diff --git a/.github/workflows/pages.yml b/.github/workflows/pages.yml index 94419103c..77357707a 100644 --- a/.github/workflows/pages.yml +++ b/.github/workflows/pages.yml @@ -1,9 +1,9 @@ -on: +on: push: branches: - main pull_request: - branches: + branches: - "**" jobs: run: @@ -33,9 +33,7 @@ jobs: if-no-files-found: error - name: Deploy if: ${{ github.ref == 'refs/heads/main'}} - uses: JamesIves/github-pages-deploy-action@4.1.1 - env: - ACCESS_TOKEN: ${{ secrets.GITHUB_TOKEN }} - BASE_BRANCH: main # The branch the action should deploy from. - BRANCH: gh-pages # The branch the action should deploy to. - FOLDER: build + uses: peaceiris/actions-gh-pages@v3 + with: + github_token: ${{ secrets.GITHUB_TOKEN }} + publish_dir: ./build diff --git a/spec/API_specification/array_object.md b/spec/API_specification/array_object.md index c5ea23491..fb711feda 100644 --- a/spec/API_specification/array_object.md +++ b