indices (array) – array indices. The array must be one-dimensional and have an integer data type.
axis (Optional[int]) –
axis over which to select values. If axis is negative, the function must determine the axis along which to select values by counting from the last dimension.

If x is a one-dimensional array, providing an axis is optional; however, if x has more than one dimension, providing an axis is required.

Returns:

out – an array having the same data type as x. The output array must have the same rank (i.e., number of dimensions) as x and must have the same shape as x, except for the axis specified by axis whose size must equal the number of elements in indices.

Return type:

array

tan: tan¶

Calculates an implementation-dependent approximation to the tangent for each element x_i of the input array x.

Each element x_i is assumed to be expressed in radians.

Note

Tangent is an analytical function on the complex plane and has no branch cuts. The function is periodic, with period $\pi j$, with respect to the real component and has first order poles along the real line at coordinates $(\pi (\frac{1}{2} + n), 0)$. However, IEEE 754 binary floating-point representation cannot represent the value $\pi / 2$ exactly, and, thus, no argument value is possible for which a pole error occurs.

Note

For complex arguments, the mathematical definition of tangent is

\[\begin{split}\begin{align} \operatorname{tan}(x) &= \frac{j(e^{-jx} - e^{jx})}{e^{-jx} + e^{jx}} \\ &= (-1) \frac{j(e^{jx} - e^{-jx})}{e^{jx} + e^{-jx}} \\ &= -j \cdot \operatorname{tanh}(jx) \end{align}\end{split}\]

where $\operatorname{tanh}$ is the hyperbolic tangent.

Parameters:: x (array) – input array whose elements are expressed in radians. Should have a floating-point data type.
Returns:: out – an array containing the tangent of each element in x. The returned array must have a floating-point data type determined by type-promotion.
Return type:: array

Notes

Special cases

For real-valued floating-point operands,

If x_i is NaN, the result is NaN.
If x_i is +0, the result is +0.
If x_i is -0, the result is -0.
If x_i is either +infinity or -infinity, the result is NaN.

For complex floating-point operands, special cases must be handled as if the operation is implemented as -1j * tanh(x*1j).

Changed in version 2022.12: Added complex data type support.

tanh: tanh¶

Calculates an implementation-dependent approximation to the hyperbolic tangent for each element x_i of the input array x.

The mathematical definition of the hyperbolic tangent is

\[\begin{split}\begin{align} \operatorname{tanh}(x) &= \frac{\operatorname{sinh}(x)}{\operatorname{cosh}(x)} \\ &= \frac{e^x - e^{-x}}{e^x + e^{-x}} \end{align}\end{split}\]

where $\operatorname{sinh}(x)$ is the hyperbolic sine and $\operatorname{cosh}(x)$ is the hyperbolic cosine.

Note

The hyperbolic tangent is an analytical function on the complex plane and has no branch cuts. The function is periodic, with period $\pi j$, with respect to the imaginary component and has first order poles along the imaginary line at coordinates $(0, \pi (\frac{1}{2} + n))$. However, IEEE 754 binary floating-point representation cannot represent $\pi / 2$ exactly, and, thus, no argument value is possible such that a pole error occurs.

Parameters:: x (array) – input array whose elements each represent a hyperbolic angle. Should have a floating-point data type.
Returns:: out – an array containing the hyperbolic tangent of each element in x. The returned array must have a floating-point data type determined by type-promotion.
Return type:: array

Notes

Special cases

Note

For all operands, tanh(-x) must equal -tanh(x).

For real-valued floating-point operands,

If x_i is NaN, the result is NaN.
If x_i is +0, the result is +0.
If x_i is -0, the result is -0.
If x_i is +infinity, the result is +1.
If x_i is -infinity, the result is -1.

For complex floating-point operands, let a = real(x_i), b = imag(x_i), and

Note

For complex floating-point operands, tanh(conj(x)) must equal conj(tanh(x)).

If a is +0 and b is +0, the result is +0 + 0j.
If a is a nonzero finite number and b is +infinity, the result is NaN + NaN j.
If a is +0 and b is +infinity, the result is +0 + NaN j.
If a is a nonzero finite number and b is NaN, the result is NaN + NaN j.
If a is +0 and b is NaN, the result is +0 + NaN j.
If a is +infinity and b is a positive (i.e., greater than 0) finite number, the result is 1 + 0j.
If a is +infinity and b is +infinity, the result is 1 + 0j (sign of the imaginary component is unspecified).
If a is +infinity and b is NaN, the result is 1 + 0j (sign of the imaginary component is unspecified).
If a is NaN and b is +0, the result is NaN + 0j.
If a is NaN and b is a nonzero number, the result is NaN + NaN j.
If a is NaN and b is NaN, the result is NaN + NaN j.

Warning

For historical reasons stemming from the C standard, array libraries may not return the expected result when a is +0 and b is either +infinity or NaN. The result should be +0 + NaN j in both cases; however, for libraries compiled against older C versions, the result may be NaN + NaN j.

Array libraries are not required to patch these older C versions, and, thus, users are advised that results may vary across array library implementations for these special cases.

Changed in version 2022.12: Added complex data type support.

tensordot: tensordot¶

Returns a tensor contraction of x1 and x2 over specific axes.

Note

The tensordot function corresponds to the generalized matrix product.

Parameters:

x1 (array) – first input array. Should have a numeric data type.
x2 (array) –
second input array. Should have a numeric data type. Corresponding contracted axes of x1 and x2 must be equal.

Note

Contracted axes (dimensions) must not be broadcasted.
axes (Union[int, Tuple[Sequence[int], Sequence[int]]]) –
number of axes (dimensions) to contract or explicit sequences of axes (dimensions) for x1 and x2, respectively.

If axes is an int equal to N, then contraction must be performed over the last N axes of x1 and the first N axes of x2 in order. The size of each corresponding axis (dimension) must match. Must be nonnegative.
- If N equals 0, the result is the tensor (outer) product.
- If N equals 1, the result is the tensor dot product.
- If N equals 2, the result is the tensor double contraction (default).
If axes is a tuple of two sequences (x1_axes, x2_axes), the first sequence must apply to x1 and the second sequence to x2. Both sequences must have the same length. Each axis (dimension) x1_axes[i] for x1 must have the same size as the respective axis (dimension) x2_axes[i] for x2. Each sequence must consist of unique (nonnegative) integers that specify valid axes for each respective array.

Note

Returns:: out – an array containing the tensor contraction whose shape consists of the non-contracted axes (dimensions) of the first array x1, followed by the non-contracted axes (dimensions) of the second array x2. The returned array must have a data type determined by type-promotion.
Return type:: array

Notes

Changed in version 2022.12: Added complex data type support.

tril: tril¶

Returns the lower triangular part of a matrix (or a stack of matrices) x.

Note

The lower triangular part of the matrix is defined as the elements on and below the specified diagonal k.

Parameters:

x (array) – input array having shape (..., M, N) and whose innermost two dimensions form MxN matrices.
k (int) –
diagonal above which to zero elements. If k = 0, the diagonal is the main diagonal. If k < 0, the diagonal is below the main diagonal. If k > 0, the diagonal is above the main diagonal. Default: 0.

Note

The main diagonal is defined as the set of indices {(i, i)} for i on the interval [0, min(M, N) - 1].

Returns:

out – an array containing the lower triangular part(s). The returned array must have the same shape and data type as x. All elements above the specified diagonal k must be zeroed. The returned array should be allocated on the same device as x.

Return type:

array

triu: triu¶

Returns the upper triangular part of a matrix (or a stack of matrices) x.

Note

The upper triangular part of the matrix is defined as the elements on and above the specified diagonal k.

Parameters:

x (array) – input array having shape (..., M, N) and whose innermost two dimensions form MxN matrices.
k (int) –
diagonal below which to zero elements. If k = 0, the diagonal is the main diagonal. If k < 0, the diagonal is below the main diagonal. If k > 0, the diagonal is above the main diagonal. Default: 0.

Note

The main diagonal is defined as the set of indices {(i, i)} for i on the interval [0, min(M, N) - 1].

Returns:

out – an array containing the upper triangular part(s). The returned array must have the same shape and data type as x. All elements below the specified diagonal k must be zeroed. The returned array should be allocated on the same device as x.

Return type:

array

trunc: trunc¶

Rounds each element x_i of the input array x to the nearest integer-valued number that is closer to zero than x_i.

Parameters:: x (array) – input array. Should have a real-valued data type.
Returns:: out – an array containing the rounded result for each element in x. The returned array must have the same data type as x.
Return type:: array

Notes

Special cases

If x_i is already integer-valued, the result is x_i.

For floating-point operands,

If x_i is +infinity, the result is +infinity.
If x_i is -infinity, the result is -infinity.
If x_i is +0, the result is +0.
If x_i is -0, the result is -0.
If x_i is NaN, the result is NaN.

uint16: TDtype¶

uint32: TDtype¶

uint64: TDtype¶

uint8: TDtype¶

unique_all: unique_all¶

Returns the unique elements of an input array x, the first occurring indices for each unique element in x, the indices from the set of unique elements that reconstruct x, and the corresponding counts for each unique element in x.

Data-dependent output shape

The shapes of two of the output arrays for this function depend on the data values in the input array; hence, array libraries which build computation graphs (e.g., JAX, Dask, etc.) may find this function difficult to implement without knowing array values. Accordingly, such libraries may choose to omit this function. See data-dependent-output-shapes section for more details.

Note

Uniqueness should be determined based on value equality (see equal()). For input arrays having floating-point data types, value-based equality implies the following behavior.

As nan values compare as False, nan values should be considered distinct.
As complex floating-point values having at least one nan component compare as False, complex floating-point values having nan components should be considered distinct.
As -0 and +0 compare as True, signed zeros should not be considered distinct, and the corresponding unique element will be implementation-dependent (e.g., an implementation could choose to return -0 if -0 occurs before +0).

As signed zeros are not distinct, using inverse_indices to reconstruct the input array is not guaranteed to return an array having the exact same values.

Each nan value and each complex floating-point value having a nan component should have a count of one, while the counts for signed zeros should be aggregated as a single count.

Parameters:

x (array) – input array. If x has more than one dimension, the function must flatten x and return the unique elements of the flattened array.

Returns:

out – a namedtuple (values, indices, inverse_indices, counts) whose

first element must have the field name values and must be an array containing the unique elements of x. The array must have the same data type as x.
second element must have the field name indices and must be an array containing the indices (first occurrences) of x that result in values. The array must have the same shape as values and must have the default array index data type.
third element must have the field name inverse_indices and must be an array containing the indices of values that reconstruct x. The array must have the same shape as x and must have the default array index data type.
fourth element must have the field name counts and must be an array containing the number of times each unique element occurs in x. The returned array must have same shape as values and must have the default array index data type.

Note

The order of unique elements is not specified and may vary between implementations.

Return type:

Tuple[array, array, array, array]

Notes

Changed in version 2022.12: Added complex data type support.

unique_counts: unique_counts¶

Returns the unique elements of an input array x and the corresponding counts for each unique element in x.

Data-dependent output shape

Note

Uniqueness should be determined based on value equality (see equal()). For input arrays having floating-point data types, value-based equality implies the following behavior.

As nan values compare as False, nan values should be considered distinct.
As complex floating-point values having at least one nan component compare as False, complex floating-point values having nan components should be considered distinct.
As -0 and +0 compare as True, signed zeros should not be considered distinct, and the corresponding unique element will be implementation-dependent (e.g., an implementation could choose to return -0 if -0 occurs before +0).

Each nan value and each complex floating-point value having a nan component should have a count of one, while the counts for signed zeros should be aggregated as a single count.

Parameters:

x (array) – input array. If x has more than one dimension, the function must flatten x and return the unique elements of the flattened array.

Returns:

out – a namedtuple (values, counts) whose

first element must have the field name values and must be an array containing the unique elements of x. The array must have the same data type as x.
second element must have the field name counts and must be an array containing the number of times each unique element occurs in x. The returned array must have same shape as values and must have the default array index data type.

Note

The order of unique elements is not specified and may vary between implementations.

Return type:

Tuple[array, array]

Notes

Changed in version 2022.12: Added complex data type support.

unique_inverse: unique_inverse¶

Returns the unique elements of an input array x and the indices from the set of unique elements that reconstruct x.

Data-dependent output shape

Note

Uniqueness should be determined based on value equality (see equal()). For input arrays having floating-point data types, value-based equality implies the following behavior.

As nan values compare as False, nan values should be considered distinct.
As complex floating-point values having at least one nan component compare as False, complex floating-point values having nan components should be considered distinct.
As -0 and +0 compare as True, signed zeros should not be considered distinct, and the corresponding unique element will be implementation-dependent (e.g., an implementation could choose to return -0 if -0 occurs before +0).

As signed zeros are not distinct, using inverse_indices to reconstruct the input array is not guaranteed to return an array having the exact same values.

Parameters:

x (array) – input array. If x has more than one dimension, the function must flatten x and return the unique elements of the flattened array.

Returns:

out – a namedtuple (values, inverse_indices) whose

first element must have the field name values and must be an array containing the unique elements of x. The array must have the same data type as x.
second element must have the field name inverse_indices and must be an array containing the indices of values that reconstruct x. The array must have the same shape as x and have the default array index data type.

Note

The order of unique elements is not specified and may vary between implementations.

Return type:

Tuple[array, array]

Notes

Changed in version 2022.12: Added complex data type support.

unique_values: unique_values¶

Returns the unique elements of an input array x.

Data-dependent output shape

Note

Uniqueness should be determined based on value equality (see equal()). For input arrays having floating-point data types, value-based equality implies the following behavior.

As nan values compare as False, nan values should be considered distinct.
As complex floating-point values having at least one nan component compare as False, complex floating-point values having nan components should be considered distinct.
As -0 and +0 compare as True, signed zeros should not be considered distinct, and the corresponding unique element will be implementation-dependent (e.g., an implementation could choose to return -0 if -0 occurs before +0).

Parameters:

x (array) – input array. If x has more than one dimension, the function must flatten x and return the unique elements of the flattened array.

Returns:

out – an array containing the set of unique elements in x. The returned array must have the same data type as x.

Note

The order of unique elements is not specified and may vary between implementations.

Return type:

array

Notes

Changed in version 2022.12: Added complex data type support.

var: var¶

Calculates the variance of the input array x.

Parameters:

x (array) – input array. Should have a real-valued floating-point data type.
axis (Optional[Union[int, Tuple[int, ...]]]) – axis or axes along which variances must be computed. By default, the variance must be computed over the entire array. If a tuple of integers, variances must be computed over multiple axes. Default: None.
correction (Union[int, float]) – degrees of freedom adjustment. Setting this parameter to a value other than 0 has the effect of adjusting the divisor during the calculation of the variance according to N-c where N corresponds to the total number of elements over which the variance is computed and c corresponds to the provided degrees of freedom adjustment. When computing the variance of a population, setting this parameter to 0 is the standard choice (i.e., the provided array contains data constituting an entire population). When computing the unbiased sample variance, setting this parameter to 1 is the standard choice (i.e., the provided array contains data sampled from a larger population; this is commonly referred to as Bessel’s correction). Default: 0.
keepdims (bool) – if True, the reduced axes (dimensions) must be included in the result as singleton dimensions, and, accordingly, the result must be compatible with the input array (see broadcasting). Otherwise, if False, the reduced axes (dimensions) must not be included in the result. Default: False.

Returns:

out – if the variance was computed over the entire array, a zero-dimensional array containing the variance; otherwise, a non-zero-dimensional array containing the variances. The returned array must have the same data type as x.

Return type:

array

Note

Notes

Special Cases

Let N equal the number of elements over which to compute the variance.

If N - correction is less than or equal to 0, the variance is NaN.
If x_i is NaN, the variance is NaN (i.e., NaN values propagate).

vecdot: vecdot¶

Computes the (vector) dot product of two arrays.

Let $\mathbf{a}$ be a vector in x1 and $\mathbf{b}$ be a corresponding vector in x2. The dot product is defined as

\[\mathbf{a} \cdot \mathbf{b} = \sum_{i=0}^{n-1} \overline{a_i}b_i\]

over the dimension specified by axis and where $n$ is the dimension size and $\overline{a_i}$ denotes the complex conjugate if $a_i$ is complex and the identity if $a_i$ is real-valued.

Parameters:

x1 (array) – first input array. Should have a floating-point data type.
x2 (array) –
second input array. Must be compatible with x1 for all non-contracted axes (see broadcasting). The size of the axis over which to compute the dot product must be the same size as the respective axis in x1. Should have a floating-point data type.

Note

The contracted axis (dimension) must not be broadcasted.
axis (int) – axis over which to compute the dot product. Must be an integer on the interval [-N, N), where N is the rank (number of dimensions) of the shape determined according to broadcasting. If specified as a negative integer, the function must determine the axis along which to compute the dot product by counting backward from the last dimension (where -1 refers to the last dimension). By default, the function must compute the dot product over the last axis. Default: -1.

Returns:

out – if x1 and x2 are both one-dimensional arrays, a zero-dimensional containing the dot product; otherwise, a non-zero-dimensional array containing the dot products and having rank N-1, where N is the rank (number of dimensions) of the shape determined according to broadcasting along the non-contracted axes. The returned array must have a data type determined by type-promotion.

Return type:

array

Notes

Changed in version 2022.12: Added complex data type support.

Raises

if provided an invalid axis.
if the size of the axis over which to compute the dot product is not the same (before broadcasting) for both x1 and x2.

where: where¶

Returns elements chosen from x1 or x2 depending on condition.

Parameters:

condition (array) – when True, yield x1_i; otherwise, yield x2_i. Should have a boolean data type. Must be compatible with x1 and x2 (see broadcasting).
x1 (array) – first input array. Must be compatible with condition and x2 (see broadcasting).
x2 (array) – second input array. Must be compatible with condition and x1 (see broadcasting).

Returns:

out – an array with elements from x1 where condition is True, and elements from x2 elsewhere. The returned array must have a data type determined by type-promotion rules with the arrays x1 and x2.

Return type:

array

zeros: zeros¶

Returns a new array having a specified shape and filled with zeros.

Parameters:

shape (Union[int, Tuple[int, ...]]) – output array shape.
dtype (Optional[dtype]) – output array data type. If dtype is None, the output array data type must be the default real-valued floating-point data type. Default: None.
device (Optional[device]) – device on which to place the created array. Default: None.

Returns:

out – an array containing zeros.

Return type:

array

zeros_like: zeros_like¶

Returns a new array filled with zeros and having the same shape as an input array x.

Parameters:

x (array) – input array from which to derive the output array shape.
dtype (Optional[dtype]) – output array data type. If dtype is None, the output array data type must be inferred from x. Default: None.
device (Optional[device]) – device on which to place the created array. If device is None, the output array device must be inferred from x. Default: None.

Returns:

out – an array having the same shape as x and filled with zeros.

Return type:

array

class array_api._2022_12.ArrayNamespaceFull(*args, **kwargs)[source]¶

Bases: ArrayNamespace, Protocol, Generic

_abc_impl = <_abc._abc_data object>¶

_is_protocol = True¶

_is_runtime_protocol = True¶

fft: FftNamespace¶

linalg: LinalgNamespace¶

class array_api._2022_12.FftNamespace(*args, **kwargs)[source]¶

Bases: Protocol, Generic

_abc_impl = <_abc._abc_data object>¶

_is_protocol = True¶

_is_runtime_protocol = True¶

fft: fft¶

Computes the one-dimensional discrete Fourier transform.

Note

Applying the one-dimensional inverse discrete Fourier transform to the output of this function must return the original (i.e., non-transformed) input array within numerical accuracy (i.e., ifft(fft(x)) == x), provided that the transform and inverse transform are performed with the same arguments (number of elements, axis, and normalization mode).

Parameters:

x (array) – input array. Should have a complex floating-point data type.
n (Optional[int]) –
number of elements over which to compute the transform along the axis (dimension) specified by axis. Let M be the size of the input array along the axis specified by axis. When n is None, the function must set n equal to M.
- If n is greater than M, the axis specified by axis must be zero-padded to size n.
- If n is less than M, the axis specified by axis must be trimmed to size n.
- If n equals M, all elements along the axis specified by axis must be used when computing the transform.
Default: None.
axis (int) – axis (dimension) of the input array over which to compute the transform. A valid axis must be an integer on the interval [-N, N), where N is the rank (number of dimensions) of x. If an axis is specified as a negative integer, the function must determine the axis along which to compute the transform by counting backward from the last dimension (where -1 refers to the last dimension). Default: -1.
norm (Literal['backward', 'ortho', 'forward']) –
normalization mode. Should be one of the following modes:
- 'backward': no normalization.
- 'ortho': normalize by 1/sqrt(n) (i.e., make the FFT orthonormal).
- 'forward': normalize by 1/n.
Default: 'backward'.

Returns:

out – an array transformed along the axis (dimension) specified by axis. The returned array must have the same data type as x and must have the same shape as x, except for the axis specified by axis which must have size n.

Return type:

array

Notes

Added in version 2022.12.

fftfreq: fftfreq¶

Computes the discrete Fourier transform sample frequencies.

For a Fourier transform of length n and length unit of d, the frequencies are described as:

f = [0, 1, ..., n/2-1, -n/2, ..., -1] / (d*n)        # if n is even
f = [0, 1, ..., (n-1)/2, -(n-1)/2, ..., -1] / (d*n)  # if n is odd

Parameters:

n (int) – window length.
d (float) – sample spacing between individual samples of the Fourier transform input. Default: 1.0.
device (Optional[device]) – device on which to place the created array. Default: None.

Returns:

out – an array of shape (n,) containing the sample frequencies. The returned array must have the default real-valued floating-point data type.

Return type:

array

Notes

Added in version 2022.12.

fftn: fftn¶

Computes the n-dimensional discrete Fourier transform.

Note

Applying the n-dimensional inverse discrete Fourier transform to the output of this function must return the original (i.e., non-transformed) input array within numerical accuracy (i.e., ifftn(fftn(x)) == x), provided that the transform and inverse transform are performed with the same arguments (sizes, axes, and normalization mode).

Parameters:

x (array) – input array. Should have a complex floating-point data type.
s (Optional[Sequence[int]]) –
number of elements over which to compute the transform along the axes (dimensions) specified by axes. Let i be the index of the n-th axis specified by axes (i.e., i = axes[n]) and M[i] be the size of the input array along axis i. When s is None, the function must set s equal to a sequence of integers such that s[i] equals M[i] for all i.
- If s[i] is greater than M[i], axis i must be zero-padded to size s[i].
- If s[i] is less than M[i], axis i must be trimmed to size s[i].
- If s[i] equals M[i] or -1, all elements along axis i must be used when computing the transform.
If s is not None, axes must not be None. Default: None.
axes (Optional[Sequence[int]]) –
axes (dimensions) over which to compute the transform. A valid axis in axes must be an integer on the interval [-N, N), where N is the rank (number of dimensions) of x. If an axis is specified as a negative integer, the function must determine the axis along which to compute the transform by counting backward from the last dimension (where -1 refers to the last dimension).

If s is provided, the corresponding axes to be transformed must also be provided. If axes is None, the function must compute the transform over all axes. Default: None.

If axes contains two or more entries which resolve to the same axis (i.e., resolved axes are not unique), the behavior is unspecified and thus implementation-defined.
norm (Literal['backward', 'ortho', 'forward']) –
normalization mode. Should be one of the following modes:
- 'backward': no normalization.
- 'ortho': normalize by 1/sqrt(n) (i.e., make the FFT orthonormal).
- 'forward': normalize by 1/n.
where n = prod(s) is the logical FFT size.

Default: 'backward'.

Returns:

out – an array transformed along the axes (dimensions) specified by axes. The returned array must have the same data type as x and must have the same shape as x, except for the axes specified by axes which must have size s[i].

Return type:

array

Notes

Added in version 2022.12.

fftshift: fftshift¶

Shifts the zero-frequency component to the center of the spectrum.

This function swaps half-spaces for all axes (dimensions) specified by axes.

Note

out[0] is the Nyquist component only if the length of the input is even.

Parameters:

x (array) – input array. Should have a floating-point data type.
axes (Optional[Union[int, Sequence[int]]]) –
axes over which to shift. If None, the function must shift all axes. Default: None.

If axes contains two or more entries which resolve to the same axis (i.e., resolved axes are not unique), the behavior is unspecified and thus implementation-defined.

Returns:

out – the shifted array. The returned array must have the same data type and shape as x.

Return type:

array

Notes

Added in version 2022.12.

hfft: hfft¶

Computes the one-dimensional discrete Fourier transform of a signal with Hermitian symmetry.

Parameters:

x (array) – input array. Should have a complex floating-point data type.
n (Optional[int]) –
number of elements along the transformed axis (dimension) specified by axis in the output array. Let M be the size of the input array along the axis specified by axis. When n is None, the function must set n equal to 2*(M-1).
- If n//2+1 is greater than M, the axis of the input array specified by axis must be zero-padded to length n//2+1.
- If n//2+1 is less than M, the axis of the input array specified by axis must be trimmed to size n//2+1.
- If n//2+1 equals M, all elements along the axis of the input array specified by axis must be used when computing the transform.
Default: None.
axis (int) – axis (dimension) of the input array over which to compute the transform. A valid axis must be an integer on the interval [-N, N), where N is the rank (number of dimensions) of x. If an axis is specified as a negative integer, the function must determine the axis along which to compute the transform by counting backward from the last dimension (where -1 refers to the last dimension). Default: -1.
norm (Literal['backward', 'ortho', 'forward']) –
normalization mode. Should be one of the following modes:
- 'backward': no normalization.
- 'ortho': normalize by 1/sqrt(n) (i.e., make the FFT orthonormal).
- 'forward': normalize by 1/n.
Default: 'backward'.

Returns:

out – an array transformed along the axis (dimension) specified by axis. The returned array must have a real-valued floating-point data type whose precision matches the precision of x (e.g., if x is complex128, then the returned array must have a float64 data type). The returned array must have the same shape as x, except for the axis specified by axis which must have size n.

Return type:

array

Notes

Added in version 2022.12.

ifft: ifft¶

Computes the one-dimensional inverse discrete Fourier transform.

Note

Parameters:

x (array) – input array. Should have a complex floating-point data type.
n (Optional[int]) –
number of elements over which to compute the transform along the axis (dimension) specified by axis. Let M be the size of the input array along the axis specified by axis. When n is None, the function must set n equal to M.
- If n is greater than M, the axis specified by axis must be zero-padded to size n.
- If n is less than M, the axis specified by axis must be trimmed to size n.
- If n equals M, all elements along the axis specified by axis must be used when computing the transform.
Default: None.
axis (int) – axis (dimension) of the input array over which to compute the transform. A valid axis must be an integer on the interval [-N, N), where N is the rank (number of dimensions) of x. If an axis is specified as a negative integer, the function must determine the axis along which to compute the transform by counting backward from the last dimension (where -1 refers to the last dimension). Default: -1.
norm (Literal['backward', 'ortho', 'forward']) –
normalization mode. Should be one of the following modes:
- 'backward': normalize by 1/n.
- 'ortho': normalize by 1/sqrt(n) (i.e., make the FFT orthonormal).
- 'forward': no normalization.
Default: 'backward'.

Returns:

Return type:

array

Notes

Added in version 2022.12.

ifftn: ifftn¶

Computes the n-dimensional inverse discrete Fourier transform.

Note

Parameters:

x (array) – input array. Should have a complex floating-point data type.
s (Optional[Sequence[int]]) –
number of elements over which to compute the transform along the axes (dimensions) specified by axes. Let i be the index of the n-th axis specified by axes (i.e., i = axes[n]) and M[i] be the size of the input array along axis i. When s is None, the function must set s equal to a sequence of integers such that s[i] equals M[i] for all i.
- If s[i] is greater than M[i], axis i must be zero-padded to size s[i].
- If s[i] is less than M[i], axis i must be trimmed to size s[i].
- If s[i] equals M[i] or -1, all elements along axis i must be used when computing the transform.
If s is not None, axes must not be None. Default: None.
axes (Optional[Sequence[int]]) –
axes (dimensions) over which to compute the transform. A valid axis in axes must be an integer on the interval [-N, N), where N is the rank (number of dimensions) of x. If an axis is specified as a negative integer, the function must determine the axis along which to compute the transform by counting backward from the last dimension (where -1 refers to the last dimension).

If s is provided, the corresponding axes to be transformed must also be provided. If axes is None, the function must compute the transform over all axes. Default: None.

If axes contains two or more entries which resolve to the same axis (i.e., resolved axes are not unique), the behavior is unspecified and thus implementation-defined.
norm (Literal['backward', 'ortho', 'forward']) –
specify the normalization mode. Should be one of the following modes:
- 'backward': normalize by 1/n.
- 'ortho': normalize by 1/sqrt(n) (i.e., make the FFT orthonormal).
- 'forward': no normalization.
where n = prod(s) is the logical FFT size.

Default: 'backward'.

Returns:

Return type:

array

Notes

Added in version 2022.12.

ifftshift: ifftshift¶

Inverse of fftshift.

Note

Although identical for even-length x, fftshift and ifftshift differ by one sample for odd-length x.

Parameters:

x (array) – input array. Should have a floating-point data type.
axes (Optional[Union[int, Sequence[int]]]) –
axes over which to perform the inverse shift. If None, the function must shift all axes. Default: None.

If axes contains two or more entries which resolve to the same axis (i.e., resolved axes are not unique), the behavior is unspecified and thus implementation-defined.

Returns:

out – the shifted array. The returned array must have the same data type and shape as x.

Return type:

array

Notes

Added in version 2022.12.

ihfft: ihfft¶

Computes the one-dimensional inverse discrete Fourier transform of a signal with Hermitian symmetry.

Parameters:

x (array) – input array. Must have a real-valued floating-point data type.
n (Optional[int]) –
number of elements over which to compute the transform along the axis (dimension) specified by axis. Let M be the size of the input array along the axis specified by axis. When n is None, the function must set n equal to M.
- If n is greater than M, the axis specified by axis must be zero-padded to size n.
- If n is less than M, the axis specified by axis must be trimmed to size n.
- If n equals M, all elements along the axis specified by axis must be used when computing the transform.
Default: None.
axis (int) – axis (dimension) of the input array over which to compute the transform. A valid axis must be an integer on the interval [-N, N), where N is the rank (number of dimensions) of x. If an axis is specified as a negative integer, the function must determine the axis along which to compute the transform by counting backward from the last dimension (where -1 refers to the last dimension). Default: -1.
norm (Literal['backward', 'ortho', 'forward']) –
normalization mode. Should be one of the following modes:
- 'backward': normalize by 1/n.
- 'ortho': normalize by 1/sqrt(n) (i.e., make the FFT orthonormal).
- 'forward': no normalization.
Default: 'backward'.

Returns:

out – an array transformed along the axis (dimension) specified by axis. The returned array must have a complex floating-point data type whose precision matches the precision of x (e.g., if x is float64, then the returned array must have a complex128 data type). The returned array must have the same shape as x, except for the axis specified by axis which must have size n//2 + 1.

Return type:

array

Notes

Added in version 2022.12.

irfft: irfft¶

Computes the one-dimensional inverse of rfft for complex-valued input.

Note

Applying the one-dimensional inverse discrete Fourier transform for real-valued input to the output of this function must return the original (i.e., non-transformed) input array within numerical accuracy (i.e., irfft(rfft(x)) == x), provided that the transform and inverse transform are performed with the same arguments (axis and normalization mode) and consistent values for the number of elements over which to compute the transforms.

Parameters:

x (array) – input array. Should have a complex floating-point data type.
n (Optional[int]) –
number of elements along the transformed axis (dimension) specified by axis in the output array. Let M be the size of the input array along the axis specified by axis. When n is None, the function must set n equal to 2*(M-1).
- If n//2+1 is greater than M, the axis of the input array specified by axis must be zero-padded to size n//2+1.
- If n//2+1 is less than M, the axis of the input array specified by axis must be trimmed to size n//2+1.
- If n//2+1 equals M, all elements along the axis of the input array specified by axis must be used when computing the transform.
Default: None.
axis (int) – axis (dimension) of the input array over which to compute the transform. A valid axis must be an integer on the interval [-N, N), where N is the rank (number of dimensions) of x. If an axis is specified as a negative integer, the function must determine the axis along which to compute the transform by counting backward from the last dimension (where -1 refers to the last dimension). Default: -1.
norm (Literal['backward', 'ortho', 'forward']) –
normalization mode. Should be one of the following modes:
- 'backward': normalize by 1/n.
- 'ortho': normalize by 1/sqrt(n) (i.e., make the FFT orthonormal).
- 'forward': no normalization.
Default: 'backward'.

Returns:

Return type:

array

Notes

In order to return an array having an odd number of elements along the transformed axis, the function must be provided an odd integer for n.

Added in version 2022.12.

irfftn: irfftn¶

Computes the n-dimensional inverse of rfftn for complex-valued input.

Note

Applying the n-dimensional inverse discrete Fourier transform for real-valued input to the output of this function must return the original (i.e., non-transformed) input array within numerical accuracy (i.e., irfftn(rfftn(x)) == x), provided that the transform and inverse transform are performed with the same arguments (axes and normalization mode) and consistent sizes.

Parameters:

x (array) – input array. Should have a complex floating-point data type.
s (Optional[Sequence[int]]) –
number of elements along the transformed axes (dimensions) specified by axes in the output array. Let i be the index of the n-th axis specified by axes (i.e., i = axes[n]) and M[i] be the size of the input array along axis i. When s is None, the function must set s equal to a sequence of integers such that s[i] equals M[i] for all i, except for the last transformed axis in which s[i] equals 2*(M[i]-1). For each i, let n equal s[i], except for the last transformed axis in which n equals s[i]//2+1.
- If n is greater than M[i], axis i of the input array must be zero-padded to size n.
- If n is less than M[i], axis i of the input array must be trimmed to size n.
- If n equals M[i] or -1, all elements along axis i of the input array must be used when computing the transform.
If s is not None, axes must not be None. Default: None.
axes (Optional[Sequence[int]]) –
axes (dimensions) over which to compute the transform. A valid axis in axes must be an integer on the interval [-N, N), where N is the rank (number of dimensions) of x. If an axis is specified as a negative integer, the function must determine the axis along which to compute the transform by counting backward from the last dimension (where -1 refers to the last dimension).

If s is provided, the corresponding axes to be transformed must also be provided. If axes is None, the function must compute the transform over all axes. Default: None.

If axes contains two or more entries which resolve to the same axis (i.e., resolved axes are not unique), the behavior is unspecified and thus implementation-defined.
norm (Literal['backward', 'ortho', 'forward']) –
normalization mode. Should be one of the following modes:
- 'backward': normalize by 1/n.
- 'ortho': normalize by 1/sqrt(n) (i.e., make the FFT orthonormal).
- 'forward': no normalization.
where n = prod(s) is the logical FFT size.

Default: 'backward'.

Returns:

out – an array transformed along the axes (dimension) specified by axes. The returned array must have a real-valued floating-point data type whose precision matches the precision of x (e.g., if x is complex128, then the returned array must have a float64 data type). The returned array must have the same shape as x, except for the transformed axes which must have size s[i].

Return type:

array

Notes

In order to return an array having an odd number of elements along the last transformed axis, the function must be provided an odd integer for s[-1].

Added in version 2022.12.

rfft: rfft¶

Computes the one-dimensional discrete Fourier transform for real-valued input.

Note

Parameters:

x (array) – input array. Must have a real-valued floating-point data type.
n (Optional[int]) –
number of elements over which to compute the transform along the axis (dimension) specified by axis. Let M be the size of the input array along the axis specified by axis. When n is None, the function must set n equal to M.
- If n is greater than M, the axis specified by axis must be zero-padded to size n.
- If n is less than M, the axis specified by axis must be trimmed to size n.
- If n equals M, all elements along the axis specified by axis must be used when computing the transform.
Default: None.
axis (int) – axis (dimension) of the input array over which to compute the transform. A valid axis must be an integer on the interval [-N, N), where N is the rank (number of dimensions) of x. If an axis is specified as a negative integer, the function must determine the axis along which to compute the transform by counting backward from the last dimension (where -1 refers to the last dimension). Default: -1.
norm (Literal['backward', 'ortho', 'forward']) –
normalization mode. Should be one of the following modes:
- 'backward': no normalization.
- 'ortho': normalize by 1/sqrt(n) (i.e., make the FFT orthonormal).
- 'forward': normalize by 1/n.
Default: 'backward'.

Returns:

Return type:

array

Notes

Added in version 2022.12.

rfftfreq: rfftfreq¶

Computes the discrete Fourier transform sample frequencies (for rfft and irfft).

For a Fourier transform of length n and length unit of d, the frequencies are described as:

f = [0, 1, ...,     n/2-1,     n/2] / (d*n)  # if n is even
f = [0, 1, ..., (n-1)/2-1, (n-1)/2] / (d*n)  # if n is odd

The Nyquist frequency component is considered to be positive.

Parameters:

n (int) – window length.
d (float) – sample spacing between individual samples of the Fourier transform input. Default: 1.0.
device (Optional[device]) – device on which to place the created array. Default: None.

Returns:

out – an array of shape (n//2+1,) containing the sample frequencies. The returned array must have the default real-valued floating-point data type.

Return type:

array

Notes

Added in version 2022.12.

rfftn: rfftn¶

Computes the n-dimensional discrete Fourier transform for real-valued input.

Note

Parameters:

x (array) – input array. Must have a real-valued floating-point data type.
s (Optional[Sequence[int]]) –
number of elements over which to compute the transform along axes (dimensions) specified by axes. Let i be the index of the n-th axis specified by axes (i.e., i = axes[n]) and M[i] be the size of the input array along axis i. When s is None, the function must set s equal to a sequence of integers such that s[i] equals M[i] for all i.
- If s[i] is greater than M[i], axis i must be zero-padded to size s[i].
- If s[i] is less than M[i], axis i must be trimmed to size s[i].
- If s[i] equals M[i] or -1, all elements along axis i must be used when computing the transform.
If s is not None, axes must not be None. Default: None.
axes (Optional[Sequence[int]]) –
axes (dimensions) over which to compute the transform. A valid axis in axes must be an integer on the interval [-N, N), where N is the rank (number of dimensions) of x. If an axis is specified as a negative integer, the function must determine the axis along which to compute the transform by counting backward from the last dimension (where -1 refers to the last dimension).

If s is provided, the corresponding axes to be transformed must also be provided. If axes is None, the function must compute the transform over all axes. Default: None.

If axes contains two or more entries which resolve to the same axis (i.e., resolved axes are not unique), the behavior is unspecified and thus implementation-defined.
norm (Literal['backward', 'ortho', 'forward']) –
normalization mode. Should be one of the following modes:
- 'backward': no normalization.
- 'ortho': normalize by 1/sqrt(n) (i.e., make the FFT orthonormal).
- 'forward': normalize by 1/n.
where n = prod(s), the logical FFT size.

Default: 'backward'.

Returns:

out – an array transformed along the axes (dimension) specified by axes. The returned array must have a complex floating-point data type whose precision matches the precision of x (e.g., if x is float64, then the returned array must have a complex128 data type). The returned array must have the same shape as x, except for the last transformed axis which must have size s[-1]//2 + 1 and the remaining transformed axes which must have size s[i].

Return type:

array

Notes

Added in version 2022.12.

class array_api._2022_12.LinalgNamespace(*args, **kwargs)[source]¶

Bases: Protocol, Generic

_abc_impl = <_abc._abc_data object>¶

_is_protocol = True¶

_is_runtime_protocol = True¶

cholesky: cholesky¶

Returns the lower (upper) Cholesky decomposition of a complex Hermitian or real symmetric positive-definite matrix x.

If x is real-valued, let $\mathbb{K}$ be the set of real numbers $\mathbb{R}$, and, if x is complex-valued, let $\mathbb{K}$ be the set of complex numbers $\mathbb{C}$.

The lower Cholesky decomposition of a complex Hermitian or real symmetric positive-definite matrix $x \in\ \mathbb{K}^{n \times n}$ is defined as

\[x = LL^{H} \qquad \text{L $\in\ \mathbb{K}^{n \times n}$}\]

where $L$ is a lower triangular matrix and $L^{H}$ is the conjugate transpose when $L$ is complex-valued and the transpose when $L$ is real-valued.

The upper Cholesky decomposition is defined similarly

\[x = U^{H}U \qquad \text{U $\in\ \mathbb{K}^{n \times n}$}\]

where $U$ is an upper triangular matrix.

When x is a stack of matrices, the function must compute the Cholesky decomposition for each matrix in the stack.

Note

Whether an array library explicitly checks whether an input array is Hermitian or a symmetric positive-definite matrix (or a stack of matrices) is implementation-defined.

Parameters:

x (array) – input array having shape (..., M, M) and whose innermost two dimensions form square complex Hermitian or real symmetric positive-definite matrices. Should have a floating-point data type.
upper (bool) – If True, the result must be the upper-triangular Cholesky factor $U$. If False, the result must be the lower-triangular Cholesky factor $L$. Default: False.

Returns:

out – an array containing the Cholesky factors for each square matrix. If upper is False, the returned array must contain lower-triangular matrices; otherwise, the returned array must contain upper-triangular matrices. The returned array must have a floating-point data type determined by type-promotion and must have the same shape as x.

Return type:

array

Notes

Changed in version 2022.12: Added complex data type support.

cross: cross¶

Returns the cross product of 3-element vectors.

If x1 and/or x2 are multi-dimensional arrays (i.e., the broadcasted result has a rank greater than 1), then the cross-product of each pair of corresponding 3-element vectors is independently computed.

Parameters:

x1 (array) – first input array. Must have a numeric data type.
x2 (array) –
second input array. Must be compatible with x1 for all non-compute axes (see broadcasting). The size of the axis over which to compute the cross product must be the same size as the respective axis in x1. Must have a numeric data type.

Note

The compute axis (dimension) must not be broadcasted.
axis (int) – the axis (dimension) of x1 and x2 containing the vectors for which to compute the cross product. Must be an integer on the interval [-N, N), where N is the rank (number of dimensions) of the shape determined according to broadcasting. If specified as a negative integer, the function must determine the axis along which to compute the cross product by counting backward from the last dimension (where -1 refers to the last dimension). By default, the function must compute the cross product over the last axis. Default: -1.

Returns:

out – an array containing the cross products. The returned array must have a data type determined by type-promotion.

Return type:

array

Notes

Changed in version 2022.12: Added support for broadcasting.

Changed in version 2022.12: Added complex data type support.

Raises

if provided an invalid axis.
if the size of the axis over which to compute the cross product is not equal to 3.
if the size of the axis over which to compute the cross product is not the same (before broadcasting) for both x1 and x2.

det: det¶

Returns the determinant of a square matrix (or a stack of square matrices) x.

Parameters:: x (array) – input array having shape (..., M, M) and whose innermost two dimensions form square matrices. Should have a floating-point data type.
Returns:: out – if x is a two-dimensional array, a zero-dimensional array containing the determinant; otherwise, a non-zero dimensional array containing the determinant for each square matrix. The returned array must have the same data type as x.
Return type:: array

Notes

Changed in version 2022.12: Added complex data type support.

diagonal: diagonal¶

Returns the specified diagonals of a matrix (or a stack of matrices) x.

Parameters:

x (array) – input array having shape (..., M, N) and whose innermost two dimensions form MxN matrices.
offset (int) –
offset specifying the off-diagonal relative to the main diagonal.
- offset = 0: the main diagonal.
- offset > 0: off-diagonal above the main diagonal.
- offset < 0: off-diagonal below the main diagonal.
Default: 0.

Returns:

out – an array containing the diagonals and whose shape is determined by removing the last two dimensions and appending a dimension equal to the size of the resulting diagonals. The returned array must have the same data type as x.

Return type:

array

eigh: eigh¶

Returns an eigenvalue decomposition of a complex Hermitian or real symmetric matrix (or a stack of matrices) x.

If x is real-valued, let $\mathbb{K}$ be the set of real numbers $\mathbb{R}$, and, if x is complex-valued, let $\mathbb{K}$ be the set of complex numbers $\mathbb{C}$.

The eigenvalue decomposition of a complex Hermitian or real symmetric matrix $x \in\ \mathbb{K}^{n \times n}$ is defined as

\[x = Q \Lambda Q^H\]

with $Q \in \mathbb{K}^{n \times n}$ and $\Lambda \in \mathbb{R}^n$ and where $Q^H$ is the conjugate transpose when $Q$ is complex and the transpose when $Q$ is real-valued and $\Lambda$ is a diagonal matrix whose diagonal elements are the corresponding eigenvalues. When x is real-valued, $Q$ is orthogonal, and, when x is complex, $Q$ is unitary.

Note

The eigenvalues of a complex Hermitian or real symmetric matrix are always real.

Warning

The eigenvectors of a symmetric matrix are not unique and are not continuous with respect to x. Because eigenvectors are not unique, different hardware and software may compute different eigenvectors.

Non-uniqueness stems from the fact that multiplying an eigenvector by $-1$ when x is real-valued and by $e^{\phi j}$ ($\phi \in \mathbb{R}$) when x is complex produces another set of valid eigenvectors.

Note

Whether an array library explicitly checks whether an input array is Hermitian or a symmetric matrix (or a stack of matrices) is implementation-defined.

Note

The function eig will be added in a future version of the specification.

Parameters:

x (array) – input array having shape (..., M, M) and whose innermost two dimensions form square matrices. Should have a floating-point data type.

Returns:

out – a namedtuple (eigenvalues, eigenvectors) whose

first element must have the field name eigenvalues (corresponding to $\operatorname{diag}\Lambda$ above) and must be an array consisting of computed eigenvalues. The array containing the eigenvalues must have shape (..., M) and must have a real-valued floating-point data type whose precision matches the precision of x (e.g., if x is complex128, then eigenvalues must be float64).
second element have have the field name eigenvectors (corresponding to $Q$ above) and must be an array where the columns of the inner most matrices contain the computed eigenvectors. These matrices must be orthogonal. The array containing the eigenvectors must have shape (..., M, M) and must have the same data type as x.

Return type:

Tuple[array, array]

Notes

Note

Eigenvalue sort order is left unspecified and is thus implementation-dependent.

Changed in version 2022.12: Added complex data type support.

eigvalsh: eigvalsh¶

Returns the eigenvalues of a complex Hermitian or real symmetric matrix (or a stack of matrices) x.

If x is real-valued, let $\mathbb{K}$ be the set of real numbers $\mathbb{R}$, and, if x is complex-valued, let $\mathbb{K}$ be the set of complex numbers $\mathbb{C}$.

The eigenvalues of a complex Hermitian or real symmetric matrix $x \in\ \mathbb{K}^{n \times n}$ are defined as the roots (counted with multiplicity) of the polynomial $p$ of degree $n$ given by

\[p(\lambda) = \operatorname{det}(x - \lambda I_n)\]

where $\lambda \in \mathbb{R}$ and where $I_n$ is the n-dimensional identity matrix.

Note

Whether an array library explicitly checks whether an input array is Hermitian or a symmetric matrix (or a stack of matrices) is implementation-defined.

Note

The function eigvals will be added in a future version of the specification.

Parameters:: x (array) – input array having shape (..., M, M) and whose innermost two dimensions form square matrices. Should have a floating-point data type.
Returns:: out – an array containing the computed eigenvalues. The returned array must have shape (..., M) and have a real-valued floating-point data type whose precision matches the precision of x (e.g., if x is complex128, then must have a float64 data type).
Return type:: array

Notes

Note

Eigenvalue sort order is left unspecified and is thus implementation-dependent.

Changed in version 2022.12: Added complex data type support.

inv: inv¶

Returns the multiplicative inverse of a square matrix (or a stack of square matrices) x.

If x is real-valued, let $\mathbb{K}$ be the set of real numbers $\mathbb{R}$, and, if x is complex-valued, let $\mathbb{K}$ be the set of complex numbers $\mathbb{C}$.

The inverse matrix $x^{-1} \in\ \mathbb{K}^{n \times n}$ of a square matrix $x \in\ \mathbb{K}^{n \times n}$ is defined as

\[x^{-1}x = xx^{-1} = I_n\]

where $I_n$ is the n-dimensional identity matrix.

The inverse matrix exists if and only if x is invertible. When x is invertible, the inverse is unique.

When x is a stack of matrices, the function must compute the inverse for each matrix in the stack.

Parameters:: x (array) – input array having shape (..., M, M) and whose innermost two dimensions form square matrices. Should have a floating-point data type.
Returns:: out – an array containing the multiplicative inverses. The returned array must have a floating-point data type determined by type-promotion and must have the same shape as x.
Return type:: array

Notes

Changed in version 2022.12: Added complex data type support.

matmul: matmul¶: Alias for matmul().

matrix_norm: matrix_norm¶

Computes the matrix norm of a matrix (or a stack of matrices) x.

Parameters:

x (array) – input array having shape (..., M, N) and whose innermost two dimensions form MxN matrices. Should have a floating-point data type.
keepdims (bool) – If True, the last two axes (dimensions) must be included in the result as singleton dimensions, and, accordingly, the result must be compatible with the input array (see broadcasting). Otherwise, if False, the last two axes (dimensions) must not be included in the result. Default: False.
ord (Optional[Union[int, float, Literal[inf, -inf, 'fro', 'nuc']]]) –
order of the norm. The following mathematical norms must be supported:

ord

description

’fro’

Frobenius norm

’nuc’

nuclear norm

1

max(sum(abs(x), axis=0))

2

largest singular value

inf

max(sum(abs(x), axis=1))

The following non-mathematical “norms” must be supported:

ord

description

-1

min(sum(abs(x), axis=0))

-2

smallest singular value

-inf

min(sum(abs(x), axis=1))

If ord=1, the norm corresponds to the induced matrix norm where p=1 (i.e., the maximum absolute value column sum).

If ord=2, the norm corresponds to the induced matrix norm where p=inf (i.e., the maximum absolute value row sum).

If ord=inf, the norm corresponds to the induced matrix norm where p=2 (i.e., the largest singular value).

Default: 'fro'.

Returns:

ord	description
’fro’	Frobenius norm
’nuc’	nuclear norm
1	max(sum(abs(x), axis=0))
2	largest singular value
inf	max(sum(abs(x), axis=1))

ord	description
-1	min(sum(abs(x), axis=0))
-2	smallest singular value
-inf	min(sum(abs(x), axis=1))

out – an array containing the norms for each MxN matrix. If keepdims is False, the returned array must have a rank which is two less than the rank of x. If x has a real-valued data type, the returned array must have a real-valued floating-point data type determined by type-promotion. If x has a complex-valued data type, the returned array must have a real-valued floating-point data type whose precision matches the precision of x (e.g., if x is complex128, then the returned array must have a float64 data type).

Return type:

array

Notes

Changed in version 2022.12: Added complex data type support.

matrix_power: matrix_power¶

Raises a square matrix (or a stack of square matrices) x to an integer power n.

Parameters:

x (array) – input array having shape (..., M, M) and whose innermost two dimensions form square matrices. Should have a floating-point data type.
n (int) – integer exponent.

Returns:

out – if n is equal to zero, an array containing the identity matrix for each square matrix. If n is less than zero, an array containing the inverse of each square matrix raised to the absolute value of n, provided that each square matrix is invertible. If n is greater than zero, an array containing the result of raising each square matrix to the power n. The returned array must have the same shape as x and a floating-point data type determined by type-promotion.

Return type:

array

Notes

Changed in version 2022.12: Added complex data type support.

matrix_rank: matrix_rank¶

Returns the rank (i.e., number of non-zero singular values) of a matrix (or a stack of matrices).

When x is a stack of matrices, the function must compute the number of non-zero singular values for each matrix in the stack.

Parameters:

x (array) – input array having shape (..., M, N) and whose innermost two dimensions form MxN matrices. Should have a floating-point data type.
rtol (Optional[Union[float, array]]) – relative tolerance for small singular values. Singular values approximately less than or equal to rtol * largest_singular_value are set to zero. If a float, the value is equivalent to a zero-dimensional array having a real-valued floating-point data type determined by type-promotion (as applied to x) and must be broadcast against each matrix. If an array, must have a real-valued floating-point data type and must be compatible with shape(x)[:-2] (see broadcasting). If None, the default value is max(M, N) * eps, where eps must be the machine epsilon associated with the real-valued floating-point data type determined by type-promotion (as applied to x). Default: None.

Returns:

out – an array containing the ranks. The returned array must have the default integer data type and must have shape (...) (i.e., must have a shape equal to shape(x)[:-2]).

Return type:

array

Notes

Changed in version 2022.12: Added complex data type support.

matrix_transpose: matrix_transpose¶: Alias for matrix_transpose().

outer: outer¶

Returns the outer product of two vectors x1 and x2.

Parameters:

x1 (array) – first one-dimensional input array of size N. Must have a numeric data type.
x2 (array) – second one-dimensional input array of size M. Must have a numeric data type.

Returns:

out – a two-dimensional array containing the outer product and whose shape is (N, M). The returned array must have a data type determined by type-promotion.

Return type:

array

Notes

Changed in version 2022.12: Added complex data type support.

pinv: pinv¶

Returns the (Moore-Penrose) pseudo-inverse of a matrix (or a stack of matrices) x.

The pseudo-inverse of a matrix $A$, denoted $A^{+}$, is defined as the matrix that “solves” the least-squares problem $Ax = b$ (i.e., if $\overline{x}$ is a solution, then $A^{+}$ is the matrix such that $\overline{x} = A^{+}b$).

While the pseudo-inverse can be defined algebraically, one can understand the pseudo-inverse via singular value decomposition (SVD). Namely, if

\[A = U \Sigma V^H\]

is a singular decomposition of $A$, then

\[A^{+} = U \Sigma^{+} V^H\]

where $U$ and $V^H$ are orthogonal matrices, $\Sigma$ is a diagonal matrix consisting of $A$’s singular values, and $\Sigma^{+}$ is then a diagonal matrix consisting of the reciprocals of $A$’s singular values, leaving zeros in place. During numerical computation, only elements larger than a small tolerance are considered nonzero, and all others replaced by zeros.

When x is a stack of matrices, the function must compute the pseudo-inverse for each matrix in the stack.

Parameters:

x (array) – input array having shape (..., M, N) and whose innermost two dimensions form MxN matrices. Should have a floating-point data type.
rtol (Optional[Union[float, array]]) – relative tolerance for small singular values. Singular values approximately less than or equal to rtol * largest_singular_value are set to zero. If a float, the value is equivalent to a zero-dimensional array having a real-valued floating-point data type determined by type-promotion (as applied to x) and must be broadcast against each matrix. If an array, must have a real-valued floating-point data type and must be compatible with shape(x)[:-2] (see broadcasting). If None, the default value is max(M, N) * eps, where eps must be the machine epsilon associated with the real-valued floating-point data type determined by type-promotion (as applied to x). Default: None.

Returns:

out – an array containing the pseudo-inverse(s). The returned array must have a floating-point data type determined by type-promotion and must have shape (..., N, M) (i.e., must have the same shape as x, except the innermost two dimensions must be transposed).

Return type:

array

Notes

Changed in version 2022.12: Added complex data type support.

qr: qr¶

Returns the QR decomposition of a full column rank matrix (or a stack of matrices).

If x is real-valued, let $\mathbb{K}$ be the set of real numbers $\mathbb{R}$, and, if x is complex-valued, let $\mathbb{K}$ be the set of complex numbers $\mathbb{C}$.

The complete QR decomposition of a matrix $x \in\ \mathbb{K}^{n \times n}$ is defined as

\[x = QR\]

where $Q \in\ \mathbb{K}^{m \times m}$ is orthogonal when x is real-valued and unitary when x is complex-valued and where $R \in\ \mathbb{K}^{m \times n}$ is an upper triangular matrix with real diagonal (even when x is complex-valued).

When $m \gt n$ (tall matrix), as $R$ is upper triangular, the last $m - n$ rows are zero. In this case, the last $m - n$ columns of $Q$ can be dropped to form the reduced QR decomposition.

\[x = QR\]

where $Q \in\ \mathbb{K}^{m \times n}$ and $R \in\ \mathbb{K}^{n \times n}$.

The reduced QR decomposition equals with the complete QR decomposition when $n \geq m$ (wide matrix).

When x is a stack of matrices, the function must compute the QR decomposition for each matrix in the stack.

Note

Whether an array library explicitly checks whether an input array is a full column rank matrix (or a stack of full column rank matrices) is implementation-defined.

Warning

The elements in the diagonal of $R$ are not necessarily positive. Accordingly, the returned QR decomposition is only unique up to the sign of the diagonal of $R$, and different libraries or inputs on different devices may produce different valid decompositions.

Warning

The QR decomposition is only well-defined if the first k = min(m,n) columns of every matrix in x are linearly independent.

Parameters:

x (array) – input array having shape (..., M, N) and whose innermost two dimensions form MxN matrices of rank N. Should have a floating-point data type.
mode (Literal['reduced', 'complete']) –
decomposition mode. Should be one of the following modes:
- 'reduced': compute only the leading K columns of q, such that q and r have dimensions (..., M, K) and (..., K, N), respectively, and where K = min(M, N).
- 'complete': compute q and r with dimensions (..., M, M) and (..., M, N), respectively.
Default: 'reduced'.

Returns:

out – a namedtuple (Q, R) whose

first element must have the field name Q and must be an array whose shape depends on the value of mode and contain matrices with orthonormal columns. If mode is 'complete', the array must have shape (..., M, M). If mode is 'reduced', the array must have shape (..., M, K), where K = min(M, N). The first x.ndim-2 dimensions must have the same size as those of the input array x.
second element must have the field name R and must be an array whose shape depends on the value of mode and contain upper-triangular matrices. If mode is 'complete', the array must have shape (..., M, N). If mode is 'reduced', the array must have shape (..., K, N), where K = min(M, N). The first x.ndim-2 dimensions must have the same size as those of the input x.

Each returned array must have a floating-point data type determined by type-promotion.

Return type:

Tuple[array, array]

Notes

Changed in version 2022.12: Added complex data type support.

slogdet: slogdet¶

Returns the sign and the natural logarithm of the absolute value of the determinant of a square matrix (or a stack of square matrices) x.

Note

The purpose of this function is to calculate the determinant more accurately when the determinant is either very small or very large, as calling det may overflow or underflow.

The sign of the determinant is given by

\[\begin{split}\operatorname{sign}(\det x) = \begin{cases} 0 & \textrm{if } \det x = 0 \\ \frac{\det x}{|\det x|} & \textrm{otherwise} \end{cases}\end{split}\]

where $|\det x|$ is the absolute value of the determinant of x.

When x is a stack of matrices, the function must compute the sign and natural logarithm of the absolute value of the determinant for each matrix in the stack.

Special Cases

For real-valued floating-point operands,

If the determinant is zero, the sign should be 0 and logabsdet should be -infinity.

For complex floating-point operands,

If the determinant is 0 + 0j, the sign should be 0 + 0j and logabsdet should be -infinity + 0j.

Note

Depending on the underlying algorithm, when the determinant is zero, the returned result may differ from -infinity (or -infinity + 0j). In all cases, the determinant should be equal to sign * exp(logabsdet) (although, again, the result may be subject to numerical precision errors).

Parameters:

x (array) – input array having shape (..., M, M) and whose innermost two dimensions form square matrices. Should have a floating-point data type.

Returns:

out – a namedtuple (sign, logabsdet) whose

first element must have the field name sign and must be an array containing a number representing the sign of the determinant for each square matrix. Must have the same data type as x.
second element must have the field name logabsdet and must be an array containing the natural logarithm of the absolute value of the determinant for each square matrix. If x is real-valued, the returned array must have a real-valued floating-point data type determined by type-promotion. If x is complex, the returned array must have a real-valued floating-point data type having the same precision as x (e.g., if x is complex64, logabsdet must have a float32 data type).

Each returned array must have shape shape(x)[:-2].

Return type:

Tuple[array, array]

Notes

Changed in version 2022.12: Added complex data type support.

solve: solve¶

Returns the solution of a square system of linear equations with a unique solution.

Let x1 equal $A$ and x2 equal $B$. If the promoted data type of x1 and x2 is real-valued, let $\mathbb{K}$ be the set of real numbers $\mathbb{R}$, and, if the promoted data type of x1 and x2 is complex-valued, let $\mathbb{K}$ be the set of complex numbers $\mathbb{C}$.

This function computes the solution $X \in\ \mathbb{K}^{m \times k}$ of the linear system associated to $A \in\ \mathbb{K}^{m \times m}$ and $B \in\ \mathbb{K}^{m \times k}$ and is defined as

\[AX = B\]

This system of linear equations has a unique solution if and only if $A$ is invertible.

Note

Whether an array library explicitly checks whether x1 is invertible is implementation-defined.

When x1 and/or x2 is a stack of matrices, the function must compute a solution for each matrix in the stack.

Parameters:

x1 (array) – coefficient array A having shape (..., M, M) and whose innermost two dimensions form square matrices. Must be of full rank (i.e., all rows or, equivalently, columns must be linearly independent). Should have a floating-point data type.
x2 (array) – ordinate (or “dependent variable”) array B. If x2 has shape (M,), x2 is equivalent to an array having shape (..., M, 1). If x2 has shape (..., M, K), each column k defines a set of ordinate values for which to compute a solution, and shape(x2)[:-2] must be compatible with shape(x1)[:-2] (see broadcasting). Should have a floating-point data type.

Returns:

out – an array containing the solution to the system AX = B for each square matrix. If x2 has shape (M,), the returned array must have shape equal to shape(x1)[:-2] + shape(x2)[-1:]. Otherwise, if x2 has shape (..., M, K)`, the returned array must have shape equal to (..., M, K), where ... refers to the result of broadcasting shape(x1)[:-2] and shape(x2)[:-2]. The returned array must have a floating-point data type determined by type-promotion.

Return type:

array

Notes

Changed in version 2022.12: Added complex data type support.

svd: svd¶

Returns a singular value decomposition (SVD) of a matrix (or a stack of matrices) x.

If x is real-valued, let $\mathbb{K}$ be the set of real numbers $\mathbb{R}$, and, if x is complex-valued, let $\mathbb{K}$ be the set of complex numbers $\mathbb{C}$.

The full singular value decomposition of an $m \times n$ matrix $x \in\ \mathbb{K}^{m \times n}$ is a factorization of the form

\[x = U \Sigma V^H\]

where $U \in\ \mathbb{K}^{m \times m}$, $\Sigma \in\ \mathbb{K}^{m \times\ n}$, $\operatorname{diag}(\Sigma) \in\ \mathbb{R}^{k}$ with $k = \operatorname{min}(m, n)$, $V^H \in\ \mathbb{K}^{n \times n}$, and where $V^H$ is the conjugate transpose when $V$ is complex and the transpose when $V$ is real-valued. When x is real-valued, $U$, $V$ (and thus $V^H$) are orthogonal, and, when x is complex, $U$, $V$ (and thus $V^H$) are unitary.

When $m \gt n$ (tall matrix), we can drop the last $m - n$ columns of $U$ to form the reduced SVD

\[x = U \Sigma V^H\]

where $U \in\ \mathbb{K}^{m \times k}$, $\Sigma \in\ \mathbb{K}^{k \times\ k}$, $\operatorname{diag}(\Sigma) \in\ \mathbb{R}^{k}$, and $V^H \in\ \mathbb{K}^{k \times n}$. In this case, $U$ and $V$ have orthonormal columns.

Similarly, when $n \gt m$ (wide matrix), we can drop the last $n - m$ columns of $V$ to also form a reduced SVD.

This function returns the decomposition $U$, $S$, and $V^H$, where $S = \operatorname{diag}(\Sigma)$.

When x is a stack of matrices, the function must compute the singular value decomposition for each matrix in the stack.

Warning

The returned arrays $U$ and $V$ are neither unique nor continuous with respect to x. Because $U$ and $V$ are not unique, different hardware and software may compute different singular vectors.

Non-uniqueness stems from the fact that multiplying any pair of singular vectors $u_k$, $v_k$ by $-1$ when x is real-valued and by $e^{\phi j}$ ($\phi \in \mathbb{R}$) when x is complex produces another two valid singular vectors of the matrix.

Parameters:

x (array) – input array having shape (..., M, N) and whose innermost two dimensions form matrices on which to perform singular value decomposition. Should have a floating-point data type.
full_matrices (bool) – If True, compute full-sized U and Vh, such that U has shape (..., M, M) and Vh has shape (..., N, N). If False, compute on the leading K singular vectors, such that U has shape (..., M, K) and Vh has shape (..., K, N) and where K = min(M, N). Default: True.

Returns:

out – a namedtuple (U, S, Vh) whose

first element must have the field name U and must be an array whose shape depends on the value of full_matrices and contain matrices with orthonormal columns (i.e., the columns are left singular vectors). If full_matrices is True, the array must have shape (..., M, M). If full_matrices is False, the array must have shape (..., M, K), where K = min(M, N). The first x.ndim-2 dimensions must have the same shape as those of the input x. Must have the same data type as x.
second element must have the field name S and must be an array with shape (..., K) that contains the vector(s) of singular values of length K, where K = min(M, N). For each vector, the singular values must be sorted in descending order by magnitude, such that s[..., 0] is the largest value, s[..., 1] is the second largest value, et cetera. The first x.ndim-2 dimensions must have the same shape as those of the input x. Must have a real-valued floating-point data type having the same precision as x (e.g., if x is complex64, S must have a float32 data type).
third element must have the field name Vh and must be an array whose shape depends on the value of full_matrices and contain orthonormal rows (i.e., the rows are the right singular vectors and the array is the adjoint). If full_matrices is True, the array must have shape (..., N, N). If full_matrices is False, the array must have shape (..., K, N) where K = min(M, N). The first x.ndim-2 dimensions must have the same shape as those of the input x. Must have the same data type as x.

Return type:

Tuple[array, array, array]

Notes

Changed in version 2022.12: Added complex data type support.

svdvals: svdvals¶

Returns the singular values of a matrix (or a stack of matrices) x.

When x is a stack of matrices, the function must compute the singular values for each matrix in the stack.

Parameters:: x (array) – input array having shape (..., M, N) and whose innermost two dimensions form matrices on which to perform singular value decomposition. Should have a floating-point data type.
Returns:: out – an array with shape (..., K) that contains the vector(s) of singular values of length K, where K = min(M, N). For each vector, the singular values must be sorted in descending order by magnitude, such that s[..., 0] is the largest value, s[..., 1] is the second largest value, et cetera. The first x.ndim-2 dimensions must have the same shape as those of the input x. The returned array must have a real-valued floating-point data type having the same precision as x (e.g., if x is complex64, the returned array must have a float32 data type).
Return type:: array

Notes

Changed in version 2022.12: Added complex data type support.

tensordot: tensordot¶: Alias for tensordot().

trace: trace¶

Returns the sum along the specified diagonals of a matrix (or a stack of matrices) x.

Parameters:

x (array) – input array having shape (..., M, N) and whose innermost two dimensions form MxN matrices. Should have a numeric data type.
offset (int) –
offset specifying the off-diagonal relative to the main diagonal.
- offset = 0: the main diagonal.
- offset > 0: off-diagonal above the main diagonal.
- offset < 0: off-diagonal below the main diagonal.
Default: 0.
dtype (Optional[dtype]) –
data type of the returned array. If None,
- if the default data type corresponding to the data type “kind” (integer, real-valued floating-point, or complex floating-point) of x has a smaller range of values than the data type of x (e.g., x has data type int64 and the default data type is int32, or x has data type uint64 and the default data type is int64), the returned array must have the same data type as x.
- if x has a real-valued floating-point data type, the returned array must have the default real-valued floating-point data type.
- if x has a complex floating-point data type, the returned array must have the default complex floating-point data type.
- if x has a signed integer data type (e.g., int16), the returned array must have the default integer data type.
- if x has an unsigned integer data type (e.g., uint16), the returned array must have an unsigned integer data type having the same number of bits as the default integer data type (e.g., if the default integer data type is int32, the returned array must have a uint32 data type).
If the data type (either specified or resolved) differs from the data type of x, the input array should be cast to the specified data type before computing the sum. Default: None.

Note

keyword argument is intended to help prevent data type overflows.

Returns:

out – an array containing the traces and whose shape is determined by removing the last two dimensions and storing the traces in the last array dimension. For example, if x has rank k and shape (I, J, K, ..., L, M, N), then an output array has rank k-2 and shape (I, J, K, ..., L) where

out[i, j, k, ..., l] = trace(a[i, j, k, ..., l, :, :])

The returned array must have a data type as described by the dtype parameter above.

Return type:

array

Notes

Special Cases

Let N equal the number of elements over which to compute the sum.

If N is 0, the sum is 0 (i.e., the empty sum).

For both real-valued and complex floating-point operands, special cases must be handled as if the operation is implemented by successive application of add().

Changed in version 2022.12: Added complex data type support.

vecdot: vecdot¶: Alias for vecdot().

vector_norm: vector_norm¶

Computes the vector norm of a vector (or batch of vectors) x.

Parameters:

x (array) – input array. Should have a floating-point data type.
axis (Optional[Union[int, Tuple[int, ...]]]) – If an integer, axis specifies the axis (dimension) along which to compute vector norms. If an n-tuple, axis specifies the axes (dimensions) along which to compute batched vector norms. If None, the vector norm must be computed over all array values (i.e., equivalent to computing the vector norm of a flattened array). Negative indices must be supported. Default: None.
keepdims (bool) – If True, the axes (dimensions) specified by axis must be included in the result as singleton dimensions, and, accordingly, the result must be compatible with the input array (see broadcasting). Otherwise, if False, the axes (dimensions) specified by axis must not be included in the result. Default: False.
ord (Union[int, float, Literal[inf, -inf]]) –
order of the norm. The following mathematical norms must be supported:

ord

description

1

L1-norm (Manhattan)

2

L2-norm (Euclidean)

inf

infinity norm

(int,float >= 1)

p-norm

The following non-mathematical “norms” must be supported:

ord

description

0

sum(a != 0)

-1

1./sum(1./abs(a))

-2

1./sqrt(sum(1./abs(a)**2))

-inf

min(abs(a))

(int,float < 1)

sum(abs(a)**ord)**(1./ord)

Default: 2.

Returns:

ord	description
1	L1-norm (Manhattan)
2	L2-norm (Euclidean)
inf	infinity norm
(int,float >= 1)	p-norm

ord	description
0	sum(a != 0)
-1	1./sum(1./abs(a))
-2	1./sqrt(sum(1./abs(a)**2))
-inf	min(abs(a))
(int,float < 1)	sum(abs(a)ord)(1./ord)

out – an array containing the vector norms. If axis is None, the returned array must be a zero-dimensional array containing a vector norm. If axis is a scalar value (int or float), the returned array must have a rank which is one less than the rank of x. If axis is a n-tuple, the returned array must have a rank which is n less than the rank of x. If x has a real-valued data type, the returned array must have a real-valued floating-point data type determined by type-promotion. If x has a complex-valued data type, the returned array must have a real-valued floating-point data type whose precision matches the precision of x (e.g., if x is complex128, then the returned array must have a float64 data type).

Return type:

array

Notes

Changed in version 2022.12: Added complex data type support.

class array_api._2022_12.NestedSequence(*args, **kwargs)[source]¶

Bases: Protocol, Generic

_abc_impl = <_abc._abc_data object>¶

_is_protocol = True¶

_is_runtime_protocol = True¶

type array_api._2022_12.ShapedAnyArray = ShapedArray[Unpack[T], Any, Any]¶

class array_api._2022_12.ShapedArray[source]¶

Bases: Array, Protocol, Generic[Unpack[T], TDevice, TDtype]

_abc_impl = <_abc._abc_data object>¶

_is_protocol = True¶

_is_runtime_protocol = True¶

property shape: tuple[*T]¶

Array dimensions.

Returns:: out – array dimensions. An array dimension must be None if and only if a dimension is unknown.
Return type:: Tuple[Optional[int], …]

Note

class array_api._2022_12.abs(*args, **kwargs)[source]¶

Bases: Protocol, Generic

Calculates the absolute value for each element x_i of the input array x.

For real-valued input arrays, the element-wise result has the same magnitude as the respective element in x but has positive sign.

Note

For signed integer data types, the absolute value of the minimum representable integer is implementation-dependent.

Note

For complex floating-point operands, the complex absolute value is known as the norm, modulus, or magnitude and, for a complex number $z = a + bj$ is computed as

\[\operatorname{abs}(z) = \sqrt{a^2 + b^2}\]

Note

For complex floating-point operands, conforming implementations should take care to avoid undue overflow or underflow during intermediate stages of computation.

Parameters:: x (array) – input array. Should have a numeric data type.
Returns:: out – an array containing the absolute value of each element in x. If x has a real-valued data type, the returned array must have the same data type as x. If x has a complex floating-point data type, the returned array must have a real-valued floating-point data type whose precision matches the precision of x (e.g., if x is complex128, then the returned array must have a float64 data type).
Return type:: array

Notes

Special Cases

For real-valued floating-point operands,

If x_i is NaN, the result is NaN.
If x_i is -0, the result is +0.
If x_i is -infinity, the result is +infinity.

For complex floating-point operands, let a = real(x_i), b = imag(x_i), and

If a is either +infinity or -infinity and b is any value (including NaN), the result is +infinity.
If a is any value (including NaN) and b is either +infinity or -infinity, the result is +infinity.
If a is either +0 or -0, the result is equal to abs(b).
If b is either +0 or -0, the result is equal to abs(a).
If a is NaN and b is a finite number, the result is NaN.
If a is a finite number and b is NaN, the result is NaN.
If a is NaN and b is NaN, the result is NaN.

Changed in version 2022.12: Added complex data type support.

_abc_impl = <_abc._abc_data object>¶

_is_protocol = True¶

_is_runtime_protocol = True¶

class array_api._2022_12.acos(*args, **kwargs)[source]¶

Bases: Protocol, Generic

Calculates an implementation-dependent approximation of the principal value of the inverse cosine for each element x_i of the input array x.

Each element-wise result is expressed in radians.

Note

The principal value of the arc cosine of a complex number $z$ is

\[\operatorname{acos}(z) = \frac{1}{2}\pi + j\ \ln(zj + \sqrt{1-z^2})\]

For any $z$,

\[\operatorname{acos}(z) = \pi - \operatorname{acos}(-z)\]

Note

For complex floating-point operands, acos(conj(x)) must equal conj(acos(x)).

Note

Accordingly, for complex arguments, the function returns the inverse cosine in the range of a strip unbounded along the imaginary axis and in the interval $[0, \pi]$ along the real axis.

Note: branch cuts follow C99 and have provisional status (see branch-cuts).

Parameters:: x (array) – input array. Should have a floating-point data type.
Returns:: out – an array containing the inverse cosine of each element in x. The returned array must have a floating-point data type determined by type-promotion.
Return type:: array

Notes

Special cases

For real-valued floating-point operands,

If x_i is NaN, the result is NaN.
If x_i is greater than 1, the result is NaN.
If x_i is less than -1, the result is NaN.
If x_i is 1, the result is +0.

For complex floating-point operands, let a = real(x_i), b = imag(x_i), and

If a is either +0 or -0 and b is +0, the result is π/2 - 0j.
If a is either +0 or -0 and b is NaN, the result is π/2 + NaN j.
If a is a finite number and b is +infinity, the result is π/2 - infinity j.
If a is a nonzero finite number and b is NaN, the result is NaN + NaN j.
If a is -infinity and b is a positive (i.e., greater than 0) finite number, the result is π - infinity j.
If a is +infinity and b is a positive (i.e., greater than 0) finite number, the result is +0 - infinity j.
If a is -infinity and b is +infinity, the result is 3π/4 - infinity j.
If a is +infinity and b is +infinity, the result is π/4 - infinity j.
If a is either +infinity or -infinity and b is NaN, the result is NaN ± infinity j (sign of the imaginary component is unspecified).
If a is NaN and b is a finite number, the result is NaN + NaN j.
If a is NaN and b is +infinity, the result is NaN - infinity j.
If a is NaN and b is NaN, the result is NaN + NaN j.

Changed in version 2022.12: Added complex data type support.

_abc_impl = <_abc._abc_data object>¶

_is_protocol = True¶

_is_runtime_protocol = True¶

class array_api._2022_12.acosh(*args, **kwargs)[source]¶

Bases: Protocol, Generic

Calculates an implementation-dependent approximation to the inverse hyperbolic cosine for each element x_i of the input array x.

Note

The principal value of the inverse hyperbolic cosine of a complex number $z$ is

\[\operatorname{acosh}(z) = \ln(z + \sqrt{z+1}\sqrt{z-1})\]

For any $z$,

\[\operatorname{acosh}(z) = \frac{\sqrt{z-1}}{\sqrt{1-z}}\operatorname{acos}(z)\]

or simply

\[\operatorname{acosh}(z) = j\ \operatorname{acos}(z)\]

in the upper half of the complex plane.

Note

For complex floating-point operands, acosh(conj(x)) must equal conj(acosh(x)).

Note

The inverse hyperbolic cosine is a multi-valued function and requires a branch cut on the complex plane. By convention, a branch cut is placed at the line segment $(-\infty, 1)$ of the real axis.

Note: branch cuts follow C99 and have provisional status (see branch-cuts).

Parameters:: x (array) – input array whose elements each represent the area of a hyperbolic sector. Should have a floating-point data type.
Returns:: out – an array containing the inverse hyperbolic cosine of each element in x. The returned array must have a floating-point data type determined by type-promotion.
Return type:: array

Notes

Special cases

For real-valued floating-point operands,

If x_i is NaN, the result is NaN.
If x_i is less than 1, the result is NaN.
If x_i is 1, the result is +0.
If x_i is +infinity, the result is +infinity.

For complex floating-point operands, let a = real(x_i), b = imag(x_i), and

If a is either +0 or -0 and b is +0, the result is +0 + πj/2.
If a is a finite number and b is +infinity, the result is +infinity + πj/2.
If a is a nonzero finite number and b is NaN, the result is NaN + NaN j.
If a is +0 and b is NaN, the result is NaN ± πj/2 (sign of imaginary component is unspecified).
If a is -infinity and b is a positive (i.e., greater than 0) finite number, the result is +infinity + πj.
If a is +infinity and b is a positive (i.e., greater than 0) finite number, the result is +infinity + 0j.
If a is -infinity and b is +infinity, the result is +infinity + 3πj/4.
If a is +infinity and b is +infinity, the result is +infinity + πj/4.
If a is either +infinity or -infinity and b is NaN, the result is +infinity + NaN j.
If a is NaN and b is a finite number, the result is NaN + NaN j.
If a is NaN and b is +infinity, the result is +infinity + NaN j.
If a is NaN and b is NaN, the result is NaN + NaN j.

Changed in version 2022.12: Added complex data type support.

_abc_impl = <_abc._abc_data object>¶

_is_protocol = True¶

_is_runtime_protocol = True¶

class array_api._2022_12.add(*args, **kwargs)[source]¶

Bases: Protocol, Generic

Calculates the sum for each element x1_i of the input array x1 with the respective element x2_i of the input array x2.

Parameters:

x1 (array) – first input array. Should have a numeric data type.
x2 (array) – second input array. Must be compatible with x1 (see broadcasting). Should have a numeric data type.

Returns:

out – an array containing the element-wise sums. The returned array must have a data type determined by type-promotion.

Return type:

array

Notes

Special cases

For real-valued floating-point operands,

If either x1_i or x2_i is NaN, the result is NaN.
If x1_i is +infinity and x2_i is -infinity, the result is NaN.
If x1_i is -infinity and x2_i is +infinity, the result is NaN.
If x1_i is +infinity and x2_i is +infinity, the result is +infinity.
If x1_i is -infinity and x2_i is -infinity, the result is -infinity.
If x1_i is +infinity and x2_i is a finite number, the result is +infinity.
If x1_i is -infinity and x2_i is a finite number, the result is -infinity.
If x1_i is a finite number and x2_i is +infinity, the result is +infinity.
If x1_i is a finite number and x2_i is -infinity, the result is -infinity.
If x1_i is -0 and x2_i is -0, the result is -0.
If x1_i is -0 and x2_i is +0, the result is +0.
If x1_i is +0 and x2_i is -0, the result is +0.
If x1_i is +0 and x2_i is +0, the result is +0.
If x1_i is either +0 or -0 and x2_i is a nonzero finite number, the result is x2_i.
If x1_i is a nonzero finite number and x2_i is either +0 or -0, the result is x1_i.
If x1_i is a nonzero finite number and x2_i is -x1_i, the result is +0.
In the remaining cases, when neither infinity, +0, -0, nor a NaN is involved, and the operands have the same mathematical sign or have different magnitudes, the sum must be computed and rounded to the nearest representable value according to IEEE 754-2019 and a supported round mode. If the magnitude is too large to represent, the operation overflows and the result is an infinity of appropriate mathematical sign.

Note

Floating-point addition is a commutative operation, but not always associative.

For complex floating-point operands, addition is defined according to the following table. For real components a and c and imaginary components b and d,

	c	dj	c + dj
a	a + c	a + dj	(a+c) + dj
bj	c + bj	(b+d)j	c + (b+d)j
a + bj	(a+c) + bj	a + (b+d)j	(a+c) + (b+d)j

If a is -0 and c is -0, the real component of the result is -0.
Similarly, if b is +0 and d is -0, the imaginary component of the result is +0.

Hence, if z1 = a + bj = -0 + 0j and z2 = c + dj = -0 - 0j, the result of z1 + z2 is -0 + 0j.

Changed in version 2022.12: Added complex data type support.

_abc_impl = <_abc._abc_data object>¶

_is_protocol = True¶

_is_runtime_protocol = True¶

class array_api._2022_12.all(*args, **kwargs)[source]¶

Bases: Protocol, Generic

Tests whether all input array elements evaluate to True along a specified axis.

Note

Positive infinity, negative infinity, and NaN must evaluate to True.

Note

If x has a complex floating-point data type, elements having a non-zero component (real or imaginary) must evaluate to True.

Note

If x is an empty array or the size of the axis (dimension) along which to evaluate elements is zero, the test result must be True.

Parameters:

x (array) – input array.
axis (Optional[Union[int, Tuple[int, ...]]]) – axis or axes along which to perform a logical AND reduction. By default, a logical AND reduction must be performed over the entire array. If a tuple of integers, logical AND reductions must be performed over multiple axes. A valid axis must be an integer on the interval [-N, N), where N is the rank (number of dimensions) of x. If an axis is specified as a negative integer, the function must determine the axis along which to perform a reduction by counting backward from the last dimension (where -1 refers to the last dimension). If provided an invalid axis, the function must raise an exception. Default: None.
keepdims (bool) – If True, the reduced axes (dimensions) must be included in the result as singleton dimensions, and, accordingly, the result must be compatible with the input array (see broadcasting). Otherwise, if False, the reduced axes (dimensions) must not be included in the result. Default: False.

Returns:

Return type:

array

Notes

Changed in version 2022.12: Added complex data type support.

_abc_impl = <_abc._abc_data object>¶

_is_protocol = True¶

_is_runtime_protocol = True¶

class array_api._2022_12.any(*args, **kwargs)[source]¶

Bases: Protocol, Generic

Tests whether any input array element evaluates to True along a specified axis.

Note

Positive infinity, negative infinity, and NaN must evaluate to True.

Note

If x has a complex floating-point data type, elements having a non-zero component (real or imaginary) must evaluate to True.

Note

If x is an empty array or the size of the axis (dimension) along which to evaluate elements is zero, the test result must be False.

Parameters:

x (array) – input array.
axis (Optional[Union[int, Tuple[int, ...]]]) – axis or axes along which to perform a logical OR reduction. By default, a logical OR reduction must be performed over the entire array. If a tuple of integers, logical OR reductions must be performed over multiple axes. A valid axis must be an integer on the interval [-N, N), where N is the rank (number of dimensions) of x. If an axis is specified as a negative integer, the function must determine the axis along which to perform a reduction by counting backward from the last dimension (where -1 refers to the last dimension). If provided an invalid axis, the function must raise an exception. Default: None.
keepdims (bool) – If True, the reduced axes (dimensions) must be included in the result as singleton dimensions, and, accordingly, the result must be compatible with the input array (see broadcasting). Otherwise, if False, the reduced axes (dimensions) must not be included in the result. Default: False.

Returns:

Return type:

array

Notes

Changed in version 2022.12: Added complex data type support.

_abc_impl = <_abc._abc_data object>¶

_is_protocol = True¶

_is_runtime_protocol = True¶

class array_api._2022_12.arange(*args, **kwargs)[source]¶

Bases: Protocol, Generic

Returns evenly spaced values within the half-open interval [start, stop) as a one-dimensional array.

Parameters:

start (Union[int, float]) – if stop is specified, the start of interval (inclusive); otherwise, the end of the interval (exclusive). If stop is not specified, the default starting value is 0.
stop (Optional[Union[int, float]]) – the end of the interval. Default: None.
step (Union[int, float]) – the distance between two adjacent elements (out[i+1] - out[i]). Must not be 0; may be negative, this results in an empty array if stop >= start. Default: 1.
dtype (Optional[dtype]) – output array data type. If dtype is None, the output array data type must be inferred from start, stop and step. If those are all integers, the output array dtype must be the default integer dtype; if one or more have type float, then the output array dtype must be the default real-valued floating-point data type. Default: None.
device (Optional[device]) – device on which to place the created array. Default: None.

Note

Returns:: out – a one-dimensional array containing evenly spaced values. The length of the output array must be ceil((stop-start)/step) if stop - start and step have the same sign, and length 0 otherwise.
Return type:: array

_abc_impl = <_abc._abc_data object>¶

_is_protocol = True¶

_is_runtime_protocol = True¶

class array_api._2022_12.argmax(*args, **kwargs)[source]¶

Bases: Protocol, Generic

Returns the indices of the maximum values along a specified axis.

When the maximum value occurs multiple times, only the indices corresponding to the first occurrence are returned.

Note

Parameters:

x (array) – input array. Should have a real-valued data type.
axis (Optional[int]) – axis along which to search. If None, the function must return the index of the maximum value of the flattened array. Default: None.
keepdims (bool) – if True, the reduced axes (dimensions) must be included in the result as singleton dimensions, and, accordingly, the result must be compatible with the input array (see broadcasting). Otherwise, if False, the reduced axes (dimensions) must not be included in the result. Default: False.

Returns:

Return type:

array

_abc_impl = <_abc._abc_data object>¶

_is_protocol = True¶

_is_runtime_protocol = True¶

class array_api._2022_12.argmin(*args, **kwargs)[source]¶

Bases: Protocol, Generic

Returns the indices of the minimum values along a specified axis.

When the minimum value occurs multiple times, only the indices corresponding to the first occurrence are returned.

Note

Parameters:

x (array) – input array. Should have a real-valued data type.
axis (Optional[int]) – axis along which to search. If None, the function must return the index of the minimum value of the flattened array. Default: None.
keepdims (bool) – if True, the reduced axes (dimensions) must be included in the result as singleton dimensions, and, accordingly, the result must be compatible with the input array (see broadcasting). Otherwise, if False, the reduced axes (dimensions) must not be included in the result. Default: False.

Returns:

Return type:

array

_abc_impl = <_abc._abc_data object>¶

_is_protocol = True¶

_is_runtime_protocol = True¶

class array_api._2022_12.argsort(*args, **kwargs)[source]¶

Bases: Protocol, Generic

Returns the indices that sort an array x along a specified axis.

Note

Parameters:

x (array) – input array. Should have a real-valued data type.
axis (int) – axis along which to sort. If set to -1, the function must sort along the last axis. Default: -1.
descending (bool) – sort order. If True, the returned indices sort x in descending order (by value). If False, the returned indices sort x in ascending order (by value). Default: False.
stable (bool) – sort stability. If True, the returned indices must maintain the relative order of x values which compare as equal. If False, the returned indices may or may not maintain the relative order of x values which compare as equal (i.e., the relative order of x values which compare as equal is implementation-dependent). Default: True.

Returns:

out – an array of indices. The returned array must have the same shape as x. The returned array must have the default array index data type.

Return type:

array

_abc_impl = <_abc._abc_data object>¶

_is_protocol = True¶

_is_runtime_protocol = True¶

class array_api._2022_12.asarray(*args, **kwargs)[source]¶

Bases: Protocol, Generic

Convert the input to an array.

Parameters:

obj (Union[array, bool, int, float, complex, NestedSequence[bool | int | float | complex], SupportsBufferProtocol]) –
object to be converted to an array. May be a Python scalar, a (possibly nested) sequence of Python scalars, or an object supporting the Python buffer protocol.

Tip

An object supporting the buffer protocol can be turned into a memoryview through memoryview(obj).
dtype (Optional[dtype]) –
output array data type. If dtype is None, the output array data type must be inferred from the data type(s) in obj. If all input values are Python scalars, then, in order of precedence,
- if all values are of type bool, the output data type must be bool.
- if all values are of type int or are a mixture of bool and int, the output data type must be the default integer data type.
- if one or more values are complex numbers, the output data type must be the default complex floating-point data type.
- if one or more values are floats, the output data type must be the default real-valued floating-point data type.
Default: None.

Note

If dtype is not None, then array conversions should obey type-promotion rules. Conversions not specified according to type-promotion rules may or may not be permitted by a conforming array library. To perform an explicit cast, use array_api.astype().

Note

If an input value exceeds the precision of the resolved output array data type, behavior is left unspecified and, thus, implementation-defined.
device (Optional[device]) – device on which to place the created array. If device is None and obj is an array, the output array device must be inferred from obj. Default: None.
copy (Optional[bool]) – boolean indicating whether or not to copy the input. If True, the function must always copy. If False, the function must never copy for input which supports the buffer protocol and must raise a ValueError in case a copy would be necessary. If None, the function must reuse existing memory buffer if possible and copy otherwise. Default: None.

Returns:

out – an array containing the data from obj.

Return type:

array

Notes

Changed in version 2022.12: Added complex data type support.

_abc_impl = <_abc._abc_data object>¶

_is_protocol = True¶

_is_runtime_protocol = True¶

class array_api._2022_12.asin(*args, **kwargs)[source]¶

Bases: Protocol, Generic

Calculates an implementation-dependent approximation of the principal value of the inverse sine for each element x_i of the input array x.

Each element-wise result is expressed in radians.

Note

The principal value of the arc sine of a complex number $z$ is

\[\operatorname{asin}(z) = -j\ \ln(zj + \sqrt{1-z^2})\]

For any $z$,

\[\operatorname{asin}(z) = \operatorname{acos}(-z) - \frac{\pi}{2}\]

Note

For complex floating-point operands, asin(conj(x)) must equal conj(asin(x)).

Note

Accordingly, for complex arguments, the function returns the inverse sine in the range of a strip unbounded along the imaginary axis and in the interval $[-\pi/2, +\pi/2]$ along the real axis.

Note: branch cuts follow C99 and have provisional status (see branch-cuts).

Parameters:: x (array) – input array. Should have a floating-point data type.
Returns:: out – an array containing the inverse sine of each element in x. The returned array must have a floating-point data type determined by type-promotion.
Return type:: array

Notes

Special cases

For real-valued floating-point operands,

If x_i is NaN, the result is NaN.
If x_i is greater than 1, the result is NaN.
If x_i is less than -1, the result is NaN.
If x_i is +0, the result is +0.
If x_i is -0, the result is -0.

For complex floating-point operands, special cases must be handled as if the operation is implemented as -1j * asinh(x*1j).

Changed in version 2022.12: Added complex data type support.

_abc_impl = <_abc._abc_data object>¶

_is_protocol = True¶

_is_runtime_protocol = True¶

class array_api._2022_12.asinh(*args, **kwargs)[source]¶

Bases: Protocol, Generic

Calculates an implementation-dependent approximation to the inverse hyperbolic sine for each element x_i in the input array x.

Note

The principal value of the inverse hyperbolic sine of a complex number $z$ is

\[\operatorname{asinh}(z) = \ln(z + \sqrt{1+z^2})\]

For any $z$,

\[\operatorname{asinh}(z) = \frac{\operatorname{asin}(zj)}{j}\]

Note

For complex floating-point operands, asinh(conj(x)) must equal conj(asinh(x)) and asinh(-z) must equal -asinh(z).

Note

Note: branch cuts follow C99 and have provisional status (see branch-cuts).

Parameters:: x (array) – input array whose elements each represent the area of a hyperbolic sector. Should have a floating-point data type.
Returns:: out – an array containing the inverse hyperbolic sine of each element in x. The returned array must have a floating-point data type determined by type-promotion.
Return type:: array

Notes

Special cases

For real-valued floating-point operands,

If x_i is NaN, the result is NaN.
If x_i is +0, the result is +0.
If x_i is -0, the result is -0.
If x_i is +infinity, the result is +infinity.
If x_i is -infinity, the result is -infinity.

For complex floating-point operands, let a = real(x_i), b = imag(x_i), and

If a is +0 and b is +0, the result is +0 + 0j.
If a is a positive (i.e., greater than 0) finite number and b is +infinity, the result is +infinity + πj/2.
If a is a finite number and b is NaN, the result is NaN + NaN j.
If a is +infinity and b is a positive (i.e., greater than 0) finite number, the result is +infinity + 0j.
If a is +infinity and b is +infinity, the result is +infinity + πj/4.
If a is NaN and b is +0, the result is NaN + 0j.
If a is NaN and b is a nonzero finite number, the result is NaN + NaN j.
If a is NaN and b is +infinity, the result is ±infinity + NaN j (sign of the real component is unspecified).
If a is NaN and b is NaN, the result is NaN + NaN j.

Changed in version 2022.12: Added complex data type support.

_abc_impl = <_abc._abc_data object>¶

_is_protocol = True¶

_is_runtime_protocol = True¶

class array_api._2022_12.astype(*args, **kwargs)[source]¶

Bases: Protocol, Generic

Copies an array to a specified data type irrespective of type-promotion rules.

Note

Casting floating-point NaN and infinity values to integral data types is not specified and is implementation-dependent.

Note

Casting a complex floating-point array to a real-valued data type should not be permitted.

Note

When casting a boolean input array to a real-valued data type, a value of True must cast to a real-valued number equal to 1, and a value of False must cast to a real-valued number equal to 0.

Note

When casting a real-valued input array to bool, a value of 0 must cast to False, and a non-zero value must cast to True.

When casting a complex floating-point array to bool, a value of 0 + 0j must cast to False, and all other values must cast to True.

Parameters:

x (array) – array to cast.
dtype (dtype) – desired data type.
copy (bool) – specifies whether to copy an array when the specified dtype matches the data type of the input array x. If True, a newly allocated array must always be returned. If False and the specified dtype matches the data type of the input array, the input array must be returned; otherwise, a newly allocated array must be returned. Default: True.

Returns:

out – an array having the specified data type. The returned array must have the same shape as x.

Return type:

array

Notes

Changed in version 2022.12: Added complex data type support.

_abc_impl = <_abc._abc_data object>¶

_is_protocol = True¶

_is_runtime_protocol = True¶

class array_api._2022_12.atan(*args, **kwargs)[source]¶

Bases: Protocol, Generic

Calculates an implementation-dependent approximation of the principal value of the inverse tangent for each element x_i of the input array x.

Each element-wise result is expressed in radians.

Note

The principal value of the inverse tangent of a complex number $z$ is

\[\operatorname{atan}(z) = -\frac{\ln(1 - zj) - \ln(1 + zj)}{2}j\]

Note

For complex floating-point operands, atan(conj(x)) must equal conj(atan(x)).

Note

Accordingly, for complex arguments, the function returns the inverse tangent in the range of a strip unbounded along the imaginary axis and in the interval $[-\pi/2, +\pi/2]$ along the real axis.

Note: branch cuts follow C99 and have provisional status (see branch-cuts).

Parameters:: x (array) – input array. Should have a floating-point data type.
Returns:: out – an array containing the inverse tangent of each element in x. The returned array must have a floating-point data type determined by type-promotion.
Return type:: array

Notes

Special cases

For real-valued floating-point operands,

If x_i is NaN, the result is NaN.
If x_i is +0, the result is +0.
If x_i is -0, the result is -0.
If x_i is +infinity, the result is an implementation-dependent approximation to +π/2.
If x_i is -infinity, the result is an implementation-dependent approximation to -π/2.

For complex floating-point operands, special cases must be handled as if the operation is implemented as -1j * atanh(x*1j).

Changed in version 2022.12: Added complex data type support.

_abc_impl = <_abc._abc_data object>¶

_is_protocol = True¶

_is_runtime_protocol = True¶

class array_api._2022_12.atan2(*args, **kwargs)[source]¶

Bases: Protocol, Generic

Note

Parameters:

x1 (array) – input array corresponding to the y-coordinates. Should have a real-valued floating-point data type.
x2 (array) – input array corresponding to the x-coordinates. Must be compatible with x1 (see broadcasting). Should have a real-valued floating-point data type.

Returns:

out – an array containing the inverse tangent of the quotient x1/x2. The returned array must have a real-valued floating-point data type determined by type-promotion.

Return type:

array

Notes

Special cases

For floating-point operands,

If either x1_i or x2_i is NaN, the result is NaN.
If x1_i is greater than 0 and x2_i is +0, the result is an implementation-dependent approximation to +π/2.
If x1_i is greater than 0 and x2_i is -0, the result is an implementation-dependent approximation to +π/2.
If x1_i is +0 and x2_i is greater than 0, the result is +0.
If x1_i is +0 and x2_i is +0, the result is +0.
If x1_i is +0 and x2_i is -0, the result is an implementation-dependent approximation to +π.
If x1_i is +0 and x2_i is less than 0, the result is an implementation-dependent approximation to +π.
If x1_i is -0 and x2_i is greater than 0, the result is -0.
If x1_i is -0 and x2_i is +0, the result is -0.
If x1_i is -0 and x2_i is -0, the result is an implementation-dependent approximation to -π.
If x1_i is -0 and x2_i is less than 0, the result is an implementation-dependent approximation to -π.
If x1_i is less than 0 and x2_i is +0, the result is an implementation-dependent approximation to -π/2.
If x1_i is less than 0 and x2_i is -0, the result is an implementation-dependent approximation to -π/2.
If x1_i is greater than 0, x1_i is a finite number, and x2_i is +infinity, the result is +0.
If x1_i is greater than 0, x1_i is a finite number, and x2_i is -infinity, the result is an implementation-dependent approximation to +π.
If x1_i is less than 0, x1_i is a finite number, and x2_i is +infinity, the result is -0.
If x1_i is less than 0, x1_i is a finite number, and x2_i is -infinity, the result is an implementation-dependent approximation to -π.
If x1_i is +infinity and x2_i is a finite number, the result is an implementation-dependent approximation to +π/2.
If x1_i is -infinity and x2_i is a finite number, the result is an implementation-dependent approximation to -π/2.
If x1_i is +infinity and x2_i is +infinity, the result is an implementation-dependent approximation to +π/4.
If x1_i is +infinity and x2_i is -infinity, the result is an implementation-dependent approximation to +3π/4.
If x1_i is -infinity and x2_i is +infinity, the result is an implementation-dependent approximation to -π/4.
If x1_i is -infinity and x2_i is -infinity, the result is an implementation-dependent approximation to -3π/4.

_abc_impl = <_abc._abc_data object>¶

_is_protocol = True¶

_is_runtime_protocol = True¶

class array_api._2022_12.atanh(*args, **kwargs)[source]¶

Bases: Protocol, Generic

Calculates an implementation-dependent approximation to the inverse hyperbolic tangent for each element x_i of the input array x.

Note

The principal value of the inverse hyperbolic tangent of a complex number $z$ is

\[\operatorname{atanh}(z) = \frac{\ln(1+z)-\ln(z-1)}{2}\]

For any $z$,

\[\operatorname{atanh}(z) = \frac{\operatorname{atan}(zj)}{j}\]

Note

For complex floating-point operands, atanh(conj(x)) must equal conj(atanh(x)) and atanh(-x) must equal -atanh(x).

Note

Note: branch cuts follow C99 and have provisional status (see branch-cuts).

Parameters:: x (array) – input array whose elements each represent the area of a hyperbolic sector. Should have a floating-point data type.
Returns:: out – an array containing the inverse hyperbolic tangent of each element in x. The returned array must have a floating-point data type determined by type-promotion.
Return type:: array

Notes

Special cases

For real-valued floating-point operands,

If x_i is NaN, the result is NaN.
If x_i is less than -1, the result is NaN.
If x_i is greater than 1, the result is NaN.
If x_i is -1, the result is -infinity.
If x_i is +1, the result is +infinity.
If x_i is +0, the result is +0.
If x_i is -0, the result is -0.

For complex floating-point operands, let a = real(x_i), b = imag(x_i), and

If a is +0 and b is +0, the result is +0 + 0j.
If a is +0 and b is NaN, the result is +0 + NaN j.
If a is 1 and b is +0, the result is +infinity + 0j.
If a is a positive (i.e., greater than 0) finite number and b is +infinity, the result is +0 + πj/2.
If a is a nonzero finite number and b is NaN, the result is NaN + NaN j.
If a is +infinity and b is a positive (i.e., greater than 0) finite number, the result is +0 + πj/2.
If a is +infinity and b is +infinity, the result is +0 + πj/2.
If a is +infinity and b is NaN, the result is +0 + NaN j.
If a is NaN and b is a finite number, the result is NaN + NaN j.
If a is NaN and b is +infinity, the result is ±0 + πj/2 (sign of the real component is unspecified).
If a is NaN and b is NaN, the result is NaN + NaN j.

Changed in version 2022.12: Added complex data type support.

_abc_impl = <_abc._abc_data object>¶

_is_protocol = True¶

_is_runtime_protocol = True¶

class array_api._2022_12.bitwise_and(*args, **kwargs)[source]¶

Bases: Protocol, Generic

Computes the bitwise AND of the underlying binary representation of each element x1_i of the input array x1 with the respective element x2_i of the input array x2.

Parameters:

x1 (array) – first input array. Should have an integer or boolean data type.
x2 (array) – second input array. Must be compatible with x1 (see broadcasting). Should have an integer or boolean data type.

Returns:

out – an array containing the element-wise results. The returned array must have a data type determined by type-promotion.

Return type:

array

_abc_impl = <_abc._abc_data object>¶

_is_protocol = True¶

_is_runtime_protocol = True¶

class array_api._2022_12.bitwise_invert(*args, **kwargs)[source]¶

Bases: Protocol, Generic

Inverts (flips) each bit for each element x_i of the input array x.

Parameters:: x (array) – input array. Should have an integer or boolean data type.
Returns:: out – an array containing the element-wise results. The returned array must have the same data type as x.
Return type:: array

_abc_impl = <_abc._abc_data object>¶

_is_protocol = True¶

_is_runtime_protocol = True¶

class array_api._2022_12.bitwise_left_shift(*args, **kwargs)[source]¶

Bases: Protocol, Generic

Shifts the bits of each element x1_i of the input array x1 to the left by appending x2_i (i.e., the respective element in the input array x2) zeros to the right of x1_i.

Parameters:

x1 (array) – first input array. Should have an integer data type.
x2 (array) – second input array. Must be compatible with x1 (see broadcasting). Should have an integer data type. Each element must be greater than or equal to 0.

Returns:

out – an array containing the element-wise results. The returned array must have a data type determined by type-promotion.

Return type:

array

_abc_impl = <_abc._abc_data object>¶

_is_protocol = True¶

_is_runtime_protocol = True¶

class array_api._2022_12.bitwise_or(*args, **kwargs)[source]¶

Bases: Protocol, Generic

Computes the bitwise OR of the underlying binary representation of each element x1_i of the input array x1 with the respective element x2_i of the input array x2.

Parameters:

x1 (array) – first input array. Should have an integer or boolean data type.
x2 (array) – second input array. Must be compatible with x1 (see broadcasting). Should have an integer or boolean data type.

Returns:

out – an array containing the element-wise results. The returned array must have a data type determined by type-promotion.

Return type:

array

_abc_impl = <_abc._abc_data object>¶

_is_protocol = True¶

_is_runtime_protocol = True¶

class array_api._2022_12.bitwise_right_shift(*args, **kwargs)[source]¶

Bases: Protocol, Generic

Shifts the bits of each element x1_i of the input array x1 to the right according to the respective element x2_i of the input array x2.

Note

This operation must be an arithmetic shift (i.e., sign-propagating) and thus equivalent to floor division by a power of two.

Parameters:

x1 (array) – first input array. Should have an integer data type.
x2 (array) – second input array. Must be compatible with x1 (see broadcasting). Should have an integer data type. Each element must be greater than or equal to 0.

Returns:

out – an array containing the element-wise results. The returned array must have a data type determined by type-promotion.

Return type:

array

_abc_impl = <_abc._abc_data object>¶

_is_protocol = True¶

_is_runtime_protocol = True¶

class array_api._2022_12.bitwise_xor(*args, **kwargs)[source]¶

Bases: Protocol, Generic

Computes the bitwise XOR of the underlying binary representation of each element x1_i of the input array x1 with the respective element x2_i of the input array x2.

Parameters:

x1 (array) – first input array. Should have an integer or boolean data type.
x2 (array) – second input array. Must be compatible with x1 (see broadcasting). Should have an integer or boolean data type.

Returns:

out – an array containing the element-wise results. The returned array must have a data type determined by type-promotion.

Return type:

array

_abc_impl = <_abc._abc_data object>¶

_is_protocol = True¶

_is_runtime_protocol = True¶

class array_api._2022_12.broadcast_arrays(*args, **kwargs)[source]¶

Bases: Protocol, Generic

Broadcasts one or more arrays against one another.

Parameters:: arrays (array) – an arbitrary number of to-be broadcasted arrays.
Returns:: out – a list of broadcasted arrays. Each array must have the same shape. Each array must have the same dtype as its corresponding input array.
Return type:: List[array]

_abc_impl = <_abc._abc_data object>¶

_is_protocol = True¶

_is_runtime_protocol = True¶

class array_api._2022_12.broadcast_to(*args, **kwargs)[source]¶

Bases: Protocol, Generic

Broadcasts an array to a specified shape.

Parameters:

x (array) – array to broadcast.
shape (Tuple[int, ...]) – array shape. Must be compatible with x (see broadcasting). If the array is incompatible with the specified shape, the function should raise an exception.

Returns:

out – an array having a specified shape. Must have the same data type as x.

Return type:

array

_abc_impl = <_abc._abc_data object>¶

_is_protocol = True¶

_is_runtime_protocol = True¶

class array_api._2022_12.can_cast(*args, **kwargs)[source]¶

Bases: Protocol, Generic

Determines if one data type can be cast to another data type according type-promotion rules.

Parameters:

from (Union[dtype, array]) – input data type or array from which to cast.
to (dtype) – desired data type.

Returns:

out – True if the cast can occur according to type-promotion rules; otherwise, False.

Return type:

bool

_abc_impl = <_abc._abc_data object>¶

_is_protocol = True¶

_is_runtime_protocol = True¶

class array_api._2022_12.ceil(*args, **kwargs)[source]¶

Bases: Protocol, Generic

Rounds each element x_i of the input array x to the smallest (i.e., closest to -infinity) integer-valued number that is not less than x_i.

Parameters:: x (array) – input array. Should have a real-valued data type.
Returns:: out – an array containing the rounded result for each element in x. The returned array must have the same data type as x.
Return type:: array

Notes

Special cases

If x_i is already integer-valued, the result is x_i.

For floating-point operands,

If x_i is +infinity, the result is +infinity.
If x_i is -infinity, the result is -infinity.
If x_i is +0, the result is +0.
If x_i is -0, the result is -0.
If x_i is NaN, the result is NaN.

_abc_impl = <_abc._abc_data object>¶

_is_protocol = True¶

_is_runtime_protocol = True¶

class array_api._2022_12.cholesky(*args, **kwargs)[source]¶

Bases: Protocol, Generic

Returns the lower (upper) Cholesky decomposition of a complex Hermitian or real symmetric positive-definite matrix x.

If x is real-valued, let $\mathbb{K}$ be the set of real numbers $\mathbb{R}$, and, if x is complex-valued, let $\mathbb{K}$ be the set of complex numbers $\mathbb{C}$.

The lower Cholesky decomposition of a complex Hermitian or real symmetric positive-definite matrix $x \in\ \mathbb{K}^{n \times n}$ is defined as

\[x = LL^{H} \qquad \text{L $\in\ \mathbb{K}^{n \times n}$}\]

where $L$ is a lower triangular matrix and $L^{H}$ is the conjugate transpose when $L$ is complex-valued and the transpose when $L$ is real-valued.

The upper Cholesky decomposition is defined similarly

\[x = U^{H}U \qquad \text{U $\in\ \mathbb{K}^{n \times n}$}\]

where $U$ is an upper triangular matrix.

When x is a stack of matrices, the function must compute the Cholesky decomposition for each matrix in the stack.

Note

Whether an array library explicitly checks whether an input array is Hermitian or a symmetric positive-definite matrix (or a stack of matrices) is implementation-defined.

Parameters:

x (array) – input array having shape (..., M, M) and whose innermost two dimensions form square complex Hermitian or real symmetric positive-definite matrices. Should have a floating-point data type.
upper (bool) – If True, the result must be the upper-triangular Cholesky factor $U$. If False, the result must be the lower-triangular Cholesky factor $L$. Default: False.

Returns:

Return type:

array

Notes

Changed in version 2022.12: Added complex data type support.

_abc_impl = <_abc._abc_data object>¶

_is_protocol = True¶

_is_runtime_protocol = True¶

class array_api._2022_12.concat(*args, **kwargs)[source]¶

Bases: Protocol, Generic

Joins a sequence of arrays along an existing axis.

Parameters:

arrays (Union[Tuple[array, ...], List[array]]) – input arrays to join. The arrays must have the same shape, except in the dimension specified by axis.
axis (Optional[int]) – axis along which the arrays will be joined. If axis is None, arrays must be flattened before concatenation. If axis is negative, the function must determine the axis along which to join by counting from the last dimension. Default: 0.

Returns:

Note

This specification leaves type promotion between data type families (i.e., intxx and floatxx) unspecified.

Return type:

array

_abc_impl = <_abc._abc_data object>¶

_is_protocol = True¶

_is_runtime_protocol = True¶

class array_api._2022_12.conj(*args, **kwargs)[source]¶

Bases: Protocol, Generic

Returns the complex conjugate for each element x_i of the input array x.

For complex numbers of the form

\[a + bj\]

the complex conjugate is defined as

\[a - bj\]

Hence, the returned complex conjugates must be computed by negating the imaginary component of each element x_i.

Parameters:

x (array) – input array. Should have a complex floating-point data type.

Returns:

out (array) – an array containing the element-wise results. The returned array must have the same data type as x.
.. versionadded:: 2022.12

_abc_impl = <_abc._abc_data object>¶

_is_protocol = True¶

_is_runtime_protocol = True¶

class array_api._2022_12.cos(*args, **kwargs)[source]¶

Bases: Protocol, Generic

Calculates an implementation-dependent approximation to the cosine for each element x_i of the input array x.

Each element x_i is assumed to be expressed in radians.

Note

The cosine is an entire function on the complex plane and has no branch cuts.

Note

For complex arguments, the mathematical definition of cosine is

\[\begin{split}\begin{align} \operatorname{cos}(x) &= \sum_{n=0}^\infty \frac{(-1)^n}{(2n)!} x^{2n} \\ &= \frac{e^{jx} + e^{-jx}}{2} \\ &= \operatorname{cosh}(jx) \end{align}\end{split}\]

where $\operatorname{cosh}$ is the hyperbolic cosine.

Parameters:: x (array) – input array whose elements are each expressed in radians. Should have a floating-point data type.
Returns:: out – an array containing the cosine of each element in x. The returned array must have a floating-point data type determined by type-promotion.
Return type:: array

Notes

Special cases

For real-valued floating-point operands,

If x_i is NaN, the result is NaN.
If x_i is +0, the result is 1.
If x_i is -0, the result is 1.
If x_i is +infinity, the result is NaN.
If x_i is -infinity, the result is NaN.

For complex floating-point operands, special cases must be handled as if the operation is implemented as cosh(x*1j).

Changed in version 2022.12: Added complex data type support.

_abc_impl = <_abc._abc_data object>¶

_is_protocol = True¶

_is_runtime_protocol = True¶

class array_api._2022_12.cosh(*args, **kwargs)[source]¶

Bases: Protocol, Generic

Calculates an implementation-dependent approximation to the hyperbolic cosine for each element x_i in the input array x.

The mathematical definition of the hyperbolic cosine is

\[\operatorname{cosh}(x) = \frac{e^x + e^{-x}}{2}\]

Note

The hyperbolic cosine is an entire function in the complex plane and has no branch cuts. The function is periodic, with period $2\pi j$, with respect to the imaginary component.

Parameters:: x (array) – input array whose elements each represent a hyperbolic angle. Should have a floating-point data type.
Returns:: out – an array containing the hyperbolic cosine of each element in x. The returned array must have a floating-point data type determined by type-promotion.
Return type:: array

Notes

Special cases

Note

For all operands, cosh(x) must equal cosh(-x).

For real-valued floating-point operands,

If x_i is NaN, the result is NaN.
If x_i is +0, the result is 1.
If x_i is -0, the result is 1.
If x_i is +infinity, the result is +infinity.
If x_i is -infinity, the result is +infinity.

For complex floating-point operands, let a = real(x_i), b = imag(x_i), and

Note

For complex floating-point operands, cosh(conj(x)) must equal conj(cosh(x)).

If a is +0 and b is +0, the result is 1 + 0j.
If a is +0 and b is +infinity, the result is NaN + 0j (sign of the imaginary component is unspecified).
If a is +0 and b is NaN, the result is NaN + 0j (sign of the imaginary component is unspecified).
If a is a nonzero finite number and b is +infinity, the result is NaN + NaN j.
If a is a nonzero finite number and b is NaN, the result is NaN + NaN j.
If a is +infinity and b is +0, the result is +infinity + 0j.
If a is +infinity and b is a nonzero finite number, the result is +infinity * cis(b).
If a is +infinity and b is +infinity, the result is +infinity + NaN j (sign of the real component is unspecified).
If a is +infinity and b is NaN, the result is +infinity + NaN j.
If a is NaN and b is either +0 or -0, the result is NaN + 0j (sign of the imaginary component is unspecified).
If a is NaN and b is a nonzero finite number, the result is NaN + NaN j.
If a is NaN and b is NaN, the result is NaN + NaN j.

where cis(v) is cos(v) + sin(v)*1j.

Changed in version 2022.12: Added complex data type support.

_abc_impl = <_abc._abc_data object>¶

_is_protocol = True¶

_is_runtime_protocol = True¶

class array_api._2022_12.cross(*args, **kwargs)[source]¶

Bases: Protocol, Generic

Returns the cross product of 3-element vectors.

Parameters:

x1 (array) – first input array. Must have a numeric data type.
x2 (array) –
second input array. Must be compatible with x1 for all non-compute axes (see broadcasting). The size of the axis over which to compute the cross product must be the same size as the respective axis in x1. Must have a numeric data type.

Note

The compute axis (dimension) must not be broadcasted.
axis (int) – the axis (dimension) of x1 and x2 containing the vectors for which to compute the cross product. Must be an integer on the interval [-N, N), where N is the rank (number of dimensions) of the shape determined according to broadcasting. If specified as a negative integer, the function must determine the axis along which to compute the cross product by counting backward from the last dimension (where -1 refers to the last dimension). By default, the function must compute the cross product over the last axis. Default: -1.

Returns:

out – an array containing the cross products. The returned array must have a data type determined by type-promotion.

Return type:

array

Notes

Changed in version 2022.12: Added support for broadcasting.

Changed in version 2022.12: Added complex data type support.

Raises

if provided an invalid axis.
if the size of the axis over which to compute the cross product is not equal to 3.
if the size of the axis over which to compute the cross product is not the same (before broadcasting) for both x1 and x2.

_abc_impl = <_abc._abc_data object>¶

_is_protocol = True¶

_is_runtime_protocol = True¶

class array_api._2022_12.det(*args, **kwargs)[source]¶

Bases: Protocol, Generic

Returns the determinant of a square matrix (or a stack of square matrices) x.

Parameters:: x (array) – input array having shape (..., M, M) and whose innermost two dimensions form square matrices. Should have a floating-point data type.
Returns:: out – if x is a two-dimensional array, a zero-dimensional array containing the determinant; otherwise, a non-zero dimensional array containing the determinant for each square matrix. The returned array must have the same data type as x.
Return type:: array

Notes

Changed in version 2022.12: Added complex data type support.

_abc_impl = <_abc._abc_data object>¶

_is_protocol = True¶

_is_runtime_protocol = True¶

class array_api._2022_12.diagonal(*args, **kwargs)[source]¶

Bases: Protocol, Generic

Returns the specified diagonals of a matrix (or a stack of matrices) x.

Parameters:

x (array) – input array having shape (..., M, N) and whose innermost two dimensions form MxN matrices.
offset (int) –
offset specifying the off-diagonal relative to the main diagonal.
- offset = 0: the main diagonal.
- offset > 0: off-diagonal above the main diagonal.
- offset < 0: off-diagonal below the main diagonal.
Default: 0.

Returns:

Return type:

array

_abc_impl = <_abc._abc_data object>¶

_is_protocol = True¶

_is_runtime_protocol = True¶

class array_api._2022_12.divide(*args, **kwargs)[source]¶

Bases: Protocol, Generic

Calculates the division of each element x1_i of the input array x1 with the respective element x2_i of the input array x2.

Note

Parameters:

x1 (array) – dividend input array. Should have a numeric data type.
x2 (array) – divisor input array. Must be compatible with x1 (see broadcasting). Should have a numeric data type.

Returns:

out – an array containing the element-wise results. The returned array must have a floating-point data type determined by type-promotion.

Return type:

array

Notes

Special cases

For real-valued floating-point operands,

If either x1_i or x2_i is NaN, the result is NaN.
If x1_i is either +infinity or -infinity and x2_i is either +infinity or -infinity, the result is NaN.
If x1_i is either +0 or -0 and x2_i is either +0 or -0, the result is NaN.
If x1_i is +0 and x2_i is greater than 0, the result is +0.
If x1_i is -0 and x2_i is greater than 0, the result is -0.
If x1_i is +0 and x2_i is less than 0, the result is -0.
If x1_i is -0 and x2_i is less than 0, the result is +0.
If x1_i is greater than 0 and x2_i is +0, the result is +infinity.
If x1_i is greater than 0 and x2_i is -0, the result is -infinity.
If x1_i is less than 0 and x2_i is +0, the result is -infinity.
If x1_i is less than 0 and x2_i is -0, the result is +infinity.
If x1_i is +infinity and x2_i is a positive (i.e., greater than 0) finite number, the result is +infinity.
If x1_i is +infinity and x2_i is a negative (i.e., less than 0) finite number, the result is -infinity.
If x1_i is -infinity and x2_i is a positive (i.e., greater than 0) finite number, the result is -infinity.
If x1_i is -infinity and x2_i is a negative (i.e., less than 0) finite number, the result is +infinity.
If x1_i is a positive (i.e., greater than 0) finite number and x2_i is +infinity, the result is +0.
If x1_i is a positive (i.e., greater than 0) finite number and x2_i is -infinity, the result is -0.
If x1_i is a negative (i.e., less than 0) finite number and x2_i is +infinity, the result is -0.
If x1_i is a negative (i.e., less than 0) finite number and x2_i is -infinity, the result is +0.
If x1_i and x2_i have the same mathematical sign and are both nonzero finite numbers, the result has a positive mathematical sign.
If x1_i and x2_i have different mathematical signs and are both nonzero finite numbers, the result has a negative mathematical sign.
In the remaining cases, where neither -infinity, +0, -0, nor NaN is involved, the quotient must be computed and rounded to the nearest representable value according to IEEE 754-2019 and a supported rounding mode. If the magnitude is too large to represent, the operation overflows and the result is an infinity of appropriate mathematical sign. If the magnitude is too small to represent, the operation underflows and the result is a zero of appropriate mathematical sign.

For complex floating-point operands, division is defined according to the following table. For real components a and c and imaginary components b and d,

	c	dj	c + dj
a	a / c	-(a/d)j	special rules
bj	(b/c)j	b/d	special rules
a + bj	(a/c) + (b/c)j	b/d - (a/d)j	special rules

\[\frac{a + bj}{c + dj} = \frac{(ac + bd) + (bc - ad)j}{c^2 + d^2}\]

When at least one of a, b, c, or d is NaN, +infinity, or -infinity,

If a, b, c, and d are all NaN, the result is NaN + NaN j.
In the remaining cases, the result is implementation dependent.

Note

Changed in version 2022.12: Added complex data type support.

_abc_impl = <_abc._abc_data object>¶

_is_protocol = True¶

_is_runtime_protocol = True¶

class array_api._2022_12.eigh(*args, **kwargs)[source]¶

Bases: Protocol, Generic

Returns an eigenvalue decomposition of a complex Hermitian or real symmetric matrix (or a stack of matrices) x.

If x is real-valued, let $\mathbb{K}$ be the set of real numbers $\mathbb{R}$, and, if x is complex-valued, let $\mathbb{K}$ be the set of complex numbers $\mathbb{C}$.

The eigenvalue decomposition of a complex Hermitian or real symmetric matrix $x \in\ \mathbb{K}^{n \times n}$ is defined as

\[x = Q \Lambda Q^H\]

Note

The eigenvalues of a complex Hermitian or real symmetric matrix are always real.

Warning

Note

Whether an array library explicitly checks whether an input array is Hermitian or a symmetric matrix (or a stack of matrices) is implementation-defined.

Note

The function eig will be added in a future version of the specification.

Parameters:

x (array) – input array having shape (..., M, M) and whose innermost two dimensions form square matrices. Should have a floating-point data type.

Returns:

out – a namedtuple (eigenvalues, eigenvectors) whose

first element must have the field name eigenvalues (corresponding to $\operatorname{diag}\Lambda$ above) and must be an array consisting of computed eigenvalues. The array containing the eigenvalues must have shape (..., M) and must have a real-valued floating-point data type whose precision matches the precision of x (e.g., if x is complex128, then eigenvalues must be float64).
second element have have the field name eigenvectors (corresponding to $Q$ above) and must be an array where the columns of the inner most matrices contain the computed eigenvectors. These matrices must be orthogonal. The array containing the eigenvectors must have shape (..., M, M) and must have the same data type as x.

Return type:

Tuple[array, array]

Notes

Note

Eigenvalue sort order is left unspecified and is thus implementation-dependent.

Changed in version 2022.12: Added complex data type support.

_abc_impl = <_abc._abc_data object>¶

_is_protocol = True¶

_is_runtime_protocol = True¶

class array_api._2022_12.eigvalsh(*args, **kwargs)[source]¶

Bases: Protocol, Generic

Returns the eigenvalues of a complex Hermitian or real symmetric matrix (or a stack of matrices) x.

If x is real-valued, let $\mathbb{K}$ be the set of real numbers $\mathbb{R}$, and, if x is complex-valued, let $\mathbb{K}$ be the set of complex numbers $\mathbb{C}$.

\[p(\lambda) = \operatorname{det}(x - \lambda I_n)\]

where $\lambda \in \mathbb{R}$ and where $I_n$ is the n-dimensional identity matrix.

Note

Whether an array library explicitly checks whether an input array is Hermitian or a symmetric matrix (or a stack of matrices) is implementation-defined.

Note

The function eigvals will be added in a future version of the specification.

Parameters:: x (array) – input array having shape (..., M, M) and whose innermost two dimensions form square matrices. Should have a floating-point data type.
Returns:: out – an array containing the computed eigenvalues. The returned array must have shape (..., M) and have a real-valued floating-point data type whose precision matches the precision of x (e.g., if x is complex128, then must have a float64 data type).
Return type:: array

Notes

Note

Eigenvalue sort order is left unspecified and is thus implementation-dependent.

Changed in version 2022.12: Added complex data type support.

_abc_impl = <_abc._abc_data object>¶

_is_protocol = True¶

_is_runtime_protocol = True¶

class array_api._2022_12.empty(*args, **kwargs)[source]¶

Bases: Protocol, Generic

Returns an uninitialized array having a specified shape.

Parameters:

shape (Union[int, Tuple[int, ...]]) – output array shape.
dtype (Optional[dtype]) – output array data type. If dtype is None, the output array data type must be the default real-valued floating-point data type. Default: None.
device (Optional[device]) – device on which to place the created array. Default: None.

Returns:

out – an array containing uninitialized data.

Return type:

array

_abc_impl = <_abc._abc_data object>¶

_is_protocol = True¶

_is_runtime_protocol = True¶

class array_api._2022_12.empty_like(*args, **kwargs)[source]¶

Bases: Protocol, Generic

Returns an uninitialized array with the same shape as an input array x.

Parameters:

x (array) – input array from which to derive the output array shape.
dtype (Optional[dtype]) – output array data type. If dtype is None, the output array data type must be inferred from x. Default: None.
device (Optional[device]) – device on which to place the created array. If device is None, the output array device must be inferred from x. Default: None.

Returns:

out – an array having the same shape as x and containing uninitialized data.

Return type:

array

_abc_impl = <_abc._abc_data object>¶

_is_protocol = True¶

_is_runtime_protocol = True¶

class array_api._2022_12.equal(*args, **kwargs)[source]¶

Bases: Protocol, Generic

Computes the truth value of x1_i == x2_i for each element x1_i of the input array x1 with the respective element x2_i of the input array x2.

Parameters:

x1 (array) – first input array. May have any data type.
x2 (array) – second input array. Must be compatible with x1 (see broadcasting). May have any data type.

Returns:

out – an array containing the element-wise results. The returned array must have a data type of bool.

Return type:

array

Notes

Special Cases

For real-valued floating-point operands,

If x1_i is NaN or x2_i is NaN, the result is False.
If x1_i is +infinity and x2_i is +infinity, the result is True.
If x1_i is -infinity and x2_i is -infinity, the result is True.
If x1_i is -0 and x2_i is either +0 or -0, the result is True.
If x1_i is +0 and x2_i is either +0 or -0, the result is True.
If x1_i is a finite number, x2_i is a finite number, and x1_i equals x2_i, the result is True.
In the remaining cases, the result is False.

For complex floating-point operands, let a = real(x1_i), b = imag(x1_i), c = real(x2_i), d = imag(x2_i), and

If a, b, c, or d is NaN, the result is False.
In the remaining cases, the result is the logical AND of the equality comparison between the real values a and c (real components) and between the real values b and d (imaginary components), as described above for real-valued floating-point operands (i.e., a == c AND b == d).

Note

For discussion of complex number equality, see complex-numbers.

Changed in version 2022.12: Added complex data type support.

_abc_impl = <_abc._abc_data object>¶

_is_protocol = True¶

_is_runtime_protocol = True¶

class array_api._2022_12.exp(*args, **kwargs)[source]¶

Bases: Protocol, Generic

Note

For complex floating-point operands, exp(conj(x)) must equal conj(exp(x)).

Note

The exponential function is an entire function in the complex plane and has no branch cuts.

Parameters:: x (array) – input array. Should have a floating-point data type.
Returns:: out – an array containing the evaluated exponential function result for each element in x. The returned array must have a floating-point data type determined by type-promotion.
Return type:: array

Notes

Special cases

For real-valued floating-point operands,

If x_i is NaN, the result is NaN.
If x_i is +0, the result is 1.
If x_i is -0, the result is 1.
If x_i is +infinity, the result is +infinity.
If x_i is -infinity, the result is +0.

For complex floating-point operands, let a = real(x_i), b = imag(x_i), and

If a is either +0 or -0 and b is +0, the result is 1 + 0j.
If a is a finite number and b is +infinity, the result is NaN + NaN j.
If a is a finite number and b is NaN, the result is NaN + NaN j.
If a is +infinity and b is +0, the result is infinity + 0j.
If a is -infinity and b is a finite number, the result is +0 * cis(b).
If a is +infinity and b is a nonzero finite number, the result is +infinity * cis(b).
If a is -infinity and b is +infinity, the result is 0 + 0j (signs of real and imaginary components are unspecified).
If a is +infinity and b is +infinity, the result is infinity + NaN j (sign of real component is unspecified).
If a is -infinity and b is NaN, the result is 0 + 0j (signs of real and imaginary components are unspecified).
If a is +infinity and b is NaN, the result is infinity + NaN j (sign of real component is unspecified).
If a is NaN and b is +0, the result is NaN + 0j.
If a is NaN and b is not equal to 0, the result is NaN + NaN j.
If a is NaN and b is NaN, the result is NaN + NaN j.

where cis(v) is cos(v) + sin(v)*1j.

Changed in version 2022.12: Added complex data type support.

_abc_impl = <_abc._abc_data object>¶

_is_protocol = True¶

_is_runtime_protocol = True¶

class array_api._2022_12.expand_dims(*args, **kwargs)[source]¶

Bases: Protocol, Generic

Expands the shape of an array by inserting a new axis (dimension) of size one at the position specified by axis.

Parameters:

x (array) – input array.
axis (int) – axis position (zero-based). If x has rank (i.e, number of dimensions) N, a valid axis must reside on the closed-interval [-N-1, N]. If provided a negative axis, the axis position at which to insert a singleton dimension must be computed as N + axis + 1. Hence, if provided -1, the resolved axis position must be N (i.e., a singleton dimension must be appended to the input array x). If provided -N-1, the resolved axis position must be 0 (i.e., a singleton dimension must be prepended to the input array x). An IndexError exception must be raised if provided an invalid axis position.

Returns:

out – an expanded output array having the same data type as x.

Return type:

array

_abc_impl = <_abc._abc_data object>¶

_is_protocol = True¶

_is_runtime_protocol = True¶

class array_api._2022_12.expm1(*args, **kwargs)[source]¶

Bases: Protocol, Generic

Calculates an implementation-dependent approximation to exp(x)-1 for each element x_i of the input array x.

Note

For complex floating-point operands, expm1(conj(x)) must equal conj(expm1(x)).

Note

The exponential function is an entire function in the complex plane and has no branch cuts.

Parameters:: x (array) – input array. Should have a floating-point data type.
Returns:: out – an array containing the evaluated result for each element in x. The returned array must have a floating-point data type determined by type-promotion.
Return type:: array

Notes

Special cases

For real-valued floating-point operands,

If x_i is NaN, the result is NaN.
If x_i is +0, the result is +0.
If x_i is -0, the result is -0.
If x_i is +infinity, the result is +infinity.
If x_i is -infinity, the result is -1.

For complex floating-point operands, let a = real(x_i), b = imag(x_i), and

If a is either +0 or -0 and b is +0, the result is 0 + 0j.
If a is a finite number and b is +infinity, the result is NaN + NaN j.
If a is a finite number and b is NaN, the result is NaN + NaN j.
If a is +infinity and b is +0, the result is +infinity + 0j.
If a is -infinity and b is a finite number, the result is +0 * cis(b) - 1.0.
If a is +infinity and b is a nonzero finite number, the result is +infinity * cis(b) - 1.0.
If a is -infinity and b is +infinity, the result is -1 + 0j (sign of imaginary component is unspecified).
If a is +infinity and b is +infinity, the result is infinity + NaN j (sign of real component is unspecified).
If a is -infinity and b is NaN, the result is -1 + 0j (sign of imaginary component is unspecified).
If a is +infinity and b is NaN, the result is infinity + NaN j (sign of real component is unspecified).
If a is NaN and b is +0, the result is NaN + 0j.
If a is NaN and b is not equal to 0, the result is NaN + NaN j.
If a is NaN and b is NaN, the result is NaN + NaN j.

where cis(v) is cos(v) + sin(v)*1j.

Changed in version 2022.12: Added complex data type support.

_abc_impl = <_abc._abc_data object>¶

_is_protocol = True¶

_is_runtime_protocol = True¶

class array_api._2022_12.eye(*args, **kwargs)[source]¶

Bases: Protocol, Generic

Returns a two-dimensional array with ones on the kth diagonal and zeros elsewhere.

Note

An output array having a complex floating-point data type must have the value 1 + 0j along the kth diagonal and 0 + 0j elsewhere.

Parameters:

n_rows (int) – number of rows in the output array.
n_cols (Optional[int]) – number of columns in the output array. If None, the default number of columns in the output array is equal to n_rows. Default: None.
k (int) – index of the diagonal. A positive value refers to an upper diagonal, a negative value to a lower diagonal, and 0 to the main diagonal. Default: 0.
dtype (Optional[dtype]) – output array data type. If dtype is None, the output array data type must be the default real-valued floating-point data type. Default: None.
device (Optional[device]) – device on which to place the created array. Default: None.

Returns:

out – an array where all elements are equal to zero, except for the kth diagonal, whose values are equal to one.

Return type:

array

Notes

Changed in version 2022.12: Added complex data type support.

_abc_impl = <_abc._abc_data object>¶

_is_protocol = True¶

_is_runtime_protocol = True¶

class array_api._2022_12.fft(*args, **kwargs)[source]¶

Bases: Protocol, Generic

Computes the one-dimensional discrete Fourier transform.

Note

Parameters:

x (array) – input array. Should have a complex floating-point data type.
n (Optional[int]) –
number of elements over which to compute the transform along the axis (dimension) specified by axis. Let M be the size of the input array along the axis specified by axis. When n is None, the function must set n equal to M.
- If n is greater than M, the axis specified by axis must be zero-padded to size n.
- If n is less than M, the axis specified by axis must be trimmed to size n.
- If n equals M, all elements along the axis specified by axis must be used when computing the transform.
Default: None.
axis (int) – axis (dimension) of the input array over which to compute the transform. A valid axis must be an integer on the interval [-N, N), where N is the rank (number of dimensions) of x. If an axis is specified as a negative integer, the function must determine the axis along which to compute the transform by counting backward from the last dimension (where -1 refers to the last dimension). Default: -1.
norm (Literal['backward', 'ortho', 'forward']) –
normalization mode. Should be one of the following modes:
- 'backward': no normalization.
- 'ortho': normalize by 1/sqrt(n) (i.e., make the FFT orthonormal).
- 'forward': normalize by 1/n.
Default: 'backward'.

Returns:

Return type:

array

Notes

Added in version 2022.12.

_abc_impl = <_abc._abc_data object>¶

_is_protocol = True¶

_is_runtime_protocol = True¶

class array_api._2022_12.fftfreq(*args, **kwargs)[source]¶

Bases: Protocol, Generic

Computes the discrete Fourier transform sample frequencies.

For a Fourier transform of length n and length unit of d, the frequencies are described as:

f = [0, 1, ..., n/2-1, -n/2, ..., -1] / (d*n)        # if n is even
f = [0, 1, ..., (n-1)/2, -(n-1)/2, ..., -1] / (d*n)  # if n is odd

Parameters:

n (int) – window length.
d (float) – sample spacing between individual samples of the Fourier transform input. Default: 1.0.
device (Optional[device]) – device on which to place the created array. Default: None.

Returns:

out – an array of shape (n,) containing the sample frequencies. The returned array must have the default real-valued floating-point data type.

Return type:

array

Notes

Added in version 2022.12.

_abc_impl = <_abc._abc_data object>¶

_is_protocol = True¶

_is_runtime_protocol = True¶

class array_api._2022_12.fftn(*args, **kwargs)[source]¶

Bases: Protocol, Generic

Computes the n-dimensional discrete Fourier transform.

Note

Parameters:

x (array) – input array. Should have a complex floating-point data type.
s (Optional[Sequence[int]]) –
number of elements over which to compute the transform along the axes (dimensions) specified by axes. Let i be the index of the n-th axis specified by axes (i.e., i = axes[n]) and M[i] be the size of the input array along axis i. When s is None, the function must set s equal to a sequence of integers such that s[i] equals M[i] for all i.
- If s[i] is greater than M[i], axis i must be zero-padded to size s[i].
- If s[i] is less than M[i], axis i must be trimmed to size s[i].
- If s[i] equals M[i] or -1, all elements along axis i must be used when computing the transform.
If s is not None, axes must not be None. Default: None.
axes (Optional[Sequence[int]]) –
axes (dimensions) over which to compute the transform. A valid axis in axes must be an integer on the interval [-N, N), where N is the rank (number of dimensions) of x. If an axis is specified as a negative integer, the function must determine the axis along which to compute the transform by counting backward from the last dimension (where -1 refers to the last dimension).

If s is provided, the corresponding axes to be transformed must also be provided. If axes is None, the function must compute the transform over all axes. Default: None.

If axes contains two or more entries which resolve to the same axis (i.e., resolved axes are not unique), the behavior is unspecified and thus implementation-defined.
norm (Literal['backward', 'ortho', 'forward']) –
normalization mode. Should be one of the following modes:
- 'backward': no normalization.
- 'ortho': normalize by 1/sqrt(n) (i.e., make the FFT orthonormal).
- 'forward': normalize by 1/n.
where n = prod(s) is the logical FFT size.

Default: 'backward'.

Returns:

Return type:

array

Notes

Added in version 2022.12.

_abc_impl = <_abc._abc_data object>¶

_is_protocol = True¶

_is_runtime_protocol = True¶

class array_api._2022_12.fftshift(*args, **kwargs)[source]¶

Bases: Protocol, Generic

Shifts the zero-frequency component to the center of the spectrum.

This function swaps half-spaces for all axes (dimensions) specified by axes.

Note

out[0] is the Nyquist component only if the length of the input is even.

Parameters:

x (array) – input array. Should have a floating-point data type.
axes (Optional[Union[int, Sequence[int]]]) –
axes over which to shift. If None, the function must shift all axes. Default: None.

If axes contains two or more entries which resolve to the same axis (i.e., resolved axes are not unique), the behavior is unspecified and thus implementation-defined.

Returns:

out – the shifted array. The returned array must have the same data type and shape as x.

Return type:

array

Notes

Added in version 2022.12.

_abc_impl = <_abc._abc_data object>¶

_is_protocol = True¶

_is_runtime_protocol = True¶

class array_api._2022_12.finfo(*args, **kwargs)[source]¶

Bases: Protocol, Generic

Machine limits for floating-point data types.

Parameters:

type (Union[dtype, array]) –

the kind of floating-point data-type about which to get information. If complex, the information is about its component data type.

Note

Complex floating-point data types are specified to always use the same precision for both its real and imaginary components, so the information should be true for either component.

Returns:

out – an object having the following attributes:

bits: int

number of bits occupied by the real-valued floating-point data type.
eps: float

difference between 1.0 and the next smallest representable real-valued floating-point number larger than 1.0 according to the IEEE-754 standard.
max: float

largest representable real-valued number.
min: float

smallest representable real-valued number.
smallest_normal: float

smallest positive real-valued floating-point number with full precision.
dtype: dtype

real-valued floating-point data type.

Added in version 2022.12.

Return type:

finfo object

Notes

Changed in version 2022.12: Added complex data type support.

_abc_impl = <_abc._abc_data object>¶

_is_protocol = True¶

_is_runtime_protocol = True¶

class array_api._2022_12.finfo_object(*args, **kwargs)[source]¶

Bases: Protocol, Generic

Dataclass returned by finfo.

_abc_impl = <_abc._abc_data object>¶

_is_protocol = True¶

_is_runtime_protocol = True¶

bits: int¶

dtype: TDtype¶

eps: float¶

max: float¶

min: float¶

smallest_normal: float¶

class array_api._2022_12.flip(*args, **kwargs)[source]¶

Bases: Protocol, Generic

Reverses the order of elements in an array along the given axis. The shape of the array must be preserved.

Parameters:

x (array) – input array.
axis (Optional[Union[int, Tuple[int, ...]]]) – axis (or axes) along which to flip. If axis is None, the function must flip all input array axes. If axis is negative, the function must count from the last dimension. If provided more than one axis, the function must flip only the specified axes. Default: None.

Returns:

out – an output array having the same data type and shape as x and whose elements, relative to x, are reordered.

Return type:

array

_abc_impl = <_abc._abc_data object>¶

_is_protocol = True¶

_is_runtime_protocol = True¶

class array_api._2022_12.floor(*args, **kwargs)[source]¶

Bases: Protocol, Generic

Rounds each element x_i of the input array x to the greatest (i.e., closest to +infinity) integer-valued number that is not greater than x_i.

Parameters:: x (array) – input array. Should have a real-valued data type.
Returns:: out – an array containing the rounded result for each element in x. The returned array must have the same data type as x.
Return type:: array

Notes

Special cases

If x_i is already integer-valued, the result is x_i.

For floating-point operands,

If x_i is +infinity, the result is +infinity.
If x_i is -infinity, the result is -infinity.
If x_i is +0, the result is +0.
If x_i is -0, the result is -0.
If x_i is NaN, the result is NaN.

_abc_impl = <_abc._abc_data object>¶

_is_protocol = True¶

_is_runtime_protocol = True¶

class array_api._2022_12.floor_divide(*args, **kwargs)[source]¶

Bases: Protocol, Generic

Note

For input arrays which promote to an integer data type, the result of division by zero is unspecified and thus implementation-defined.

Parameters:

x1 (array) – dividend input array. Should have a real-valued data type.
x2 (array) – divisor input array. Must be compatible with x1 (see broadcasting). Should have a real-valued data type.

Returns:

out – an array containing the element-wise results. The returned array must have a data type determined by type-promotion.

Return type:

array

Notes

Special cases

Note

For floating-point operands,

If either x1_i or x2_i is NaN, the result is NaN.
If x1_i is either +infinity or -infinity and x2_i is either +infinity or -infinity, the result is NaN.
If x1_i is either +0 or -0 and x2_i is either +0 or -0, the result is NaN.
If x1_i is +0 and x2_i is greater than 0, the result is +0.
If x1_i is -0 and x2_i is greater than 0, the result is -0.
If x1_i is +0 and x2_i is less than 0, the result is -0.
If x1_i is -0 and x2_i is less than 0, the result is +0.
If x1_i is greater than 0 and x2_i is +0, the result is +infinity.
If x1_i is greater than 0 and x2_i is -0, the result is -infinity.
If x1_i is less than 0 and x2_i is +0, the result is -infinity.
If x1_i is less than 0 and x2_i is -0, the result is +infinity.
If x1_i is +infinity and x2_i is a positive (i.e., greater than 0) finite number, the result is +infinity. (note: libraries may return NaN to match Python behavior.)
If x1_i is +infinity and x2_i is a negative (i.e., less than 0) finite number, the result is -infinity. (note: libraries may return NaN to match Python behavior.)
If x1_i is -infinity and x2_i is a positive (i.e., greater than 0) finite number, the result is -infinity. (note: libraries may return NaN to match Python behavior.)
If x1_i is -infinity and x2_i is a negative (i.e., less than 0) finite number, the result is +infinity. (note: libraries may return NaN to match Python behavior.)
If x1_i is a positive (i.e., greater than 0) finite number and x2_i is +infinity, the result is +0.
If x1_i is a positive (i.e., greater than 0) finite number and x2_i is -infinity, the result is -0. (note: libraries may return -1.0 to match Python behavior.)
If x1_i is a negative (i.e., less than 0) finite number and x2_i is +infinity, the result is -0. (note: libraries may return -1.0 to match Python behavior.)
If x1_i is a negative (i.e., less than 0) finite number and x2_i is -infinity, the result is +0.
If x1_i and x2_i have the same mathematical sign and are both nonzero finite numbers, the result has a positive mathematical sign.
If x1_i and x2_i have different mathematical signs and are both nonzero finite numbers, the result has a negative mathematical sign.
In the remaining cases, where neither -infinity, +0, -0, nor NaN is involved, the quotient must be computed and rounded to the greatest (i.e., closest to +infinity) representable integer-value number that is not greater than the division result. If the magnitude is too large to represent, the operation overflows and the result is an infinity of appropriate mathematical sign. If the magnitude is too small to represent, the operation underflows and the result is a zero of appropriate mathematical sign.

_abc_impl = <_abc._abc_data object>¶

_is_protocol = True¶

_is_runtime_protocol = True¶

class array_api._2022_12.from_dlpack(*args, **kwargs)[source]¶

Bases: Protocol, Generic

Returns a new array containing the data from another (array) object with a __dlpack__ method.

Parameters:

x (object) – input (array) object.

Returns:

out – an array containing the data in x.

Note

The returned array may be either a copy or a view. See data-interchange for details.

Return type:

array

_abc_impl = <_abc._abc_data object>¶

_is_protocol = True¶

_is_runtime_protocol = True¶

class array_api._2022_12.full(*args, **kwargs)[source]¶

Bases: Protocol, Generic

Returns a new array having a specified shape and filled with fill_value.

Parameters:

shape (Union[int, Tuple[int, ...]]) – output array shape.
fill_value (Union[bool, int, float, complex]) – fill value.
dtype (Optional[dtype]) –
output array data type. If dtype is None, the output array data type must be inferred from fill_value according to the following rules:
- If the fill value is an int, the output array data type must be the default integer data type.
- If the fill value is a float, the output array data type must be the default real-valued floating-point data type.
- If the fill value is a complex number, the output array data type must be the default complex floating-point data type.
- If the fill value is a bool, the output array must have a boolean data type. Default: None.
Note

If the fill_value exceeds the precision of the resolved default output array data type, behavior is left unspecified and, thus, implementation-defined.
device (Optional[device]) – device on which to place the created array. Default: None.

Returns:

out – an array where every element is equal to fill_value.

Return type:

array

Notes

Changed in version 2022.12: Added complex data type support.

_abc_impl = <_abc._abc_data object>¶

_is_protocol = True¶

_is_runtime_protocol = True¶

class array_api._2022_12.full_like(*args, **kwargs)[source]¶

Bases: Protocol, Generic

Returns a new array filled with fill_value and having the same shape as an input array x.

Parameters:

x (array) – input array from which to derive the output array shape.
fill_value (Union[bool, int, float, complex]) – fill value.
dtype (Optional[dtype]) –
output array data type. If dtype is None, the output array data type must be inferred from x. Default: None.

Note

If the fill_value exceeds the precision of the resolved output array data type, behavior is unspecified and, thus, implementation-defined.

Note

If the fill_value has a data type which is not of the same data type kind (boolean, integer, or floating-point) as the resolved output array data type (see type-promotion), behavior is unspecified and, thus, implementation-defined.
device (Optional[device]) – device on which to place the created array. If device is None, the output array device must be inferred from x. Default: None.

Returns:

out – an array having the same shape as x and where every element is equal to fill_value.

Return type:

array

Notes

Changed in version 2022.12: Added complex data type support.

_abc_impl = <_abc._abc_data object>¶

_is_protocol = True¶

_is_runtime_protocol = True¶

class array_api._2022_12.greater(*args, **kwargs)[source]¶

Bases: Protocol, Generic

Computes the truth value of x1_i > x2_i for each element x1_i of the input array x1 with the respective element x2_i of the input array x2.

Note

Parameters:

x1 (array) – first input array. Should have a real-valued data type.
x2 (array) – second input array. Must be compatible with x1 (see broadcasting). Should have a real-valued data type.

Returns:

out – an array containing the element-wise results. The returned array must have a data type of bool.

Return type:

array

_abc_impl = <_abc._abc_data object>¶

_is_protocol = True¶

_is_runtime_protocol = True¶

class array_api._2022_12.greater_equal(*args, **kwargs)[source]¶

Bases: Protocol, Generic

Computes the truth value of x1_i >= x2_i for each element x1_i of the input array x1 with the respective element x2_i of the input array x2.

Note

Parameters:

x1 (array) – first input array. Should have a real-valued data type.
x2 (array) – second input array. Must be compatible with x1 (see broadcasting). Should have a real-valued data type.

Returns:

out – an array containing the element-wise results. The returned array must have a data type of bool.

Return type:

array

_abc_impl = <_abc._abc_data object>¶

_is_protocol = True¶

_is_runtime_protocol = True¶

class array_api._2022_12.hfft(*args, **kwargs)[source]¶

Bases: Protocol, Generic

Computes the one-dimensional discrete Fourier transform of a signal with Hermitian symmetry.

Parameters:

x (array) – input array. Should have a complex floating-point data type.
n (Optional[int]) –
number of elements along the transformed axis (dimension) specified by axis in the output array. Let M be the size of the input array along the axis specified by axis. When n is None, the function must set n equal to 2*(M-1).
- If n//2+1 is greater than M, the axis of the input array specified by axis must be zero-padded to length n//2+1.
- If n//2+1 is less than M, the axis of the input array specified by axis must be trimmed to size n//2+1.
- If n//2+1 equals M, all elements along the axis of the input array specified by axis must be used when computing the transform.
Default: None.
axis (int) – axis (dimension) of the input array over which to compute the transform. A valid axis must be an integer on the interval [-N, N), where N is the rank (number of dimensions) of x. If an axis is specified as a negative integer, the function must determine the axis along which to compute the transform by counting backward from the last dimension (where -1 refers to the last dimension). Default: -1.
norm (Literal['backward', 'ortho', 'forward']) –
normalization mode. Should be one of the following modes:
- 'backward': no normalization.
- 'ortho': normalize by 1/sqrt(n) (i.e., make the FFT orthonormal).
- 'forward': normalize by 1/n.
Default: 'backward'.

Returns:

Return type:

array

Notes

Added in version 2022.12.

_abc_impl = <_abc._abc_data object>¶

_is_protocol = True¶

_is_runtime_protocol = True¶

class array_api._2022_12.ifft(*args, **kwargs)[source]¶

Bases: Protocol, Generic

Computes the one-dimensional inverse discrete Fourier transform.

Note

Parameters:

x (array) – input array. Should have a complex floating-point data type.
n (Optional[int]) –
number of elements over which to compute the transform along the axis (dimension) specified by axis. Let M be the size of the input array along the axis specified by axis. When n is None, the function must set n equal to M.
- If n is greater than M, the axis specified by axis must be zero-padded to size n.
- If n is less than M, the axis specified by axis must be trimmed to size n.
- If n equals M, all elements along the axis specified by axis must be used when computing the transform.
Default: None.
axis (int) – axis (dimension) of the input array over which to compute the transform. A valid axis must be an integer on the interval [-N, N), where N is the rank (number of dimensions) of x. If an axis is specified as a negative integer, the function must determine the axis along which to compute the transform by counting backward from the last dimension (where -1 refers to the last dimension). Default: -1.
norm (Literal['backward', 'ortho', 'forward']) –
normalization mode. Should be one of the following modes:
- 'backward': normalize by 1/n.
- 'ortho': normalize by 1/sqrt(n) (i.e., make the FFT orthonormal).
- 'forward': no normalization.
Default: 'backward'.

Returns:

Return type:

array

Notes

Added in version 2022.12.

_abc_impl = <_abc._abc_data object>¶

_is_protocol = True¶

_is_runtime_protocol = True¶

class array_api._2022_12.ifftn(*args, **kwargs)[source]¶

Bases: Protocol, Generic

Computes the n-dimensional inverse discrete Fourier transform.

Note

Parameters:

x (array) – input array. Should have a complex floating-point data type.
s (Optional[Sequence[int]]) –
number of elements over which to compute the transform along the axes (dimensions) specified by axes. Let i be the index of the n-th axis specified by axes (i.e., i = axes[n]) and M[i] be the size of the input array along axis i. When s is None, the function must set s equal to a sequence of integers such that s[i] equals M[i] for all i.
- If s[i] is greater than M[i], axis i must be zero-padded to size s[i].
- If s[i] is less than M[i], axis i must be trimmed to size s[i].
- If s[i] equals M[i] or -1, all elements along axis i must be used when computing the transform.
If s is not None, axes must not be None. Default: None.
axes (Optional[Sequence[int]]) –
axes (dimensions) over which to compute the transform. A valid axis in axes must be an integer on the interval [-N, N), where N is the rank (number of dimensions) of x. If an axis is specified as a negative integer, the function must determine the axis along which to compute the transform by counting backward from the last dimension (where -1 refers to the last dimension).

If s is provided, the corresponding axes to be transformed must also be provided. If axes is None, the function must compute the transform over all axes. Default: None.

If axes contains two or more entries which resolve to the same axis (i.e., resolved axes are not unique), the behavior is unspecified and thus implementation-defined.
norm (Literal['backward', 'ortho', 'forward']) –
specify the normalization mode. Should be one of the following modes:
- 'backward': normalize by 1/n.
- 'ortho': normalize by 1/sqrt(n) (i.e., make the FFT orthonormal).
- 'forward': no normalization.
where n = prod(s) is the logical FFT size.

Default: 'backward'.

Returns:

Return type:

array

Notes

Added in version 2022.12.

_abc_impl = <_abc._abc_data object>¶

_is_protocol = True¶

_is_runtime_protocol = True¶

class array_api._2022_12.ifftshift(*args, **kwargs)[source]¶

Bases: Protocol, Generic

Inverse of fftshift.

Note

Although identical for even-length x, fftshift and ifftshift differ by one sample for odd-length x.

Parameters:

x (array) – input array. Should have a floating-point data type.
axes (Optional[Union[int, Sequence[int]]]) –
axes over which to perform the inverse shift. If None, the function must shift all axes. Default: None.

If axes contains two or more entries which resolve to the same axis (i.e., resolved axes are not unique), the behavior is unspecified and thus implementation-defined.

Returns:

out – the shifted array. The returned array must have the same data type and shape as x.

Return type:

array

Notes

Added in version 2022.12.

_abc_impl = <_abc._abc_data object>¶

_is_protocol = True¶

_is_runtime_protocol = True¶

class array_api._2022_12.ihfft(*args, **kwargs)[source]¶

Bases: Protocol, Generic

Computes the one-dimensional inverse discrete Fourier transform of a signal with Hermitian symmetry.

Parameters:

x (array) – input array. Must have a real-valued floating-point data type.
n (Optional[int]) –
number of elements over which to compute the transform along the axis (dimension) specified by axis. Let M be the size of the input array along the axis specified by axis. When n is None, the function must set n equal to M.
- If n is greater than M, the axis specified by axis must be zero-padded to size n.
- If n is less than M, the axis specified by axis must be trimmed to size n.
- If n equals M, all elements along the axis specified by axis must be used when computing the transform.
Default: None.
axis (int) – axis (dimension) of the input array over which to compute the transform. A valid axis must be an integer on the interval [-N, N), where N is the rank (number of dimensions) of x. If an axis is specified as a negative integer, the function must determine the axis along which to compute the transform by counting backward from the last dimension (where -1 refers to the last dimension). Default: -1.
norm (Literal['backward', 'ortho', 'forward']) –
normalization mode. Should be one of the following modes:
- 'backward': normalize by 1/n.
- 'ortho': normalize by 1/sqrt(n) (i.e., make the FFT orthonormal).
- 'forward': no normalization.
Default: 'backward'.

Returns:

Return type:

array

Notes

Added in version 2022.12.

_abc_impl = <_abc._abc_data object>¶

_is_protocol = True¶

_is_runtime_protocol = True¶

class array_api._2022_12.iinfo(*args, **kwargs)[source]¶

Bases: Protocol, Generic

Machine limits for integer data types.

Parameters:

type (Union[dtype, array]) – the kind of integer data-type about which to get information.

Returns:

out – an object having the following attributes:

bits: int

number of bits occupied by the type.
max: int

largest representable number.
min: int

smallest representable number.
dtype: dtype

integer data type.

Added in version 2022.12.

Return type:

iinfo object

_abc_impl = <_abc._abc_data object>¶

_is_protocol = True¶

_is_runtime_protocol = True¶

class array_api._2022_12.iinfo_object(*args, **kwargs)[source]¶

Bases: Protocol, Generic

Dataclass returned by iinfo.

_abc_impl = <_abc._abc_data object>¶

_is_protocol = True¶

_is_runtime_protocol = True¶

bits: int¶

dtype: TDtype¶

max: int¶

min: int¶

class array_api._2022_12.imag(*args, **kwargs)[source]¶

Bases: Protocol, Generic

Returns the imaginary component of a complex number for each element x_i of the input array x.

Parameters:

x (array) – input array. Should have a complex floating-point data type.

Returns:

out (array) – an array containing the element-wise results. The returned array must have a floating-point data type with the same floating-point precision as x (e.g., if x is complex64, the returned array must have the floating-point data type float32).
.. versionadded:: 2022.12

_abc_impl = <_abc._abc_data object>¶

_is_protocol = True¶

_is_runtime_protocol = True¶

class array_api._2022_12.inv(*args, **kwargs)[source]¶

Bases: Protocol, Generic

Returns the multiplicative inverse of a square matrix (or a stack of square matrices) x.

If x is real-valued, let $\mathbb{K}$ be the set of real numbers $\mathbb{R}$, and, if x is complex-valued, let $\mathbb{K}$ be the set of complex numbers $\mathbb{C}$.

The inverse matrix $x^{-1} \in\ \mathbb{K}^{n \times n}$ of a square matrix $x \in\ \mathbb{K}^{n \times n}$ is defined as

\[x^{-1}x = xx^{-1} = I_n\]

where $I_n$ is the n-dimensional identity matrix.

The inverse matrix exists if and only if x is invertible. When x is invertible, the inverse is unique.

When x is a stack of matrices, the function must compute the inverse for each matrix in the stack.

Parameters:: x (array) – input array having shape (..., M, M) and whose innermost two dimensions form square matrices. Should have a floating-point data type.
Returns:: out – an array containing the multiplicative inverses. The returned array must have a floating-point data type determined by type-promotion and must have the same shape as x.
Return type:: array

Notes

Changed in version 2022.12: Added complex data type support.

_abc_impl = <_abc._abc_data object>¶

_is_protocol = True¶

_is_runtime_protocol = True¶

class array_api._2022_12.irfft(*args, **kwargs)[source]¶

Bases: Protocol, Generic

Computes the one-dimensional inverse of rfft for complex-valued input.

Note

Parameters:

x (array) – input array. Should have a complex floating-point data type.
n (Optional[int]) –
number of elements along the transformed axis (dimension) specified by axis in the output array. Let M be the size of the input array along the axis specified by axis. When n is None, the function must set n equal to 2*(M-1).
- If n//2+1 is greater than M, the axis of the input array specified by axis must be zero-padded to size n//2+1.
- If n//2+1 is less than M, the axis of the input array specified by axis must be trimmed to size n//2+1.
- If n//2+1 equals M, all elements along the axis of the input array specified by axis must be used when computing the transform.
Default: None.
axis (int) – axis (dimension) of the input array over which to compute the transform. A valid axis must be an integer on the interval [-N, N), where N is the rank (number of dimensions) of x. If an axis is specified as a negative integer, the function must determine the axis along which to compute the transform by counting backward from the last dimension (where -1 refers to the last dimension). Default: -1.
norm (Literal['backward', 'ortho', 'forward']) –
normalization mode. Should be one of the following modes:
- 'backward': normalize by 1/n.
- 'ortho': normalize by 1/sqrt(n) (i.e., make the FFT orthonormal).
- 'forward': no normalization.
Default: 'backward'.

Returns:

Return type:

array

Notes

In order to return an array having an odd number of elements along the transformed axis, the function must be provided an odd integer for n.

Added in version 2022.12.

_abc_impl = <_abc._abc_data object>¶

_is_protocol = True¶

_is_runtime_protocol = True¶

class array_api._2022_12.irfftn(*args, **kwargs)[source]¶

Bases: Protocol, Generic

Computes the n-dimensional inverse of rfftn for complex-valued input.

Note

Parameters:

x (array) – input array. Should have a complex floating-point data type.
s (Optional[Sequence[int]]) –
number of elements along the transformed axes (dimensions) specified by axes in the output array. Let i be the index of the n-th axis specified by axes (i.e., i = axes[n]) and M[i] be the size of the input array along axis i. When s is None, the function must set s equal to a sequence of integers such that s[i] equals M[i] for all i, except for the last transformed axis in which s[i] equals 2*(M[i]-1). For each i, let n equal s[i], except for the last transformed axis in which n equals s[i]//2+1.
- If n is greater than M[i], axis i of the input array must be zero-padded to size n.
- If n is less than M[i], axis i of the input array must be trimmed to size n.
- If n equals M[i] or -1, all elements along axis i of the input array must be used when computing the transform.
If s is not None, axes must not be None. Default: None.
axes (Optional[Sequence[int]]) –
axes (dimensions) over which to compute the transform. A valid axis in axes must be an integer on the interval [-N, N), where N is the rank (number of dimensions) of x. If an axis is specified as a negative integer, the function must determine the axis along which to compute the transform by counting backward from the last dimension (where -1 refers to the last dimension).

If s is provided, the corresponding axes to be transformed must also be provided. If axes is None, the function must compute the transform over all axes. Default: None.

If axes contains two or more entries which resolve to the same axis (i.e., resolved axes are not unique), the behavior is unspecified and thus implementation-defined.
norm (Literal['backward', 'ortho', 'forward']) –
normalization mode. Should be one of the following modes:
- 'backward': normalize by 1/n.
- 'ortho': normalize by 1/sqrt(n) (i.e., make the FFT orthonormal).
- 'forward': no normalization.
where n = prod(s) is the logical FFT size.

Default: 'backward'.

Returns:

Return type:

array

Notes

In order to return an array having an odd number of elements along the last transformed axis, the function must be provided an odd integer for s[-1].

Added in version 2022.12.

_abc_impl = <_abc._abc_data object>¶

_is_protocol = True¶

_is_runtime_protocol = True¶

class array_api._2022_12.isdtype(*args, **kwargs)[source]¶

Bases: Protocol, Generic

Returns a boolean indicating whether a provided dtype is of a specified data type “kind”.

Parameters:

dtype (dtype) – the input dtype.
kind (Union[str, dtype, Tuple[Union[str, dtype], ...]]) –
data type kind.
- If kind is a dtype, the function must return a boolean indicating whether the input dtype is equal to the dtype specified by kind.
- If kind is a string, the function must return a boolean indicating whether the input dtype is of a specified data type kind. The following dtype kinds must be supported:
  - 'bool': boolean data types (e.g., bool).
  - 'signed integer': signed integer data types (e.g., int8, int16, int32, int64).
  - 'unsigned integer': unsigned integer data types (e.g., uint8, uint16, uint32, uint64).
  - 'integral': integer data types. Shorthand for ('signed integer', 'unsigned integer').
  - 'real floating': real-valued floating-point data types (e.g., float32, float64).
  - 'complex floating': complex floating-point data types (e.g., complex64, complex128).
  - 'numeric': numeric data types. Shorthand for ('integral', 'real floating', 'complex floating').
- If kind is a tuple, the tuple specifies a union of dtypes and/or kinds, and the function must return a boolean indicating whether the input dtype is either equal to a specified dtype or belongs to at least one specified data type kind.
Note

A conforming implementation of the array API standard is not limited to only including the dtypes described in this specification in the required data type kinds. For example, implementations supporting float16 and bfloat16 can include float16 and bfloat16 in the real floating data type kind. Similarly, implementations supporting int128 can include int128 in the signed integer data type kind.

In short, conforming implementations may extend data type kinds; however, data type kinds must remain consistent (e.g., only integer dtypes may belong to integer data type kinds and only floating-point dtypes may belong to floating-point data type kinds), and extensions must be clearly documented as such in library documentation.

Returns:

out – boolean indicating whether a provided dtype is of a specified data type kind.

Return type:

bool

Notes

Added in version 2022.12.

_abc_impl = <_abc._abc_data object>¶

_is_protocol = True¶

_is_runtime_protocol = True¶

class array_api._2022_12.isfinite(*args, **kwargs)[source]¶

Bases: Protocol, Generic

Tests each element x_i of the input array x to determine if finite.

Parameters:: x (array) – input array. Should have a numeric data type.
Returns:: out – an array containing test results. The returned array must have a data type of bool.
Return type:: array

Notes

Special Cases

For real-valued floating-point operands,

If x_i is either +infinity or -infinity, the result is False.
If x_i is NaN, the result is False.
If x_i is a finite number, the result is True.

For complex floating-point operands, let a = real(x_i), b = imag(x_i), and

If a is NaN or b is NaN, the result is False.
If a is either +infinity or -infinity and b is any value, the result is False.
If a is any value and b is either +infinity or -infinity, the result is False.
If a is a finite number and b is a finite number, the result is True.

Changed in version 2022.12: Added complex data type support.

_abc_impl = <_abc._abc_data object>¶

_is_protocol = True¶

_is_runtime_protocol = True¶

class array_api._2022_12.isinf(*args, **kwargs)[source]¶

Bases: Protocol, Generic

Tests each element x_i of the input array x to determine if equal to positive or negative infinity.

Parameters:: x (array) – input array. Should have a numeric data type.
Returns:: out – an array containing test results. The returned array must have a data type of bool.
Return type:: array

Notes

Special Cases

For real-valued floating-point operands,

If x_i is either +infinity or -infinity, the result is True.
In the remaining cases, the result is False.

For complex floating-point operands, let a = real(x_i), b = imag(x_i), and

If a is either +infinity or -infinity and b is any value (including NaN), the result is True.
If a is either a finite number or NaN and b is either +infinity or -infinity, the result is True.
In the remaining cases, the result is False.

Changed in version 2022.12: Added complex data type support.

_abc_impl = <_abc._abc_data object>¶

_is_protocol = True¶

_is_runtime_protocol = True¶

class array_api._2022_12.isnan(*args, **kwargs)[source]¶

Bases: Protocol, Generic

Tests each element x_i of the input array x to determine whether the element is NaN.

Parameters:: x (array) – input array. Should have a numeric data type.
Returns:: out – an array containing test results. The returned array should have a data type of bool.
Return type:: array

Notes

Special Cases

For real-valued floating-point operands,

If x_i is NaN, the result is True.
In the remaining cases, the result is False.

For complex floating-point operands, let a = real(x_i), b = imag(x_i), and

If a or b is NaN, the result is True.
In the remaining cases, the result is False.

Changed in version 2022.12: Added complex data type support.

_abc_impl = <_abc._abc_data object>¶

_is_protocol = True¶

_is_runtime_protocol = True¶

class array_api._2022_12.less(*args, **kwargs)[source]¶

Bases: Protocol, Generic

Computes the truth value of x1_i < x2_i for each element x1_i of the input array x1 with the respective element x2_i of the input array x2.

Note

Parameters:

x1 (array) – first input array. Should have a real-valued data type.
x2 (array) – second input array. Must be compatible with x1 (see broadcasting). Should have a real-valued data type.

Returns:

out – an array containing the element-wise results. The returned array must have a data type of bool.

Return type:

array

_abc_impl = <_abc._abc_data object>¶

_is_protocol = True¶

_is_runtime_protocol = True¶

class array_api._2022_12.less_equal(*args, **kwargs)[source]¶

Bases: Protocol, Generic

Computes the truth value of x1_i <= x2_i for each element x1_i of the input array x1 with the respective element x2_i of the input array x2.

Note

Parameters:

x1 (array) – first input array. Should have a real-valued data type.
x2 (array) – second input array. Must be compatible with x1 (see broadcasting). Should have a real-valued data type.

Returns:

out – an array containing the element-wise results. The returned array must have a data type of bool.

Return type:

array

_abc_impl = <_abc._abc_data object>¶

_is_protocol = True¶

_is_runtime_protocol = True¶

class array_api._2022_12.linspace(*args, **kwargs)[source]¶

Bases: Protocol, Generic

Returns evenly spaced numbers over a specified interval.

\[\Delta_{\textrm{real}} = \frac{\textrm{stop} - \textrm{start}}{N - 1}\]

For complex output arrays, let a = real(start), b = imag(start), c = real(stop), and d = imag(stop). The spacing between complex values is given by

\[\Delta_{\textrm{complex}} = \frac{c-a}{N-1} + \frac{d-b}{N-1} j\]

Parameters:

start (Union[int, float, complex]) – the start of the interval.
stop (Union[int, float, complex]) –
the end of the interval. If endpoint is False, the function must generate a sequence of num+1 evenly spaced numbers starting with start and ending with stop and exclude the stop from the returned array such that the returned array consists of evenly spaced numbers over the half-open interval [start, stop). If endpoint is True, the output array must consist of evenly spaced numbers over the closed interval [start, stop]. Default: True.

Note

The step size changes when endpoint is False.
num (int) – number of samples. Must be a nonnegative integer value.
dtype (Optional[dtype]) –
output array data type. Should be a floating-point data type. If dtype is None,
- if either start or stop is a complex number, the output data type must be the default complex floating-point data type.
- if both start and stop are real-valued, the output data type must be the default real-valued floating-point data type.
Default: None.

Note

If dtype is not None, conversion of start and stop should obey type-promotion rules. Conversions not specified according to type-promotion rules may or may not be permitted by a conforming array library.
device (Optional[device]) – device on which to place the created array. Default: None.
endpoint (bool) – boolean indicating whether to include stop in the interval. Default: True.

Returns:

out – a one-dimensional array containing evenly spaced values.

Return type:

array

Notes

Note

Changed in version 2022.12: Added complex data type support.

_abc_impl = <_abc._abc_data object>¶

_is_protocol = True¶

_is_runtime_protocol = True¶

class array_api._2022_12.log(*args, **kwargs)[source]¶

Bases: Protocol, Generic

Calculates an implementation-dependent approximation to the natural (base e) logarithm for each element x_i of the input array x.

Note

The natural logarithm of a complex number $z$ with polar coordinates $(r,\theta)$ equals $\ln r + (\theta + 2n\pi)j$ with principal value $\ln r + \theta j$.

Note

For complex floating-point operands, log(conj(x)) must equal conj(log(x)).

Note

By convention, the branch cut of the natural logarithm is the negative real axis $(-\infty, 0)$.

The natural logarithm is a continuous function from above the branch cut, taking into account the sign of the imaginary component.

Note: branch cuts follow C99 and have provisional status (see branch-cuts).

Parameters:: x (array) – input array. Should have a floating-point data type.
Returns:: out – an array containing the evaluated natural logarithm for each element in x. The returned array must have a floating-point data type determined by type-promotion.
Return type:: array

Notes

Special cases

For real-valued floating-point operands,

If x_i is NaN, the result is NaN.
If x_i is less than 0, the result is NaN.
If x_i is either +0 or -0, the result is -infinity.
If x_i is 1, the result is +0.
If x_i is +infinity, the result is +infinity.

For complex floating-point operands, let a = real(x_i), b = imag(x_i), and

If a is -0 and b is +0, the result is -infinity + πj.
If a is +0 and b is +0, the result is -infinity + 0j.
If a is a finite number and b is +infinity, the result is +infinity + πj/2.
If a is a finite number and b is NaN, the result is NaN + NaN j.
If a is -infinity and b is a positive (i.e., greater than 0) finite number, the result is +infinity + πj.
If a is +infinity and b is a positive (i.e., greater than 0) finite number, the result is +infinity + 0j.
If a is -infinity and b is +infinity, the result is +infinity + 3πj/4.
If a is +infinity and b is +infinity, the result is +infinity + πj/4.
If a is either +infinity or -infinity and b is NaN, the result is +infinity + NaN j.
If a is NaN and b is a finite number, the result is NaN + NaN j.
If a is NaN and b is +infinity, the result is +infinity + NaN j.
If a is NaN and b is NaN, the result is NaN + NaN j.

Changed in version 2022.12: Added complex data type support.

_abc_impl = <_abc._abc_data object>¶

_is_protocol = True¶

_is_runtime_protocol = True¶

class array_api._2022_12.log10(*args, **kwargs)[source]¶

Bases: Protocol, Generic

Calculates an implementation-dependent approximation to the base 10 logarithm for each element x_i of the input array x.

Note

For complex floating-point operands, log10(conj(x)) must equal conj(log10(x)).

Parameters:: x (array) – input array. Should have a floating-point data type.
Returns:: out – an array containing the evaluated base 10 logarithm for each element in x. The returned array must have a floating-point data type determined by type-promotion.
Return type:: array

Notes

Special cases

For real-valued floating-point operands,

If x_i is NaN, the result is NaN.
If x_i is less than 0, the result is NaN.
If x_i is either +0 or -0, the result is -infinity.
If x_i is 1, the result is +0.
If x_i is +infinity, the result is +infinity.

For complex floating-point operands, special cases must be handled as if the operation is implemented using the standard change of base formula

\[\log_{10} x = \frac{\log_{e} x}{\log_{e} 10}\]

where $\log_{e}$ is the natural logarithm, as implemented by log().

Changed in version 2022.12: Added complex data type support.

_abc_impl = <_abc._abc_data object>¶

_is_protocol = True¶

_is_runtime_protocol = True¶

class array_api._2022_12.log1p(*args, **kwargs)[source]¶

Bases: Protocol, Generic

Calculates an implementation-dependent approximation to log(1+x), where log refers to the natural (base e) logarithm, for each element x_i of the input array x.

Note

For complex floating-point operands, log1p(conj(x)) must equal conj(log1p(x)).

Note

By convention, the branch cut of the natural logarithm is the negative real axis $(-\infty, 0)$.

The natural logarithm is a continuous function from above the branch cut, taking into account the sign of the imaginary component.

Note: branch cuts follow C99 and have provisional status (see branch-cuts).

Parameters:: x (array) – input array. Should have a floating-point data type.
Returns:: out – an array containing the evaluated result for each element in x. The returned array must have a floating-point data type determined by type-promotion.
Return type:: array

Notes

Special cases

For real-valued floating-point operands,

If x_i is NaN, the result is NaN.
If x_i is less than -1, the result is NaN.
If x_i is -1, the result is -infinity.
If x_i is -0, the result is -0.
If x_i is +0, the result is +0.
If x_i is +infinity, the result is +infinity.

For complex floating-point operands, let a = real(x_i), b = imag(x_i), and

If a is -1 and b is +0, the result is -infinity + 0j.
If a is a finite number and b is +infinity, the result is +infinity + πj/2.
If a is a finite number and b is NaN, the result is NaN + NaN j.
If a is -infinity and b is a positive (i.e., greater than 0) finite number, the result is +infinity + πj.
If a is +infinity and b is a positive (i.e., greater than 0) finite number, the result is +infinity + 0j.
If a is -infinity and b is +infinity, the result is +infinity + 3πj/4.
If a is +infinity and b is +infinity, the result is +infinity + πj/4.
If a is either +infinity or -infinity and b is NaN, the result is +infinity + NaN j.
If a is NaN and b is a finite number, the result is NaN + NaN j.
If a is NaN and b is +infinity, the result is +infinity + NaN j.
If a is NaN and b is NaN, the result is NaN + NaN j.

Changed in version 2022.12: Added complex data type support.

_abc_impl = <_abc._abc_data object>¶

_is_protocol = True¶

_is_runtime_protocol = True¶

class array_api._2022_12.log2(*args, **kwargs)[source]¶

Bases: Protocol, Generic

Calculates an implementation-dependent approximation to the base 2 logarithm for each element x_i of the input array x.

Note

For complex floating-point operands, log2(conj(x)) must equal conj(log2(x)).

Parameters:: x (array) – input array. Should have a floating-point data type.
Returns:: out – an array containing the evaluated base 2 logarithm for each element in x. The returned array must have a floating-point data type determined by type-promotion.
Return type:: array

Notes

Special cases

For real-valued floating-point operands,

If x_i is NaN, the result is NaN.
If x_i is less than 0, the result is NaN.
If x_i is either +0 or -0, the result is -infinity.
If x_i is 1, the result is +0.
If x_i is +infinity, the result is +infinity.

For complex floating-point operands, special cases must be handled as if the operation is implemented using the standard change of base formula

\[\log_{2} x = \frac{\log_{e} x}{\log_{e} 2}\]

where $\log_{e}$ is the natural logarithm, as implemented by log().

Changed in version 2022.12: Added complex data type support.

_abc_impl = <_abc._abc_data object>¶

_is_protocol = True¶

_is_runtime_protocol = True¶

class array_api._2022_12.logaddexp(*args, **kwargs)[source]¶

Bases: Protocol, Generic

Calculates the logarithm of the sum of exponentiations log(exp(x1) + exp(x2)) for each element x1_i of the input array x1 with the respective element x2_i of the input array x2.

Parameters:

x1 (array) – first input array. Should have a real-valued floating-point data type.
x2 (array) – second input array. Must be compatible with x1 (see broadcasting). Should have a real-valued floating-point data type.

Returns:

out – an array containing the element-wise results. The returned array must have a real-valued floating-point data type determined by type-promotion.

Return type:

array

Notes

Special cases

For floating-point operands,

If either x1_i or x2_i is NaN, the result is NaN.
If x1_i is +infinity and x2_i is not NaN, the result is +infinity.
If x1_i is not NaN and x2_i is +infinity, the result is +infinity.

_abc_impl = <_abc._abc_data object>¶

_is_protocol = True¶

_is_runtime_protocol = True¶

class array_api._2022_12.logical_and(*args, **kwargs)[source]¶

Bases: Protocol, Generic

Computes the logical AND for each element x1_i of the input array x1 with the respective element x2_i of the input array x2.

Note

Parameters:

x1 (array) – first input array. Should have a boolean data type.
x2 (array) – second input array. Must be compatible with x1 (see broadcasting). Should have a boolean data type.

Returns:

out – an array containing the element-wise results. The returned array must have a data type of bool.

Return type:

array

_abc_impl = <_abc._abc_data object>¶

_is_protocol = True¶

_is_runtime_protocol = True¶

class array_api._2022_12.logical_not(*args, **kwargs)[source]¶

Bases: Protocol, Generic

Computes the logical NOT for each element x_i of the input array x.

Note

Parameters:: x (array) – input array. Should have a boolean data type.
Returns:: out – an array containing the element-wise results. The returned array must have a data type of bool.
Return type:: array

_abc_impl = <_abc._abc_data object>¶

_is_protocol = True¶

_is_runtime_protocol = True¶

class array_api._2022_12.logical_or(*args, **kwargs)[source]¶

Bases: Protocol, Generic

Computes the logical OR for each element x1_i of the input array x1 with the respective element x2_i of the input array x2.

Note

Parameters:

x1 (array) – first input array. Should have a boolean data type.
x2 (array) – second input array. Must be compatible with x1 (see broadcasting). Should have a boolean data type.

Returns:

out – an array containing the element-wise results. The returned array must have a data type of bool.

Return type:

array

_abc_impl = <_abc._abc_data object>¶

_is_protocol = True¶

_is_runtime_protocol = True¶

class array_api._2022_12.logical_xor(*args, **kwargs)[source]¶

Bases: Protocol, Generic

Computes the logical XOR for each element x1_i of the input array x1 with the respective element x2_i of the input array x2.

Note

Parameters:

x1 (array) – first input array. Should have a boolean data type.
x2 (array) – second input array. Must be compatible with x1 (see broadcasting). Should have a boolean data type.

Returns:

out – an array containing the element-wise results. The returned array must have a data type of bool.

Return type:

array

_abc_impl = <_abc._abc_data object>¶

_is_protocol = True¶

_is_runtime_protocol = True¶

class array_api._2022_12.matmul(*args, **kwargs)[source]¶

Bases: Protocol, Generic

Computes the matrix product.

Note

The matmul function must implement the same semantics as the built-in @ operator (see PEP 465).

Parameters:

x1 (array) – first input array. Should have a numeric data type. Must have at least one dimension. If x1 is one-dimensional having shape (M,) and x2 has more than one dimension, x1 must be promoted to a two-dimensional array by prepending 1 to its dimensions (i.e., must have shape (1, M)). After matrix multiplication, the prepended dimensions in the returned array must be removed. If x1 has more than one dimension (including after vector-to-matrix promotion), shape(x1)[:-2] must be compatible with shape(x2)[:-2] (after vector-to-matrix promotion) (see broadcasting). If x1 has shape (..., M, K), the innermost two dimensions form matrices on which to perform matrix multiplication.
x2 (array) – second input array. Should have a numeric data type. Must have at least one dimension. If x2 is one-dimensional having shape (N,) and x1 has more than one dimension, x2 must be promoted to a two-dimensional array by appending 1 to its dimensions (i.e., must have shape (N, 1)). After matrix multiplication, the appended dimensions in the returned array must be removed. If x2 has more than one dimension (including after vector-to-matrix promotion), shape(x2)[:-2] must be compatible with shape(x1)[:-2] (after vector-to-matrix promotion) (see broadcasting). If x2 has shape (..., K, N), the innermost two dimensions form matrices on which to perform matrix multiplication.

Note

Returns:

out –

if both x1 and x2 are one-dimensional arrays having shape (N,), a zero-dimensional array containing the inner product as its only element.
if x1 is a two-dimensional array having shape (M, K) and x2 is a two-dimensional array having shape (K, N), a two-dimensional array containing the conventional matrix product and having shape (M, N).
if x1 is a one-dimensional array having shape (K,) and x2 is an array having shape (..., K, N), an array having shape (..., N) (i.e., prepended dimensions during vector-to-matrix promotion must be removed) and containing the conventional matrix product.
if x1 is an array having shape (..., M, K) and x2 is a one-dimensional array having shape (K,), an array having shape (..., M) (i.e., appended dimensions during vector-to-matrix promotion must be removed) and containing the conventional matrix product.
if x1 is a two-dimensional array having shape (M, K) and x2 is an array having shape (..., K, N), an array having shape (..., M, N) and containing the conventional matrix product for each stacked matrix.
if x1 is an array having shape (..., M, K) and x2 is a two-dimensional array having shape (K, N), an array having shape (..., M, N) and containing the conventional matrix product for each stacked matrix.
if either x1 or x2 has more than two dimensions, an array having a shape determined by broadcasting shape(x1)[:-2] against shape(x2)[:-2] and containing the conventional matrix product for each stacked matrix.

The returned array must have a data type determined by type-promotion.

Return type:

array

Notes

Changed in version 2022.12: Added complex data type support.

Raises

if either x1 or x2 is a zero-dimensional array.
if x1 is a one-dimensional array having shape (K,), x2 is a one-dimensional array having shape (L,), and K != L.
if x1 is a one-dimensional array having shape (K,), x2 is an array having shape (..., L, N), and K != L.
if x1 is an array having shape (..., M, K), x2 is a one-dimensional array having shape (L,), and K != L.
if x1 is an array having shape (..., M, K), x2 is an array having shape (..., L, N), and K != L.

_abc_impl = <_abc._abc_data object>¶

_is_protocol = True¶

_is_runtime_protocol = True¶

class array_api._2022_12.matrix_norm(*args, **kwargs)[source]¶

Bases: Protocol, Generic

Computes the matrix norm of a matrix (or a stack of matrices) x.

Parameters:

x (array) – input array having shape (..., M, N) and whose innermost two dimensions form MxN matrices. Should have a floating-point data type.
keepdims (bool) – If True, the last two axes (dimensions) must be included in the result as singleton dimensions, and, accordingly, the result must be compatible with the input array (see broadcasting). Otherwise, if False, the last two axes (dimensions) must not be included in the result. Default: False.
ord (Optional[Union[int, float, Literal[inf, -inf, 'fro', 'nuc']]]) –
order of the norm. The following mathematical norms must be supported:

ord

description

’fro’

Frobenius norm

’nuc’

nuclear norm

1

max(sum(abs(x), axis=0))

2

largest singular value

inf

max(sum(abs(x), axis=1))

The following non-mathematical “norms” must be supported:

ord

description

-1

min(sum(abs(x), axis=0))

-2

smallest singular value

-inf

min(sum(abs(x), axis=1))

If ord=1, the norm corresponds to the induced matrix norm where p=1 (i.e., the maximum absolute value column sum).

If ord=2, the norm corresponds to the induced matrix norm where p=inf (i.e., the maximum absolute value row sum).

If ord=inf, the norm corresponds to the induced matrix norm where p=2 (i.e., the largest singular value).

Default: 'fro'.

Returns:

Return type:

ord	description
’fro’	Frobenius norm
’nuc’	nuclear norm
1	max(sum(abs(x), axis=0))
2	largest singular value
inf	max(sum(abs(x), axis=1))

ord	description
-1	min(sum(abs(x), axis=0))
-2	smallest singular value
-inf	min(sum(abs(x), axis=1))

array

Notes

Changed in version 2022.12: Added complex data type support.

_abc_impl = <_abc._abc_data object>¶

_is_protocol = True¶

_is_runtime_protocol = True¶

class array_api._2022_12.matrix_power(*args, **kwargs)[source]¶

Bases: Protocol, Generic

Raises a square matrix (or a stack of square matrices) x to an integer power n.

Parameters:

x (array) – input array having shape (..., M, M) and whose innermost two dimensions form square matrices. Should have a floating-point data type.
n (int) – integer exponent.

Returns:

Return type:

array

Notes

Changed in version 2022.12: Added complex data type support.

_abc_impl = <_abc._abc_data object>¶

_is_protocol = True¶

_is_runtime_protocol = True¶

class array_api._2022_12.matrix_rank(*args, **kwargs)[source]¶

Bases: Protocol, Generic

Returns the rank (i.e., number of non-zero singular values) of a matrix (or a stack of matrices).

When x is a stack of matrices, the function must compute the number of non-zero singular values for each matrix in the stack.

Parameters:

x (array) – input array having shape (..., M, N) and whose innermost two dimensions form MxN matrices. Should have a floating-point data type.
rtol (Optional[Union[float, array]]) – relative tolerance for small singular values. Singular values approximately less than or equal to rtol * largest_singular_value are set to zero. If a float, the value is equivalent to a zero-dimensional array having a real-valued floating-point data type determined by type-promotion (as applied to x) and must be broadcast against each matrix. If an array, must have a real-valued floating-point data type and must be compatible with shape(x)[:-2] (see broadcasting). If None, the default value is max(M, N) * eps, where eps must be the machine epsilon associated with the real-valued floating-point data type determined by type-promotion (as applied to x). Default: None.

Returns:

out – an array containing the ranks. The returned array must have the default integer data type and must have shape (...) (i.e., must have a shape equal to shape(x)[:-2]).

Return type:

array

Notes

Changed in version 2022.12: Added complex data type support.

_abc_impl = <_abc._abc_data object>¶

_is_protocol = True¶

_is_runtime_protocol = True¶

class array_api._2022_12.matrix_transpose(*args, **kwargs)[source]¶

Bases: Protocol, Generic

Transposes a matrix (or a stack of matrices) x.

Parameters:: x (array) – input array having shape (..., M, N) and whose innermost two dimensions form MxN matrices.
Returns:: out – an array containing the transpose for each matrix and having shape (..., N, M). The returned array must have the same data type as x.
Return type:: array

_abc_impl = <_abc._abc_data object>¶

_is_protocol = True¶

_is_runtime_protocol = True¶

class array_api._2022_12.max(*args, **kwargs)[source]¶

Bases: Protocol, Generic

Calculates the maximum value of the input array x.

Note

Parameters:

x (array) – input array. Should have a real-valued data type.
axis (Optional[Union[int, Tuple[int, ...]]]) – axis or axes along which maximum values must be computed. By default, the maximum value must be computed over the entire array. If a tuple of integers, maximum values must be computed over multiple axes. Default: None.
keepdims (bool) – if True, the reduced axes (dimensions) must be included in the result as singleton dimensions, and, accordingly, the result must be compatible with the input array (see broadcasting). Otherwise, if False, the reduced axes (dimensions) must not be included in the result. Default: False.

Returns:

Return type:

array

Notes

Special Cases

For floating-point operands,

If x_i is NaN, the maximum value is NaN (i.e., NaN values propagate).

_abc_impl = <_abc._abc_data object>¶

_is_protocol = True¶

_is_runtime_protocol = True¶

class array_api._2022_12.mean(*args, **kwargs)[source]¶

Bases: Protocol, Generic

Calculates the arithmetic mean of the input array x.

Parameters:

x (array) – input array. Should have a real-valued floating-point data type.
axis (Optional[Union[int, Tuple[int, ...]]]) – axis or axes along which arithmetic means must be computed. By default, the mean must be computed over the entire array. If a tuple of integers, arithmetic means must be computed over multiple axes. Default: None.
keepdims (bool) – if True, the reduced axes (dimensions) must be included in the result as singleton dimensions, and, accordingly, the result must be compatible with the input array (see broadcasting). Otherwise, if False, the reduced axes (dimensions) must not be included in the result. Default: False.

Returns:

Note

Return type:

array

Notes

Special Cases

Let N equal the number of elements over which to compute the arithmetic mean.

If N is 0, the arithmetic mean is NaN.
If x_i is NaN, the arithmetic mean is NaN (i.e., NaN values propagate).

_abc_impl = <_abc._abc_data object>¶

_is_protocol = True¶

_is_runtime_protocol = True¶

class array_api._2022_12.meshgrid(*args, **kwargs)[source]¶

Bases: Protocol, Generic

Returns coordinate matrices from coordinate vectors.

Parameters:

arrays (array) – an arbitrary number of one-dimensional arrays representing grid coordinates. Each array should have the same numeric data type.
indexing (Literal["xy", "ij"]) – Cartesian 'xy' or matrix 'ij' indexing of output. If provided zero or one one-dimensional vector(s) (i.e., the zero- and one-dimensional cases, respectively), the indexing keyword has no effect and should be ignored. Default: 'xy'.

Returns:

out – list of N arrays, where N is the number of provided one-dimensional input arrays. Each returned array must have rank N. For N one-dimensional arrays having lengths Ni = len(xi),

if matrix indexing ij, then each returned array must have the shape (N1, N2, N3, ..., Nn).
if Cartesian indexing xy, then each returned array must have shape (N2, N1, N3, ..., Nn).

Each returned array should have the same data type as the input arrays.

Return type:

List[array]

Notes

Changed in version 2022.12: Added complex data type support.

_abc_impl = <_abc._abc_data object>¶

_is_protocol = True¶

_is_runtime_protocol = True¶

class array_api._2022_12.min(*args, **kwargs)[source]¶

Bases: Protocol, Generic

Calculates the minimum value of the input array x.

Note

Parameters:

x (array) – input array. Should have a real-valued data type.
axis (Optional[Union[int, Tuple[int, ...]]]) – axis or axes along which minimum values must be computed. By default, the minimum value must be computed over the entire array. If a tuple of integers, minimum values must be computed over multiple axes. Default: None.
keepdims (bool) – if True, the reduced axes (dimensions) must be included in the result as singleton dimensions, and, accordingly, the result must be compatible with the input array (see broadcasting). Otherwise, if False, the reduced axes (dimensions) must not be included in the result. Default: False.

Returns:

Return type:

array

Notes

Special Cases

For floating-point operands,

If x_i is NaN, the minimum value is NaN (i.e., NaN values propagate).

_abc_impl = <_abc._abc_data object>¶

_is_protocol = True¶

_is_runtime_protocol = True¶

class array_api._2022_12.multiply(*args, **kwargs)[source]¶

Bases: Protocol, Generic

Calculates the product for each element x1_i of the input array x1 with the respective element x2_i of the input array x2.

Note

Floating-point multiplication is not always associative due to finite precision.

Parameters:

x1 (array) – first input array. Should have a numeric data type.
x2 (array) – second input array. Must be compatible with x1 (see broadcasting). Should have a numeric data type.

Returns:

out – an array containing the element-wise products. The returned array must have a data type determined by type-promotion.

Return type:

array

Notes

Special cases

For real-valued floating-point operands,

If either x1_i or x2_i is NaN, the result is NaN.
If x1_i is either +infinity or -infinity and x2_i is either +0 or -0, the result is NaN.
If x1_i is either +0 or -0 and x2_i is either +infinity or -infinity, the result is NaN.
If x1_i and x2_i have the same mathematical sign, the result has a positive mathematical sign, unless the result is NaN. If the result is NaN, the “sign” of NaN is implementation-defined.
If x1_i and x2_i have different mathematical signs, the result has a negative mathematical sign, unless the result is NaN. If the result is NaN, the “sign” of NaN is implementation-defined.
If x1_i is either +infinity or -infinity and x2_i is either +infinity or -infinity, the result is a signed infinity with the mathematical sign determined by the rule already stated above.
If x1_i is either +infinity or -infinity and x2_i is a nonzero finite number, the result is a signed infinity with the mathematical sign determined by the rule already stated above.
If x1_i is a nonzero finite number and x2_i is either +infinity or -infinity, the result is a signed infinity with the mathematical sign determined by the rule already stated above.
In the remaining cases, where neither infinity nor NaN is involved, the product must be computed and rounded to the nearest representable value according to IEEE 754-2019 and a supported rounding mode. If the magnitude is too large to represent, the result is an infinity of appropriate mathematical sign. If the magnitude is too small to represent, the result is a zero of appropriate mathematical sign.

For complex floating-point operands, multiplication is defined according to the following table. For real components a and c and imaginary components b and d,

	c	dj	c + dj
a	a * c	(a*d)j	(ac) + (ad)j
bj	(b*c)j	-(b*d)	-(bd) + (bc)j
a + bj	(ac) + (bc)j	-(bd) + (ad)j	special rules

\[(a + bj) \cdot (c + dj) = (ac - bd) + (bc + ad)j\]

When at least one of a, b, c, or d is NaN, +infinity, or -infinity,

If a, b, c, and d are all NaN, the result is NaN + NaN j.
In the remaining cases, the result is implementation dependent.

Note

For complex floating-point operands, the results of special cases may be implementation dependent depending on how an implementation chooses to model complex numbers and complex infinity (e.g., complex plane versus Riemann sphere). For those implementations following C99 and its one-infinity model, when at least one component is infinite, even if the other component is NaN, the complex value is infinite, and the usual arithmetic rules do not apply to complex-complex multiplication. In the interest of performance, other implementations may want to avoid the complex branching logic necessary to implement the one-infinity model and choose to implement all complex-complex multiplication according to the textbook formula. Accordingly, special case behavior is unlikely to be consistent across implementations.

Changed in version 2022.12: Added complex data type support.

_abc_impl = <_abc._abc_data object>¶

_is_protocol = True¶

_is_runtime_protocol = True¶

class array_api._2022_12.negative(*args, **kwargs)[source]¶

Bases: Protocol, Generic

Computes the numerical negative of each element x_i (i.e., y_i = -x_i) of the input array x.

Note

For signed integer data types, the numerical negative of the minimum representable integer is implementation-dependent.

Note

If x has a complex floating-point data type, both the real and imaginary components for each x_i must be negated (a result which follows from the rules of complex number multiplication).

Parameters:: x (array) – input array. Should have a numeric data type.
Returns:: out – an array containing the evaluated result for each element in x. The returned array must have a data type determined by type-promotion.
Return type:: array

Notes

Changed in version 2022.12: Added complex data type support.

_abc_impl = <_abc._abc_data object>¶

_is_protocol = True¶

_is_runtime_protocol = True¶

class array_api._2022_12.nonzero(*args, **kwargs)[source]¶

Bases: Protocol, Generic

Returns the indices of the array elements which are non-zero.

Note

If x has a complex floating-point data type, non-zero elements are those elements having at least one component (real or imaginary) which is non-zero.

Note

If x has a boolean data type, non-zero elements are those elements which are equal to True.

Data-dependent output shape

Parameters:: x (array) – input array. Must have a positive rank. If x is zero-dimensional, the function must raise an exception.
Returns:: out – a tuple of k arrays, one for each dimension of x and each of size n (where n is the total number of non-zero elements), containing the indices of the non-zero elements in that dimension. The indices must be returned in row-major, C-style order. The returned array must have the default array index data type.
Return type:: Tuple[array, …]

Notes

Changed in version 2022.12: Added complex data type support.

_abc_impl = <_abc._abc_data object>¶

_is_protocol = True¶

_is_runtime_protocol = True¶

class array_api._2022_12.not_equal(*args, **kwargs)[source]¶

Bases: Protocol, Generic

Computes the truth value of x1_i != x2_i for each element x1_i of the input array x1 with the respective element x2_i of the input array x2.

Parameters:

x1 (array) – first input array. May have any data type.
x2 (array) – second input array. Must be compatible with x1 (see broadcasting).

Returns:

out – an array containing the element-wise results. The returned array must have a data type of bool.

Return type:

array

Notes

Special Cases

For real-valued floating-point operands,

If x1_i is NaN or x2_i is NaN, the result is True.
If x1_i is +infinity and x2_i is -infinity, the result is True.
If x1_i is -infinity and x2_i is +infinity, the result is True.
If x1_i is a finite number, x2_i is a finite number, and x1_i does not equal x2_i, the result is True.
In the remaining cases, the result is False.

For complex floating-point operands, let a = real(x1_i), b = imag(x1_i), c = real(x2_i), d = imag(x2_i), and

If a, b, c, or d is NaN, the result is True.
In the remaining cases, the result is the logical OR of the equality comparison between the real values a and c (real components) and between the real values b and d (imaginary components), as described above for real-valued floating-point operands (i.e., a != c OR b != d).

Note

For discussion of complex number equality, see complex-numbers.

Changed in version 2022.12: Added complex data type support.

_abc_impl = <_abc._abc_data object>¶

_is_protocol = True¶

_is_runtime_protocol = True¶

class array_api._2022_12.ones(*args, **kwargs)[source]¶

Bases: Protocol, Generic

Returns a new array having a specified shape and filled with ones.

Note

An output array having a complex floating-point data type must contain complex numbers having a real component equal to one and an imaginary component equal to zero (i.e., 1 + 0j).

Parameters:

shape (Union[int, Tuple[int, ...]]) – output array shape.
dtype (Optional[dtype]) – output array data type. If dtype is None, the output array data type must be the default real-valued floating-point data type. Default: None.
device (Optional[device]) – device on which to place the created array. Default: None.

Returns:

out – an array containing ones.

Return type:

array

Notes

Changed in version 2022.12: Added complex data type support.

_abc_impl = <_abc._abc_data object>¶

_is_protocol = True¶

_is_runtime_protocol = True¶

class array_api._2022_12.ones_like(*args, **kwargs)[source]¶

Bases: Protocol, Generic

Returns a new array filled with ones and having the same shape as an input array x.

Note

An output array having a complex floating-point data type must contain complex numbers having a real component equal to one and an imaginary component equal to zero (i.e., 1 + 0j).

Parameters:

x (array) – input array from which to derive the output array shape.
dtype (Optional[dtype]) – output array data type. If dtype is None, the output array data type must be inferred from x. Default: None.
device (Optional[device]) – device on which to place the created array. If device is None, the output array device must be inferred from x. Default: None.

Returns:

out – an array having the same shape as x and filled with ones.

Return type:

array

Notes

Changed in version 2022.12: Added complex data type support.

_abc_impl = <_abc._abc_data object>¶

_is_protocol = True¶

_is_runtime_protocol = True¶

class array_api._2022_12.outer(*args, **kwargs)[source]¶

Bases: Protocol, Generic

Returns the outer product of two vectors x1 and x2.

Parameters:

x1 (array) – first one-dimensional input array of size N. Must have a numeric data type.
x2 (array) – second one-dimensional input array of size M. Must have a numeric data type.

Returns:

out – a two-dimensional array containing the outer product and whose shape is (N, M). The returned array must have a data type determined by type-promotion.

Return type:

array

Notes

Changed in version 2022.12: Added complex data type support.

_abc_impl = <_abc._abc_data object>¶

_is_protocol = True¶

_is_runtime_protocol = True¶

class array_api._2022_12.permute_dims(*args, **kwargs)[source]¶

Bases: Protocol, Generic

Permutes the axes (dimensions) of an array x.

Parameters:

x (array) – input array.
axes (Tuple[int, ...]) – tuple containing a permutation of (0, 1, ..., N-1) where N is the number of axes (dimensions) of x.

Returns:

out – an array containing the axes permutation. The returned array must have the same data type as x.

Return type:

array

_abc_impl = <_abc._abc_data object>¶

_is_protocol = True¶

_is_runtime_protocol = True¶

class array_api._2022_12.pinv(*args, **kwargs)[source]¶

Bases: Protocol, Generic

Returns the (Moore-Penrose) pseudo-inverse of a matrix (or a stack of matrices) x.

While the pseudo-inverse can be defined algebraically, one can understand the pseudo-inverse via singular value decomposition (SVD). Namely, if

\[A = U \Sigma V^H\]

is a singular decomposition of $A$, then

\[A^{+} = U \Sigma^{+} V^H\]

When x is a stack of matrices, the function must compute the pseudo-inverse for each matrix in the stack.

Parameters:

x (array) – input array having shape (..., M, N) and whose innermost two dimensions form MxN matrices. Should have a floating-point data type.
rtol (Optional[Union[float, array]]) – relative tolerance for small singular values. Singular values approximately less than or equal to rtol * largest_singular_value are set to zero. If a float, the value is equivalent to a zero-dimensional array having a real-valued floating-point data type determined by type-promotion (as applied to x) and must be broadcast against each matrix. If an array, must have a real-valued floating-point data type and must be compatible with shape(x)[:-2] (see broadcasting). If None, the default value is max(M, N) * eps, where eps must be the machine epsilon associated with the real-valued floating-point data type determined by type-promotion (as applied to x). Default: None.

Returns:

Return type:

array

Notes

Changed in version 2022.12: Added complex data type support.

_abc_impl = <_abc._abc_data object>¶

_is_protocol = True¶

_is_runtime_protocol = True¶

class array_api._2022_12.positive(*args, **kwargs)[source]¶

Bases: Protocol, Generic

Computes the numerical positive of each element x_i (i.e., y_i = +x_i) of the input array x.

Parameters:: x (array) – input array. Should have a numeric data type.
Returns:: out – an array containing the evaluated result for each element in x. The returned array must have the same data type as x.
Return type:: array

Notes

Changed in version 2022.12: Added complex data type support.

_abc_impl = <_abc._abc_data object>¶

_is_protocol = True¶

_is_runtime_protocol = True¶

class array_api._2022_12.pow(*args, **kwargs)[source]¶

Bases: Protocol, Generic

Note

If both x1 and x2 have integer data types, the result of pow when x2_i is negative (i.e., less than zero) is unspecified and thus implementation-dependent.

Note

By convention, the branch cut of the natural logarithm is the negative real axis $(-\infty, 0)$.

Note: branch cuts follow C99 and have provisional status (see branch-cuts).

Parameters:

x1 (array) – first input array whose elements correspond to the exponentiation base. Should have a numeric data type.
x2 (array) – second input array whose elements correspond to the exponentiation exponent. Must be compatible with x1 (see broadcasting). Should have a numeric data type.

Returns:

out – an array containing the element-wise results. The returned array must have a data type determined by type-promotion.

Return type:

array

Notes

Special cases

For real-valued floating-point operands,

If x1_i is not equal to 1 and x2_i is NaN, the result is NaN.
If x2_i is +0, the result is 1, even if x1_i is NaN.
If x2_i is -0, the result is 1, even if x1_i is NaN.
If x1_i is NaN and x2_i is not equal to 0, the result is NaN.
If abs(x1_i) is greater than 1 and x2_i is +infinity, the result is +infinity.
If abs(x1_i) is greater than 1 and x2_i is -infinity, the result is +0.
If abs(x1_i) is 1 and x2_i is +infinity, the result is 1.
If abs(x1_i) is 1 and x2_i is -infinity, the result is 1.
If x1_i is 1 and x2_i is not NaN, the result is 1.
If abs(x1_i) is less than 1 and x2_i is +infinity, the result is +0.
If abs(x1_i) is less than 1 and x2_i is -infinity, the result is +infinity.
If x1_i is +infinity and x2_i is greater than 0, the result is +infinity.
If x1_i is +infinity and x2_i is less than 0, the result is +0.
If x1_i is -infinity, x2_i is greater than 0, and x2_i is an odd integer value, the result is -infinity.
If x1_i is -infinity, x2_i is greater than 0, and x2_i is not an odd integer value, the result is +infinity.
If x1_i is -infinity, x2_i is less than 0, and x2_i is an odd integer value, the result is -0.
If x1_i is -infinity, x2_i is less than 0, and x2_i is not an odd integer value, the result is +0.
If x1_i is +0 and x2_i is greater than 0, the result is +0.
If x1_i is +0 and x2_i is less than 0, the result is +infinity.
If x1_i is -0, x2_i is greater than 0, and x2_i is an odd integer value, the result is -0.
If x1_i is -0, x2_i is greater than 0, and x2_i is not an odd integer value, the result is +0.
If x1_i is -0, x2_i is less than 0, and x2_i is an odd integer value, the result is -infinity.
If x1_i is -0, x2_i is less than 0, and x2_i is not an odd integer value, the result is +infinity.
If x1_i is less than 0, x1_i is a finite number, x2_i is a finite number, and x2_i is not an integer value, the result is NaN.

For complex floating-point operands, special cases should be handled as if the operation is implemented as exp(x2*log(x1)).

Note

Conforming implementations are allowed to treat special cases involving complex floating-point operands more carefully than as described in this specification.

Changed in version 2022.12: Added complex data type support.

_abc_impl = <_abc._abc_data object>¶

_is_protocol = True¶

_is_runtime_protocol = True¶

class array_api._2022_12.prod(*args, **kwargs)[source]¶

Bases: Protocol, Generic

Calculates the product of input array x elements.

Parameters:

x (array) – input array. Should have a numeric data type.
axis (Optional[Union[int, Tuple[int, ...]]]) – axis or axes along which products must be computed. By default, the product must be computed over the entire array. If a tuple of integers, products must be computed over multiple axes. Default: None.
dtype (Optional[dtype]) –
data type of the returned array. If None,
- if the default data type corresponding to the data type “kind” (integer, real-valued floating-point, or complex floating-point) of x has a smaller range of values than the data type of x (e.g., x has data type int64 and the default data type is int32, or x has data type uint64 and the default data type is int64), the returned array must have the same data type as x.
- if x has a real-valued floating-point data type, the returned array must have the default real-valued floating-point data type.
- if x has a complex floating-point data type, the returned array must have the default complex floating-point data type.
- if x has a signed integer data type (e.g., int16), the returned array must have the default integer data type.
- if x has an unsigned integer data type (e.g., uint16), the returned array must have an unsigned integer data type having the same number of bits as the default integer data type (e.g., if the default integer data type is int32, the returned array must have a uint32 data type).
If the data type (either specified or resolved) differs from the data type of x, the input array should be cast to the specified data type before computing the product. Default: None.

Note

This keyword argument is intended to help prevent data type overflows.
keepdims (bool) – if True, the reduced axes (dimensions) must be included in the result as singleton dimensions, and, accordingly, the result must be compatible with the input array (see broadcasting). Otherwise, if False, the reduced axes (dimensions) must not be included in the result. Default: False.

Returns:

Return type:

array

Notes

Special Cases

Let N equal the number of elements over which to compute the product.

If N is 0, the product is 1 (i.e., the empty product).

For both real-valued and complex floating-point operands, special cases must be handled as if the operation is implemented by successive application of multiply().

Changed in version 2022.12: Added complex data type support.

_abc_impl = <_abc._abc_data object>¶

_is_protocol = True¶

_is_runtime_protocol = True¶

class array_api._2022_12.qr(*args, **kwargs)[source]¶

Bases: Protocol, Generic

Returns the QR decomposition of a full column rank matrix (or a stack of matrices).

If x is real-valued, let $\mathbb{K}$ be the set of real numbers $\mathbb{R}$, and, if x is complex-valued, let $\mathbb{K}$ be the set of complex numbers $\mathbb{C}$.

The complete QR decomposition of a matrix $x \in\ \mathbb{K}^{n \times n}$ is defined as

\[x = QR\]

where $Q \in\ \mathbb{K}^{m \times n}$ and $R \in\ \mathbb{K}^{n \times n}$.

The reduced QR decomposition equals with the complete QR decomposition when $n \geq m$ (wide matrix).

When x is a stack of matrices, the function must compute the QR decomposition for each matrix in the stack.

Note

Whether an array library explicitly checks whether an input array is a full column rank matrix (or a stack of full column rank matrices) is implementation-defined.

Warning

The QR decomposition is only well-defined if the first k = min(m,n) columns of every matrix in x are linearly independent.

Parameters:

x (array) – input array having shape (..., M, N) and whose innermost two dimensions form MxN matrices of rank N. Should have a floating-point data type.
mode (Literal['reduced', 'complete']) –
decomposition mode. Should be one of the following modes:
- 'reduced': compute only the leading K columns of q, such that q and r have dimensions (..., M, K) and (..., K, N), respectively, and where K = min(M, N).
- 'complete': compute q and r with dimensions (..., M, M) and (..., M, N), respectively.
Default: 'reduced'.

Returns:

out – a namedtuple (Q, R) whose

first element must have the field name Q and must be an array whose shape depends on the value of mode and contain matrices with orthonormal columns. If mode is 'complete', the array must have shape (..., M, M). If mode is 'reduced', the array must have shape (..., M, K), where K = min(M, N). The first x.ndim-2 dimensions must have the same size as those of the input array x.
second element must have the field name R and must be an array whose shape depends on the value of mode and contain upper-triangular matrices. If mode is 'complete', the array must have shape (..., M, N). If mode is 'reduced', the array must have shape (..., K, N), where K = min(M, N). The first x.ndim-2 dimensions must have the same size as those of the input x.

Each returned array must have a floating-point data type determined by type-promotion.

Return type:

Tuple[array, array]

Notes

Changed in version 2022.12: Added complex data type support.

_abc_impl = <_abc._abc_data object>¶

_is_protocol = True¶

_is_runtime_protocol = True¶

class array_api._2022_12.real(*args, **kwargs)[source]¶

Bases: Protocol, Generic

Returns the real component of a complex number for each element x_i of the input array x.

Parameters:: x (array) – input array. Should have a complex floating-point data type.
Returns:: out – an array containing the element-wise results. The returned array must have a floating-point data type with the same floating-point precision as x (e.g., if x is complex64, the returned array must have the floating-point data type float32).
Return type:: array

Notes

Added in version 2022.12.

_abc_impl = <_abc._abc_data object>¶

_is_protocol = True¶

_is_runtime_protocol = True¶

class array_api._2022_12.remainder(*args, **kwargs)[source]¶

Bases: Protocol, Generic

Returns the remainder of division for each element x1_i of the input array x1 and the respective element x2_i of the input array x2.

Note

This function is equivalent to the Python modulus operator x1_i % x2_i.

Note

For input arrays which promote to an integer data type, the result of division by zero is unspecified and thus implementation-defined.

Parameters:

x1 (array) – dividend input array. Should have a real-valued data type.
x2 (array) – divisor input array. Must be compatible with x1 (see broadcasting). Should have a real-valued data type.

Returns:

Return type:

array

Notes

Special cases

Note

For floating-point operands,

If either x1_i or x2_i is NaN, the result is NaN.
If x1_i is either +infinity or -infinity and x2_i is either +infinity or -infinity, the result is NaN.
If x1_i is either +0 or -0 and x2_i is either +0 or -0, the result is NaN.
If x1_i is +0 and x2_i is greater than 0, the result is +0.
If x1_i is -0 and x2_i is greater than 0, the result is +0.
If x1_i is +0 and x2_i is less than 0, the result is -0.
If x1_i is -0 and x2_i is less than 0, the result is -0.
If x1_i is greater than 0 and x2_i is +0, the result is NaN.
If x1_i is greater than 0 and x2_i is -0, the result is NaN.
If x1_i is less than 0 and x2_i is +0, the result is NaN.
If x1_i is less than 0 and x2_i is -0, the result is NaN.
If x1_i is +infinity and x2_i is a positive (i.e., greater than 0) finite number, the result is NaN.
If x1_i is +infinity and x2_i is a negative (i.e., less than 0) finite number, the result is NaN.
If x1_i is -infinity and x2_i is a positive (i.e., greater than 0) finite number, the result is NaN.
If x1_i is -infinity and x2_i is a negative (i.e., less than 0) finite number, the result is NaN.
If x1_i is a positive (i.e., greater than 0) finite number and x2_i is +infinity, the result is x1_i. (note: this result matches Python behavior.)
If x1_i is a positive (i.e., greater than 0) finite number and x2_i is -infinity, the result is x2_i. (note: this result matches Python behavior.)
If x1_i is a negative (i.e., less than 0) finite number and x2_i is +infinity, the result is x2_i. (note: this results matches Python behavior.)
If x1_i is a negative (i.e., less than 0) finite number and x2_i is -infinity, the result is x1_i. (note: this result matches Python behavior.)
In the remaining cases, the result must match that of the Python % operator.

_abc_impl = <_abc._abc_data object>¶

_is_protocol = True¶

_is_runtime_protocol = True¶

class array_api._2022_12.reshape(*args, **kwargs)[source]¶

Bases: Protocol, Generic

Reshapes an array without changing its data.

Parameters:

x (array) – input array to reshape.
shape (Tuple[int, ...]) – a new shape compatible with the original shape. One shape dimension is allowed to be -1. When a shape dimension is -1, the corresponding output array shape dimension must be inferred from the length of the array and the remaining dimensions.
copy (Optional[bool]) – boolean indicating whether or not to copy the input array. If True, the function must always copy. If False, the function must never copy and must raise a ValueError in case a copy would be necessary. If None, the function must reuse existing memory buffer if possible and copy otherwise. Default: None.

Returns:

out – an output array having the same data type and elements as x.

Return type:

array

_abc_impl = <_abc._abc_data object>¶

_is_protocol = True¶

_is_runtime_protocol = True¶

class array_api._2022_12.result_type(*args, **kwargs)[source]¶

Bases: Protocol, Generic

Returns the dtype that results from applying the type promotion rules (see type-promotion) to the arguments.

Note

If provided mixed dtypes (e.g., integer and floating-point), the returned dtype will be implementation-specific.

Parameters:: arrays_and_dtypes (Union[array, dtype]) – an arbitrary number of input arrays and/or dtypes.
Returns:: out – the dtype resulting from an operation involving the input arrays and dtypes.
Return type:: dtype

_abc_impl = <_abc._abc_data object>¶

_is_protocol = True¶

_is_runtime_protocol = True¶

class array_api._2022_12.rfft(*args, **kwargs)[source]¶

Bases: Protocol, Generic

Computes the one-dimensional discrete Fourier transform for real-valued input.

Note

Parameters:

x (array) – input array. Must have a real-valued floating-point data type.
n (Optional[int]) –
number of elements over which to compute the transform along the axis (dimension) specified by axis. Let M be the size of the input array along the axis specified by axis. When n is None, the function must set n equal to M.
- If n is greater than M, the axis specified by axis must be zero-padded to size n.
- If n is less than M, the axis specified by axis must be trimmed to size n.
- If n equals M, all elements along the axis specified by axis must be used when computing the transform.
Default: None.
axis (int) – axis (dimension) of the input array over which to compute the transform. A valid axis must be an integer on the interval [-N, N), where N is the rank (number of dimensions) of x. If an axis is specified as a negative integer, the function must determine the axis along which to compute the transform by counting backward from the last dimension (where -1 refers to the last dimension). Default: -1.
norm (Literal['backward', 'ortho', 'forward']) –
normalization mode. Should be one of the following modes:
- 'backward': no normalization.
- 'ortho': normalize by 1/sqrt(n) (i.e., make the FFT orthonormal).
- 'forward': normalize by 1/n.
Default: 'backward'.

Returns:

Return type:

array

Notes

Added in version 2022.12.

_abc_impl = <_abc._abc_data object>¶

_is_protocol = True¶

_is_runtime_protocol = True¶

class array_api._2022_12.rfftfreq(*args, **kwargs)[source]¶

Bases: Protocol, Generic

Computes the discrete Fourier transform sample frequencies (for rfft and irfft).

For a Fourier transform of length n and length unit of d, the frequencies are described as:

f = [0, 1, ...,     n/2-1,     n/2] / (d*n)  # if n is even
f = [0, 1, ..., (n-1)/2-1, (n-1)/2] / (d*n)  # if n is odd

The Nyquist frequency component is considered to be positive.

Parameters:

n (int) – window length.
d (float) – sample spacing between individual samples of the Fourier transform input. Default: 1.0.
device (Optional[device]) – device on which to place the created array. Default: None.

Returns:

out – an array of shape (n//2+1,) containing the sample frequencies. The returned array must have the default real-valued floating-point data type.

Return type:

array

Notes

Added in version 2022.12.

_abc_impl = <_abc._abc_data object>¶

_is_protocol = True¶

_is_runtime_protocol = True¶

class array_api._2022_12.rfftn(*args, **kwargs)[source]¶

Bases: Protocol, Generic

Computes the n-dimensional discrete Fourier transform for real-valued input.

Note

Parameters:

x (array) – input array. Must have a real-valued floating-point data type.
s (Optional[Sequence[int]]) –
number of elements over which to compute the transform along axes (dimensions) specified by axes. Let i be the index of the n-th axis specified by axes (i.e., i = axes[n]) and M[i] be the size of the input array along axis i. When s is None, the function must set s equal to a sequence of integers such that s[i] equals M[i] for all i.
- If s[i] is greater than M[i], axis i must be zero-padded to size s[i].
- If s[i] is less than M[i], axis i must be trimmed to size s[i].
- If s[i] equals M[i] or -1, all elements along axis i must be used when computing the transform.
If s is not None, axes must not be None. Default: None.
axes (Optional[Sequence[int]]) –
axes (dimensions) over which to compute the transform. A valid axis in axes must be an integer on the interval [-N, N), where N is the rank (number of dimensions) of x. If an axis is specified as a negative integer, the function must determine the axis along which to compute the transform by counting backward from the last dimension (where -1 refers to the last dimension).

If s is provided, the corresponding axes to be transformed must also be provided. If axes is None, the function must compute the transform over all axes. Default: None.

If axes contains two or more entries which resolve to the same axis (i.e., resolved axes are not unique), the behavior is unspecified and thus implementation-defined.
norm (Literal['backward', 'ortho', 'forward']) –
normalization mode. Should be one of the following modes:
- 'backward': no normalization.
- 'ortho': normalize by 1/sqrt(n) (i.e., make the FFT orthonormal).
- 'forward': normalize by 1/n.
where n = prod(s), the logical FFT size.

Default: 'backward'.

Returns:

Return type:

array

Notes

Added in version 2022.12.

_abc_impl = <_abc._abc_data object>¶

_is_protocol = True¶

_is_runtime_protocol = True¶

class array_api._2022_12.roll(*args, **kwargs)[source]¶

Bases: Protocol, Generic

Parameters:

x (array) – input array.
shift (Union[int, Tuple[int, ...]]) – number of places by which the elements are shifted. If shift is a tuple, then axis must be a tuple of the same size, and each of the given axes must be shifted by the corresponding element in shift. If shift is an int and axis a tuple, then the same shift must be used for all specified axes. If a shift is positive, then array elements must be shifted positively (toward larger indices) along the dimension of axis. If a shift is negative, then array elements must be shifted negatively (toward smaller indices) along the dimension of axis.
axis (Optional[Union[int, Tuple[int, ...]]]) – axis (or axes) along which elements to shift. If axis is None, the array must be flattened, shifted, and then restored to its original shape. Default: None.

Returns:

out – an output array having the same data type as x and whose elements, relative to x, are shifted.

Return type:

array

_abc_impl = <_abc._abc_data object>¶

_is_protocol = True¶

_is_runtime_protocol = True¶

class array_api._2022_12.round(*args, **kwargs)[source]¶

Bases: Protocol, Generic

Rounds each element x_i of the input array x to the nearest integer-valued number.

Note

For complex floating-point operands, real and imaginary components must be independently rounded to the nearest integer-valued number.

Parameters:: x (array) – input array. Should have a numeric data type.
Returns:: out – an array containing the rounded result for each element in x. The returned array must have the same data type as x.
Return type:: array

Notes

Special cases

Note

For complex floating-point operands, the following special cases apply to real and imaginary components independently (e.g., if real(x_i) is NaN, the rounded real component is NaN).

If x_i is already integer-valued, the result is x_i.

For floating-point operands,

If x_i is +infinity, the result is +infinity.
If x_i is -infinity, the result is -infinity.
If x_i is +0, the result is +0.
If x_i is -0, the result is -0.
If x_i is NaN, the result is NaN.
If two integers are equally close to x_i, the result is the even integer closest to x_i.

Changed in version 2022.12: Added complex data type support.

_abc_impl = <_abc._abc_data object>¶

_is_protocol = True¶

_is_runtime_protocol = True¶

class array_api._2022_12.sign(*args, **kwargs)[source]¶

Bases: Protocol, Generic

Returns an indication of the sign of a number for each element x_i of the input array x.

The sign function (also known as the signum function) of a number $x_i$ is defined as

\[\begin{split}\operatorname{sign}(x_i) = \begin{cases} 0 & \textrm{if } x_i = 0 \\ \frac{x_i}{|x_i|} & \textrm{otherwise} \end{cases}\end{split}\]

where $|x_i|$ is the absolute value of $x_i$.

Parameters:: x (array) – input array. Should have a numeric data type.
Returns:: out – an array containing the evaluated result for each element in x. The returned array must have the same data type as x.
Return type:: array

Notes

Special cases

For real-valued operands,

If x_i is less than 0, the result is -1.
If x_i is either -0 or +0, the result is 0.
If x_i is greater than 0, the result is +1.
If x_i is NaN, the result is NaN.

For complex floating-point operands, let a = real(x_i), b = imag(x_i), and

If a is either -0 or +0 and b is either -0 or +0, the result is 0 + 0j.
If a is NaN or b is NaN, the result is NaN + NaN j.
In the remaining cases, special cases must be handled according to the rules of complex number division (see divide()).

Changed in version 2022.12: Added complex data type support.

_abc_impl = <_abc._abc_data object>¶

_is_protocol = True¶

_is_runtime_protocol = True¶

class array_api._2022_12.sin(*args, **kwargs)[source]¶

Bases: Protocol, Generic

Calculates an implementation-dependent approximation to the sine for each element x_i of the input array x.

Each element x_i is assumed to be expressed in radians.

Note

The sine is an entire function on the complex plane and has no branch cuts.

Note

For complex arguments, the mathematical definition of sine is

where $\operatorname{sinh}$ is the hyperbolic sine.

Parameters:: x (array) – input array whose elements are each expressed in radians. Should have a floating-point data type.
Returns:: out – an array containing the sine of each element in x. The returned array must have a floating-point data type determined by type-promotion.
Return type:: array

Notes

Special cases

For real-valued floating-point operands,

If x_i is NaN, the result is NaN.
If x_i is +0, the result is +0.
If x_i is -0, the result is -0.
If x_i is either +infinity or -infinity, the result is NaN.

For complex floating-point operands, special cases must be handled as if the operation is implemented as -1j * sinh(x*1j).

Changed in version 2022.12: Added complex data type support.

_abc_impl = <_abc._abc_data object>¶

_is_protocol = True¶

_is_runtime_protocol = True¶

class array_api._2022_12.sinh(*args, **kwargs)[source]¶

Bases: Protocol, Generic

Calculates an implementation-dependent approximation to the hyperbolic sine for each element x_i of the input array x.

The mathematical definition of the hyperbolic sine is

\[\operatorname{sinh}(x) = \frac{e^x - e^{-x}}{2}\]

Note

The hyperbolic sine is an entire function in the complex plane and has no branch cuts. The function is periodic, with period $2\pi j$, with respect to the imaginary component.

Parameters:: x (array) – input array whose elements each represent a hyperbolic angle. Should have a floating-point data type.
Returns:: out – an array containing the hyperbolic sine of each element in x. The returned array must have a floating-point data type determined by type-promotion.
Return type:: array

Notes

Special cases

Note

For all operands, sinh(x) must equal -sinh(-x).

For real-valued floating-point operands,

If x_i is NaN, the result is NaN.
If x_i is +0, the result is +0.
If x_i is -0, the result is -0.
If x_i is +infinity, the result is +infinity.
If x_i is -infinity, the result is -infinity.

For complex floating-point operands, let a = real(x_i), b = imag(x_i), and

Note

For complex floating-point operands, sinh(conj(x)) must equal conj(sinh(x)).

If a is +0 and b is +0, the result is +0 + 0j.
If a is +0 and b is +infinity, the result is 0 + NaN j (sign of the real component is unspecified).
If a is +0 and b is NaN, the result is 0 + NaN j (sign of the real component is unspecified).
If a is a positive (i.e., greater than 0) finite number and b is +infinity, the result is NaN + NaN j.
If a is a positive (i.e., greater than 0) finite number and b is NaN, the result is NaN + NaN j.
If a is +infinity and b is +0, the result is +infinity + 0j.
If a is +infinity and b is a positive finite number, the result is +infinity * cis(b).
If a is +infinity and b is +infinity, the result is infinity + NaN j (sign of the real component is unspecified).
If a is +infinity and b is NaN, the result is infinity + NaN j (sign of the real component is unspecified).
If a is NaN and b is +0, the result is NaN + 0j.
If a is NaN and b is a nonzero finite number, the result is NaN + NaN j.
If a is NaN and b is NaN, the result is NaN + NaN j.

where cis(v) is cos(v) + sin(v)*1j.

Changed in version 2022.12: Added complex data type support.

_abc_impl = <_abc._abc_data object>¶

_is_protocol = True¶

_is_runtime_protocol = True¶

class array_api._2022_12.slogdet(*args, **kwargs)[source]¶

Bases: Protocol, Generic

Returns the sign and the natural logarithm of the absolute value of the determinant of a square matrix (or a stack of square matrices) x.

Note

The purpose of this function is to calculate the determinant more accurately when the determinant is either very small or very large, as calling det may overflow or underflow.

The sign of the determinant is given by

\[\begin{split}\operatorname{sign}(\det x) = \begin{cases} 0 & \textrm{if } \det x = 0 \\ \frac{\det x}{|\det x|} & \textrm{otherwise} \end{cases}\end{split}\]

where $|\det x|$ is the absolute value of the determinant of x.

When x is a stack of matrices, the function must compute the sign and natural logarithm of the absolute value of the determinant for each matrix in the stack.

Special Cases

For real-valued floating-point operands,

If the determinant is zero, the sign should be 0 and logabsdet should be -infinity.

For complex floating-point operands,

If the determinant is 0 + 0j, the sign should be 0 + 0j and logabsdet should be -infinity + 0j.

Note

Parameters:

x (array) – input array having shape (..., M, M) and whose innermost two dimensions form square matrices. Should have a floating-point data type.

Returns:

out – a namedtuple (sign, logabsdet) whose

first element must have the field name sign and must be an array containing a number representing the sign of the determinant for each square matrix. Must have the same data type as x.
second element must have the field name logabsdet and must be an array containing the natural logarithm of the absolute value of the determinant for each square matrix. If x is real-valued, the returned array must have a real-valued floating-point data type determined by type-promotion. If x is complex, the returned array must have a real-valued floating-point data type having the same precision as x (e.g., if x is complex64, logabsdet must have a float32 data type).

Each returned array must have shape shape(x)[:-2].

Return type:

Tuple[array, array]

Notes

Changed in version 2022.12: Added complex data type support.

_abc_impl = <_abc._abc_data object>¶

_is_protocol = True¶

_is_runtime_protocol = True¶

class array_api._2022_12.solve(*args, **kwargs)[source]¶

Bases: Protocol, Generic

Returns the solution of a square system of linear equations with a unique solution.

\[AX = B\]

This system of linear equations has a unique solution if and only if $A$ is invertible.

Note

Whether an array library explicitly checks whether x1 is invertible is implementation-defined.

When x1 and/or x2 is a stack of matrices, the function must compute a solution for each matrix in the stack.

Parameters:

x1 (array) – coefficient array A having shape (..., M, M) and whose innermost two dimensions form square matrices. Must be of full rank (i.e., all rows or, equivalently, columns must be linearly independent). Should have a floating-point data type.
x2 (array) – ordinate (or “dependent variable”) array B. If x2 has shape (M,), x2 is equivalent to an array having shape (..., M, 1). If x2 has shape (..., M, K), each column k defines a set of ordinate values for which to compute a solution, and shape(x2)[:-2] must be compatible with shape(x1)[:-2] (see broadcasting). Should have a floating-point data type.

Returns:

Return type:

array

Notes

Changed in version 2022.12: Added complex data type support.

_abc_impl = <_abc._abc_data object>¶

_is_protocol = True¶

_is_runtime_protocol = True¶

class array_api._2022_12.sort(*args, **kwargs)[source]¶

Bases: Protocol, Generic

Returns a sorted copy of an input array x.

Note

Parameters:

x (array) – input array. Should have a real-valued data type.
axis (int) – axis along which to sort. If set to -1, the function must sort along the last axis. Default: -1.
descending (bool) – sort order. If True, the array must be sorted in descending order (by value). If False, the array must be sorted in ascending order (by value). Default: False.
stable (bool) – sort stability. If True, the returned array must maintain the relative order of x values which compare as equal. If False, the returned array may or may not maintain the relative order of x values which compare as equal (i.e., the relative order of x values which compare as equal is implementation-dependent). Default: True.

Returns:

out – a sorted array. The returned array must have the same data type and shape as x.

Return type:

array

_abc_impl = <_abc._abc_data object>¶

_is_protocol = True¶

_is_runtime_protocol = True¶

class array_api._2022_12.sqrt(*args, **kwargs)[source]¶

Bases: Protocol, Generic

Calculates the principal square root for each element x_i of the input array x.

Note

After rounding, each result must be indistinguishable from the infinitely precise result (as required by IEEE 754).

Note

For complex floating-point operands, sqrt(conj(x)) must equal conj(sqrt(x)).

Note

By convention, the branch cut of the square root is the negative real axis $(-\infty, 0)$.

The square root is a continuous function from above the branch cut, taking into account the sign of the imaginary component.

Note: branch cuts follow C99 and have provisional status (see branch-cuts).

Parameters:: x (array) – input array. Should have a floating-point data type.
Returns:: out – an array containing the square root of each element in x. The returned array must have a floating-point data type determined by type-promotion.
Return type:: array

Notes

Special cases

For real-valued floating-point operands,

If x_i is NaN, the result is NaN.
If x_i is less than 0, the result is NaN.
If x_i is +0, the result is +0.
If x_i is -0, the result is -0.
If x_i is +infinity, the result is +infinity.

For complex floating-point operands, let a = real(x_i), b = imag(x_i), and

If a is either +0 or -0 and b is +0, the result is +0 + 0j.
If a is any value (including NaN) and b is +infinity, the result is +infinity + infinity j.
If a is a finite number and b is NaN, the result is NaN + NaN j.
If a -infinity and b is a positive (i.e., greater than 0) finite number, the result is +0 + infinity j.
If a is +infinity and b is a positive (i.e., greater than 0) finite number, the result is +infinity + 0 j.
If a is -infinity and b is NaN, the result is NaN + infinity j (sign of the imaginary component is unspecified).
If a is +infinity and b is NaN, the result is +infinity + NaN j.
If a is NaN and b is any value, the result is NaN + NaN j.
If a is NaN and b is NaN, the result is NaN + NaN j.

Changed in version 2022.12: Added complex data type support.

_abc_impl = <_abc._abc_data object>¶

_is_protocol = True¶

_is_runtime_protocol = True¶

class array_api._2022_12.square(*args, **kwargs)[source]¶

Bases: Protocol, Generic

Squares each element x_i of the input array x.

The square of a number x_i is defined as

\[x_i^2 = x_i \cdot x_i\]

Parameters:: x (array) – input array. Should have a numeric data type.
Returns:: out – an array containing the evaluated result for each element in x. The returned array must have a data type determined by type-promotion.
Return type:: array

Notes

Special cases

For floating-point operands, special cases must be handled as if the operation is implemented as x * x (see multiply()).

Changed in version 2022.12: Added complex data type support.

_abc_impl = <_abc._abc_data object>¶

_is_protocol = True¶

_is_runtime_protocol = True¶

class array_api._2022_12.squeeze(*args, **kwargs)[source]¶

Bases: Protocol, Generic

Removes singleton dimensions (axes) from x.

Parameters:

x (array) – input array.
axis (Union[int, Tuple[int, ...]]) – axis (or axes) to squeeze. If a specified axis has a size greater than one, a ValueError must be raised.

Returns:

out – an output array having the same data type and elements as x.

Return type:

array

_abc_impl = <_abc._abc_data object>¶

_is_protocol = True¶

_is_runtime_protocol = True¶

class array_api._2022_12.stack(*args, **kwargs)[source]¶

Bases: Protocol, Generic

Joins a sequence of arrays along a new axis.

Parameters:

arrays (Union[Tuple[array, ...], List[array]]) – input arrays to join. Each array must have the same shape.
axis (int) – axis along which the arrays will be joined. Providing an axis specifies the index of the new axis in the dimensions of the result. For example, if axis is 0, the new axis will be the first dimension and the output array will have shape (N, A, B, C); if axis is 1, the new axis will be the second dimension and the output array will have shape (A, N, B, C); and, if axis is -1, the new axis will be the last dimension and the output array will have shape (A, B, C, N). A valid axis must be on the interval [-N, N), where N is the rank (number of dimensions) of x. If provided an axis outside of the required interval, the function must raise an exception. Default: 0.

Returns:

Note

This specification leaves type promotion between data type families (i.e., intxx and floatxx) unspecified.

Return type:

array

_abc_impl = <_abc._abc_data object>¶

_is_protocol = True¶

_is_runtime_protocol = True¶

class array_api._2022_12.std(*args, **kwargs)[source]¶

Bases: Protocol, Generic

Calculates the standard deviation of the input array x.

Parameters:

x (array) – input array. Should have a real-valued floating-point data type.
axis (Optional[Union[int, Tuple[int, ...]]]) – axis or axes along which standard deviations must be computed. By default, the standard deviation must be computed over the entire array. If a tuple of integers, standard deviations must be computed over multiple axes. Default: None.
correction (Union[int, float]) – degrees of freedom adjustment. Setting this parameter to a value other than 0 has the effect of adjusting the divisor during the calculation of the standard deviation according to N-c where N corresponds to the total number of elements over which the standard deviation is computed and c corresponds to the provided degrees of freedom adjustment. When computing the standard deviation of a population, setting this parameter to 0 is the standard choice (i.e., the provided array contains data constituting an entire population). When computing the corrected sample standard deviation, setting this parameter to 1 is the standard choice (i.e., the provided array contains data sampled from a larger population; this is commonly referred to as Bessel’s correction). Default: 0.
keepdims (bool) – if True, the reduced axes (dimensions) must be included in the result as singleton dimensions, and, accordingly, the result must be compatible with the input array (see broadcasting). Otherwise, if False, the reduced axes (dimensions) must not be included in the result. Default: False.

Returns:

Note

Return type:

array

Notes

Special Cases

Let N equal the number of elements over which to compute the standard deviation.

If N - correction is less than or equal to 0, the standard deviation is NaN.
If x_i is NaN, the standard deviation is NaN (i.e., NaN values propagate).

_abc_impl = <_abc._abc_data object>¶

_is_protocol = True¶

_is_runtime_protocol = True¶

class array_api._2022_12.subtract(*args, **kwargs)[source]¶

Bases: Protocol, Generic

Calculates the difference for each element x1_i of the input array x1 with the respective element x2_i of the input array x2.

The result of x1_i - x2_i must be the same as x1_i + (-x2_i) and must be governed by the same floating-point rules as addition (see add()).

Parameters:

x1 (array) – first input array. Should have a numeric data type.
x2 (array) – second input array. Must be compatible with x1 (see broadcasting). Should have a numeric data type.

Returns:

out – an array containing the element-wise differences. The returned array must have a data type determined by type-promotion.

Return type:

array

Notes

Changed in version 2022.12: Added complex data type support.

_abc_impl = <_abc._abc_data object>¶

_is_protocol = True¶

_is_runtime_protocol = True¶

class array_api._2022_12.sum(*args, **kwargs)[source]¶

Bases: Protocol, Generic

Calculates the sum of the input array x.

Parameters:

x (array) – input array. Should have a numeric data type.
axis (Optional[Union[int, Tuple[int, ...]]]) – axis or axes along which sums must be computed. By default, the sum must be computed over the entire array. If a tuple of integers, sums must be computed over multiple axes. Default: None.
dtype (Optional[dtype]) –
data type of the returned array. If None,
- if the default data type corresponding to the data type “kind” (integer, real-valued floating-point, or complex floating-point) of x has a smaller range of values than the data type of x (e.g., x has data type int64 and the default data type is int32, or x has data type uint64 and the default data type is int64), the returned array must have the same data type as x.
- if x has a real-valued floating-point data type, the returned array must have the default real-valued floating-point data type.
- if x has a complex floating-point data type, the returned array must have the default complex floating-point data type.
- if x has a signed integer data type (e.g., int16), the returned array must have the default integer data type.
- if x has an unsigned integer data type (e.g., uint16), the returned array must have an unsigned integer data type having the same number of bits as the default integer data type (e.g., if the default integer data type is int32, the returned array must have a uint32 data type).
If the data type (either specified or resolved) differs from the data type of x, the input array should be cast to the specified data type before computing the sum. Default: None.

Note

keyword argument is intended to help prevent data type overflows.
keepdims (bool) – if True, the reduced axes (dimensions) must be included in the result as singleton dimensions, and, accordingly, the result must be compatible with the input array (see broadcasting). Otherwise, if False, the reduced axes (dimensions) must not be included in the result. Default: False.

Returns:

Return type:

array

Notes

Special Cases

Let N equal the number of elements over which to compute the sum.

If N is 0, the sum is 0 (i.e., the empty sum).

For both real-valued and complex floating-point operands, special cases must be handled as if the operation is implemented by successive application of add().

Changed in version 2022.12: Added complex data type support.

_abc_impl = <_abc._abc_data object>¶

_is_protocol = True¶

_is_runtime_protocol = True¶

class array_api._2022_12.svd(*args, **kwargs)[source]¶

Bases: Protocol, Generic

Returns a singular value decomposition (SVD) of a matrix (or a stack of matrices) x.

If x is real-valued, let $\mathbb{K}$ be the set of real numbers $\mathbb{R}$, and, if x is complex-valued, let $\mathbb{K}$ be the set of complex numbers $\mathbb{C}$.

The full singular value decomposition of an $m \times n$ matrix $x \in\ \mathbb{K}^{m \times n}$ is a factorization of the form

\[x = U \Sigma V^H\]

When $m \gt n$ (tall matrix), we can drop the last $m - n$ columns of $U$ to form the reduced SVD

\[x = U \Sigma V^H\]

Similarly, when $n \gt m$ (wide matrix), we can drop the last $n - m$ columns of $V$ to also form a reduced SVD.

This function returns the decomposition $U$, $S$, and $V^H$, where $S = \operatorname{diag}(\Sigma)$.

When x is a stack of matrices, the function must compute the singular value decomposition for each matrix in the stack.

Warning

Parameters:

x (array) – input array having shape (..., M, N) and whose innermost two dimensions form matrices on which to perform singular value decomposition. Should have a floating-point data type.
full_matrices (bool) – If True, compute full-sized U and Vh, such that U has shape (..., M, M) and Vh has shape (..., N, N). If False, compute on the leading K singular vectors, such that U has shape (..., M, K) and Vh has shape (..., K, N) and where K = min(M, N). Default: True.

Returns:

out – a namedtuple (U, S, Vh) whose

first element must have the field name U and must be an array whose shape depends on the value of full_matrices and contain matrices with orthonormal columns (i.e., the columns are left singular vectors). If full_matrices is True, the array must have shape (..., M, M). If full_matrices is False, the array must have shape (..., M, K), where K = min(M, N). The first x.ndim-2 dimensions must have the same shape as those of the input x. Must have the same data type as x.
second element must have the field name S and must be an array with shape (..., K) that contains the vector(s) of singular values of length K, where K = min(M, N). For each vector, the singular values must be sorted in descending order by magnitude, such that s[..., 0] is the largest value, s[..., 1] is the second largest value, et cetera. The first x.ndim-2 dimensions must have the same shape as those of the input x. Must have a real-valued floating-point data type having the same precision as x (e.g., if x is complex64, S must have a float32 data type).
third element must have the field name Vh and must be an array whose shape depends on the value of full_matrices and contain orthonormal rows (i.e., the rows are the right singular vectors and the array is the adjoint). If full_matrices is True, the array must have shape (..., N, N). If full_matrices is False, the array must have shape (..., K, N) where K = min(M, N). The first x.ndim-2 dimensions must have the same shape as those of the input x. Must have the same data type as x.

Return type:

Tuple[array, array, array]

Notes

Changed in version 2022.12: Added complex data type support.

_abc_impl = <_abc._abc_data object>¶

_is_protocol = True¶

_is_runtime_protocol = True¶

class array_api._2022_12.svdvals(*args, **kwargs)[source]¶

Bases: Protocol, Generic

Returns the singular values of a matrix (or a stack of matrices) x.

When x is a stack of matrices, the function must compute the singular values for each matrix in the stack.

Parameters:: x (array) – input array having shape (..., M, N) and whose innermost two dimensions form matrices on which to perform singular value decomposition. Should have a floating-point data type.
Returns:: out – an array with shape (..., K) that contains the vector(s) of singular values of length K, where K = min(M, N). For each vector, the singular values must be sorted in descending order by magnitude, such that s[..., 0] is the largest value, s[..., 1] is the second largest value, et cetera. The first x.ndim-2 dimensions must have the same shape as those of the input x. The returned array must have a real-valued floating-point data type having the same precision as x (e.g., if x is complex64, the returned array must have a float32 data type).
Return type:: array

Notes

Changed in version 2022.12: Added complex data type support.

_abc_impl = <_abc._abc_data object>¶

_is_protocol = True¶

_is_runtime_protocol = True¶

class array_api._2022_12.take(*args, **kwargs)[source]¶

Bases: Protocol, Generic

Returns elements of an array along an axis.

Note

Parameters:

x (array) – input array.
indices (array) – array indices. The array must be one-dimensional and have an integer data type.
axis (Optional[int]) –
axis over which to select values. If axis is negative, the function must determine the axis along which to select values by counting from the last dimension.

If x is a one-dimensional array, providing an axis is optional; however, if x has more than one dimension, providing an axis is required.

Returns:

Return type:

array

_abc_impl = <_abc._abc_data object>¶

_is_protocol = True¶

_is_runtime_protocol = True¶

class array_api._2022_12.tan(*args, **kwargs)[source]¶

Bases: Protocol, Generic

Calculates an implementation-dependent approximation to the tangent for each element x_i of the input array x.

Each element x_i is assumed to be expressed in radians.

Note

For complex arguments, the mathematical definition of tangent is

where $\operatorname{tanh}$ is the hyperbolic tangent.

Parameters:: x (array) – input array whose elements are expressed in radians. Should have a floating-point data type.
Returns:: out – an array containing the tangent of each element in x. The returned array must have a floating-point data type determined by type-promotion.
Return type:: array

Notes

Special cases

For real-valued floating-point operands,

If x_i is NaN, the result is NaN.
If x_i is +0, the result is +0.
If x_i is -0, the result is -0.
If x_i is either +infinity or -infinity, the result is NaN.

For complex floating-point operands, special cases must be handled as if the operation is implemented as -1j * tanh(x*1j).

Changed in version 2022.12: Added complex data type support.

_abc_impl = <_abc._abc_data object>¶

_is_protocol = True¶

_is_runtime_protocol = True¶

class array_api._2022_12.tanh(*args, **kwargs)[source]¶

Bases: Protocol, Generic

Calculates an implementation-dependent approximation to the hyperbolic tangent for each element x_i of the input array x.

The mathematical definition of the hyperbolic tangent is

\[\begin{split}\begin{align} \operatorname{tanh}(x) &= \frac{\operatorname{sinh}(x)}{\operatorname{cosh}(x)} \\ &= \frac{e^x - e^{-x}}{e^x + e^{-x}} \end{align}\end{split}\]

where $\operatorname{sinh}(x)$ is the hyperbolic sine and $\operatorname{cosh}(x)$ is the hyperbolic cosine.

Note

Parameters:: x (array) – input array whose elements each represent a hyperbolic angle. Should have a floating-point data type.
Returns:: out – an array containing the hyperbolic tangent of each element in x. The returned array must have a floating-point data type determined by type-promotion.
Return type:: array

Notes

Special cases

Note

For all operands, tanh(-x) must equal -tanh(x).

For real-valued floating-point operands,

If x_i is NaN, the result is NaN.
If x_i is +0, the result is +0.
If x_i is -0, the result is -0.
If x_i is +infinity, the result is +1.
If x_i is -infinity, the result is -1.

For complex floating-point operands, let a = real(x_i), b = imag(x_i), and

Note

For complex floating-point operands, tanh(conj(x)) must equal conj(tanh(x)).

If a is +0 and b is +0, the result is +0 + 0j.
If a is a nonzero finite number and b is +infinity, the result is NaN + NaN j.
If a is +0 and b is +infinity, the result is +0 + NaN j.
If a is a nonzero finite number and b is NaN, the result is NaN + NaN j.
If a is +0 and b is NaN, the result is +0 + NaN j.
If a is +infinity and b is a positive (i.e., greater than 0) finite number, the result is 1 + 0j.
If a is +infinity and b is +infinity, the result is 1 + 0j (sign of the imaginary component is unspecified).
If a is +infinity and b is NaN, the result is 1 + 0j (sign of the imaginary component is unspecified).
If a is NaN and b is +0, the result is NaN + 0j.
If a is NaN and b is a nonzero number, the result is NaN + NaN j.
If a is NaN and b is NaN, the result is NaN + NaN j.

Warning

Array libraries are not required to patch these older C versions, and, thus, users are advised that results may vary across array library implementations for these special cases.

Changed in version 2022.12: Added complex data type support.

_abc_impl = <_abc._abc_data object>¶

_is_protocol = True¶

_is_runtime_protocol = True¶

class array_api._2022_12.tensordot(*args, **kwargs)[source]¶

Bases: Protocol, Generic

Returns a tensor contraction of x1 and x2 over specific axes.

Note

The tensordot function corresponds to the generalized matrix product.

Parameters:

x1 (array) – first input array. Should have a numeric data type.
x2 (array) –
second input array. Should have a numeric data type. Corresponding contracted axes of x1 and x2 must be equal.

Note

Contracted axes (dimensions) must not be broadcasted.
axes (Union[int, Tuple[Sequence[int], Sequence[int]]]) –
number of axes (dimensions) to contract or explicit sequences of axes (dimensions) for x1 and x2, respectively.

If axes is an int equal to N, then contraction must be performed over the last N axes of x1 and the first N axes of x2 in order. The size of each corresponding axis (dimension) must match. Must be nonnegative.
- If N equals 0, the result is the tensor (outer) product.
- If N equals 1, the result is the tensor dot product.
- If N equals 2, the result is the tensor double contraction (default).
If axes is a tuple of two sequences (x1_axes, x2_axes), the first sequence must apply to x1 and the second sequence to x2. Both sequences must have the same length. Each axis (dimension) x1_axes[i] for x1 must have the same size as the respective axis (dimension) x2_axes[i] for x2. Each sequence must consist of unique (nonnegative) integers that specify valid axes for each respective array.

Note

Returns:: out – an array containing the tensor contraction whose shape consists of the non-contracted axes (dimensions) of the first array x1, followed by the non-contracted axes (dimensions) of the second array x2. The returned array must have a data type determined by type-promotion.
Return type:: array

Notes

Changed in version 2022.12: Added complex data type support.

_abc_impl = <_abc._abc_data object>¶

_is_protocol = True¶

_is_runtime_protocol = True¶

class array_api._2022_12.trace(*args, **kwargs)[source]¶

Bases: Protocol, Generic

Returns the sum along the specified diagonals of a matrix (or a stack of matrices) x.

Parameters:

x (array) – input array having shape (..., M, N) and whose innermost two dimensions form MxN matrices. Should have a numeric data type.
offset (int) –
offset specifying the off-diagonal relative to the main diagonal.
- offset = 0: the main diagonal.
- offset > 0: off-diagonal above the main diagonal.
- offset < 0: off-diagonal below the main diagonal.
Default: 0.
dtype (Optional[dtype]) –
data type of the returned array. If None,
- if the default data type corresponding to the data type “kind” (integer, real-valued floating-point, or complex floating-point) of x has a smaller range of values than the data type of x (e.g., x has data type int64 and the default data type is int32, or x has data type uint64 and the default data type is int64), the returned array must have the same data type as x.
- if x has a real-valued floating-point data type, the returned array must have the default real-valued floating-point data type.
- if x has a complex floating-point data type, the returned array must have the default complex floating-point data type.
- if x has a signed integer data type (e.g., int16), the returned array must have the default integer data type.
- if x has an unsigned integer data type (e.g., uint16), the returned array must have an unsigned integer data type having the same number of bits as the default integer data type (e.g., if the default integer data type is int32, the returned array must have a uint32 data type).
If the data type (either specified or resolved) differs from the data type of x, the input array should be cast to the specified data type before computing the sum. Default: None.

Note

keyword argument is intended to help prevent data type overflows.

Returns:

out[i, j, k, ..., l] = trace(a[i, j, k, ..., l, :, :])

The returned array must have a data type as described by the dtype parameter above.

Return type:

array

Notes

Special Cases

Let N equal the number of elements over which to compute the sum.

If N is 0, the sum is 0 (i.e., the empty sum).

For both real-valued and complex floating-point operands, special cases must be handled as if the operation is implemented by successive application of add().

Changed in version 2022.12: Added complex data type support.

_abc_impl = <_abc._abc_data object>¶

_is_protocol = True¶

_is_runtime_protocol = True¶

class array_api._2022_12.tril(*args, **kwargs)[source]¶

Bases: Protocol, Generic

Returns the lower triangular part of a matrix (or a stack of matrices) x.

Note

The lower triangular part of the matrix is defined as the elements on and below the specified diagonal k.

Parameters:

x (array) – input array having shape (..., M, N) and whose innermost two dimensions form MxN matrices.
k (int) –
diagonal above which to zero elements. If k = 0, the diagonal is the main diagonal. If k < 0, the diagonal is below the main diagonal. If k > 0, the diagonal is above the main diagonal. Default: 0.

Note

The main diagonal is defined as the set of indices {(i, i)} for i on the interval [0, min(M, N) - 1].

Returns:

Return type:

array

_abc_impl = <_abc._abc_data object>¶

_is_protocol = True¶

_is_runtime_protocol = True¶

class array_api._2022_12.triu(*args, **kwargs)[source]¶

Bases: Protocol, Generic

Returns the upper triangular part of a matrix (or a stack of matrices) x.

Note

The upper triangular part of the matrix is defined as the elements on and above the specified diagonal k.

Parameters:

x (array) – input array having shape (..., M, N) and whose innermost two dimensions form MxN matrices.
k (int) –
diagonal below which to zero elements. If k = 0, the diagonal is the main diagonal. If k < 0, the diagonal is below the main diagonal. If k > 0, the diagonal is above the main diagonal. Default: 0.

Note

The main diagonal is defined as the set of indices {(i, i)} for i on the interval [0, min(M, N) - 1].

Returns:

Return type:

array

_abc_impl = <_abc._abc_data object>¶

_is_protocol = True¶

_is_runtime_protocol = True¶

class array_api._2022_12.trunc(*args, **kwargs)[source]¶

Bases: Protocol, Generic

Rounds each element x_i of the input array x to the nearest integer-valued number that is closer to zero than x_i.

Parameters:: x (array) – input array. Should have a real-valued data type.
Returns:: out – an array containing the rounded result for each element in x. The returned array must have the same data type as x.
Return type:: array

Notes

Special cases

If x_i is already integer-valued, the result is x_i.

For floating-point operands,

If x_i is +infinity, the result is +infinity.
If x_i is -infinity, the result is -infinity.
If x_i is +0, the result is +0.
If x_i is -0, the result is -0.
If x_i is NaN, the result is NaN.

_abc_impl = <_abc._abc_data object>¶

_is_protocol = True¶

_is_runtime_protocol = True¶

class array_api._2022_12.unique_all(*args, **kwargs)[source]¶

Bases: Protocol, Generic

Data-dependent output shape

Note

Uniqueness should be determined based on value equality (see equal()). For input arrays having floating-point data types, value-based equality implies the following behavior.

As nan values compare as False, nan values should be considered distinct.
As complex floating-point values having at least one nan component compare as False, complex floating-point values having nan components should be considered distinct.
As -0 and +0 compare as True, signed zeros should not be considered distinct, and the corresponding unique element will be implementation-dependent (e.g., an implementation could choose to return -0 if -0 occurs before +0).

As signed zeros are not distinct, using inverse_indices to reconstruct the input array is not guaranteed to return an array having the exact same values.

Each nan value and each complex floating-point value having a nan component should have a count of one, while the counts for signed zeros should be aggregated as a single count.

Parameters:

x (array) – input array. If x has more than one dimension, the function must flatten x and return the unique elements of the flattened array.

Returns:

out – a namedtuple (values, indices, inverse_indices, counts) whose

first element must have the field name values and must be an array containing the unique elements of x. The array must have the same data type as x.
second element must have the field name indices and must be an array containing the indices (first occurrences) of x that result in values. The array must have the same shape as values and must have the default array index data type.
third element must have the field name inverse_indices and must be an array containing the indices of values that reconstruct x. The array must have the same shape as x and must have the default array index data type.
fourth element must have the field name counts and must be an array containing the number of times each unique element occurs in x. The returned array must have same shape as values and must have the default array index data type.

Note

The order of unique elements is not specified and may vary between implementations.

Return type:

Tuple[array, array, array, array]

Notes

Changed in version 2022.12: Added complex data type support.

_abc_impl = <_abc._abc_data object>¶

_is_protocol = True¶

_is_runtime_protocol = True¶

class array_api._2022_12.unique_counts(*args, **kwargs)[source]¶

Bases: Protocol, Generic

Returns the unique elements of an input array x and the corresponding counts for each unique element in x.

Data-dependent output shape

Note

Uniqueness should be determined based on value equality (see equal()). For input arrays having floating-point data types, value-based equality implies the following behavior.

As nan values compare as False, nan values should be considered distinct.
As complex floating-point values having at least one nan component compare as False, complex floating-point values having nan components should be considered distinct.
As -0 and +0 compare as True, signed zeros should not be considered distinct, and the corresponding unique element will be implementation-dependent (e.g., an implementation could choose to return -0 if -0 occurs before +0).

Each nan value and each complex floating-point value having a nan component should have a count of one, while the counts for signed zeros should be aggregated as a single count.

Parameters:

x (array) – input array. If x has more than one dimension, the function must flatten x and return the unique elements of the flattened array.

Returns:

out – a namedtuple (values, counts) whose

first element must have the field name values and must be an array containing the unique elements of x. The array must have the same data type as x.
second element must have the field name counts and must be an array containing the number of times each unique element occurs in x. The returned array must have same shape as values and must have the default array index data type.

Note

The order of unique elements is not specified and may vary between implementations.

Return type:

Tuple[array, array]

Notes

Changed in version 2022.12: Added complex data type support.

_abc_impl = <_abc._abc_data object>¶

_is_protocol = True¶

_is_runtime_protocol = True¶

class array_api._2022_12.unique_inverse(*args, **kwargs)[source]¶

Bases: Protocol, Generic

Returns the unique elements of an input array x and the indices from the set of unique elements that reconstruct x.

Data-dependent output shape

Note

Uniqueness should be determined based on value equality (see equal()). For input arrays having floating-point data types, value-based equality implies the following behavior.

As nan values compare as False, nan values should be considered distinct.
As complex floating-point values having at least one nan component compare as False, complex floating-point values having nan components should be considered distinct.
As -0 and +0 compare as True, signed zeros should not be considered distinct, and the corresponding unique element will be implementation-dependent (e.g., an implementation could choose to return -0 if -0 occurs before +0).

As signed zeros are not distinct, using inverse_indices to reconstruct the input array is not guaranteed to return an array having the exact same values.

Parameters:

x (array) – input array. If x has more than one dimension, the function must flatten x and return the unique elements of the flattened array.

Returns:

out – a namedtuple (values, inverse_indices) whose

first element must have the field name values and must be an array containing the unique elements of x. The array must have the same data type as x.
second element must have the field name inverse_indices and must be an array containing the indices of values that reconstruct x. The array must have the same shape as x and have the default array index data type.

Note

The order of unique elements is not specified and may vary between implementations.

Return type:

Tuple[array, array]

Notes

Changed in version 2022.12: Added complex data type support.

_abc_impl = <_abc._abc_data object>¶

_is_protocol = True¶

_is_runtime_protocol = True¶

class array_api._2022_12.unique_values(*args, **kwargs)[source]¶

Bases: Protocol, Generic

Returns the unique elements of an input array x.

Data-dependent output shape

Note

Uniqueness should be determined based on value equality (see equal()). For input arrays having floating-point data types, value-based equality implies the following behavior.

As nan values compare as False, nan values should be considered distinct.
As complex floating-point values having at least one nan component compare as False, complex floating-point values having nan components should be considered distinct.
As -0 and +0 compare as True, signed zeros should not be considered distinct, and the corresponding unique element will be implementation-dependent (e.g., an implementation could choose to return -0 if -0 occurs before +0).

Parameters:

x (array) – input array. If x has more than one dimension, the function must flatten x and return the unique elements of the flattened array.

Returns:

out – an array containing the set of unique elements in x. The returned array must have the same data type as x.

Note

The order of unique elements is not specified and may vary between implementations.

Return type:

array

Notes

Changed in version 2022.12: Added complex data type support.

_abc_impl = <_abc._abc_data object>¶

_is_protocol = True¶

_is_runtime_protocol = True¶

class array_api._2022_12.var(*args, **kwargs)[source]¶

Bases: Protocol, Generic

Calculates the variance of the input array x.

Parameters:

x (array) – input array. Should have a real-valued floating-point data type.
axis (Optional[Union[int, Tuple[int, ...]]]) – axis or axes along which variances must be computed. By default, the variance must be computed over the entire array. If a tuple of integers, variances must be computed over multiple axes. Default: None.
correction (Union[int, float]) – degrees of freedom adjustment. Setting this parameter to a value other than 0 has the effect of adjusting the divisor during the calculation of the variance according to N-c where N corresponds to the total number of elements over which the variance is computed and c corresponds to the provided degrees of freedom adjustment. When computing the variance of a population, setting this parameter to 0 is the standard choice (i.e., the provided array contains data constituting an entire population). When computing the unbiased sample variance, setting this parameter to 1 is the standard choice (i.e., the provided array contains data sampled from a larger population; this is commonly referred to as Bessel’s correction). Default: 0.
keepdims (bool) – if True, the reduced axes (dimensions) must be included in the result as singleton dimensions, and, accordingly, the result must be compatible with the input array (see broadcasting). Otherwise, if False, the reduced axes (dimensions) must not be included in the result. Default: False.

Returns:

Return type:

array

Note

Notes

Special Cases

Let N equal the number of elements over which to compute the variance.

If N - correction is less than or equal to 0, the variance is NaN.
If x_i is NaN, the variance is NaN (i.e., NaN values propagate).

_abc_impl = <_abc._abc_data object>¶

_is_protocol = True¶

_is_runtime_protocol = True¶

class array_api._2022_12.vecdot(*args, **kwargs)[source]¶

Bases: Protocol, Generic

Computes the (vector) dot product of two arrays.

Let $\mathbf{a}$ be a vector in x1 and $\mathbf{b}$ be a corresponding vector in x2. The dot product is defined as

\[\mathbf{a} \cdot \mathbf{b} = \sum_{i=0}^{n-1} \overline{a_i}b_i\]

over the dimension specified by axis and where $n$ is the dimension size and $\overline{a_i}$ denotes the complex conjugate if $a_i$ is complex and the identity if $a_i$ is real-valued.

Parameters:

x1 (array) – first input array. Should have a floating-point data type.
x2 (array) –
second input array. Must be compatible with x1 for all non-contracted axes (see broadcasting). The size of the axis over which to compute the dot product must be the same size as the respective axis in x1. Should have a floating-point data type.

Note

The contracted axis (dimension) must not be broadcasted.
axis (int) – axis over which to compute the dot product. Must be an integer on the interval [-N, N), where N is the rank (number of dimensions) of the shape determined according to broadcasting. If specified as a negative integer, the function must determine the axis along which to compute the dot product by counting backward from the last dimension (where -1 refers to the last dimension). By default, the function must compute the dot product over the last axis. Default: -1.

Returns:

Return type:

array

Notes

Changed in version 2022.12: Added complex data type support.

Raises

if provided an invalid axis.
if the size of the axis over which to compute the dot product is not the same (before broadcasting) for both x1 and x2.

_abc_impl = <_abc._abc_data object>¶

_is_protocol = True¶

_is_runtime_protocol = True¶

class array_api._2022_12.vector_norm(*args, **kwargs)[source]¶

Bases: Protocol, Generic

Computes the vector norm of a vector (or batch of vectors) x.

Parameters:

x (array) – input array. Should have a floating-point data type.
axis (Optional[Union[int, Tuple[int, ...]]]) – If an integer, axis specifies the axis (dimension) along which to compute vector norms. If an n-tuple, axis specifies the axes (dimensions) along which to compute batched vector norms. If None, the vector norm must be computed over all array values (i.e., equivalent to computing the vector norm of a flattened array). Negative indices must be supported. Default: None.
keepdims (bool) – If True, the axes (dimensions) specified by axis must be included in the result as singleton dimensions, and, accordingly, the result must be compatible with the input array (see broadcasting). Otherwise, if False, the axes (dimensions) specified by axis must not be included in the result. Default: False.
ord (Union[int, float, Literal[inf, -inf]]) –
order of the norm. The following mathematical norms must be supported:

ord

description

1

L1-norm (Manhattan)

2

L2-norm (Euclidean)

inf

infinity norm

(int,float >= 1)

p-norm

The following non-mathematical “norms” must be supported:

ord

description

0

sum(a != 0)

-1

1./sum(1./abs(a))

-2

1./sqrt(sum(1./abs(a)**2))

-inf

min(abs(a))

(int,float < 1)

sum(abs(a)**ord)**(1./ord)

Default: 2.

Returns:

Return type:

ord	description
1	L1-norm (Manhattan)
2	L2-norm (Euclidean)
inf	infinity norm
(int,float >= 1)	p-norm

ord	description
0	sum(a != 0)
-1	1./sum(1./abs(a))
-2	1./sqrt(sum(1./abs(a)**2))
-inf	min(abs(a))
(int,float < 1)	sum(abs(a)ord)(1./ord)

array

Notes

Changed in version 2022.12: Added complex data type support.

_abc_impl = <_abc._abc_data object>¶

_is_protocol = True¶

_is_runtime_protocol = True¶

class array_api._2022_12.where(*args, **kwargs)[source]¶

Bases: Protocol, Generic

Returns elements chosen from x1 or x2 depending on condition.

Parameters:

condition (array) – when True, yield x1_i; otherwise, yield x2_i. Should have a boolean data type. Must be compatible with x1 and x2 (see broadcasting).
x1 (array) – first input array. Must be compatible with condition and x2 (see broadcasting).
x2 (array) – second input array. Must be compatible with condition and x1 (see broadcasting).

Returns:

Return type:

array

_abc_impl = <_abc._abc_data object>¶

_is_protocol = True¶

_is_runtime_protocol = True¶

class array_api._2022_12.zeros(*args, **kwargs)[source]¶

Bases: Protocol, Generic

Returns a new array having a specified shape and filled with zeros.

Parameters:

shape (Union[int, Tuple[int, ...]]) – output array shape.
dtype (Optional[dtype]) – output array data type. If dtype is None, the output array data type must be the default real-valued floating-point data type. Default: None.
device (Optional[device]) – device on which to place the created array. Default: None.

Returns:

out – an array containing zeros.

Return type:

array

_abc_impl = <_abc._abc_data object>¶

_is_protocol = True¶

_is_runtime_protocol = True¶

class array_api._2022_12.zeros_like(*args, **kwargs)[source]¶

Bases: Protocol, Generic

Returns a new array filled with zeros and having the same shape as an input array x.

Parameters:

x (array) – input array from which to derive the output array shape.
dtype (Optional[dtype]) – output array data type. If dtype is None, the output array data type must be inferred from x. Default: None.
device (Optional[device]) – device on which to place the created array. If device is None, the output array device must be inferred from x. Default: None.

Returns:

out – an array having the same shape as x and filled with zeros.

Return type:

array

_abc_impl = <_abc._abc_data object>¶

_is_protocol = True¶

_is_runtime_protocol = True¶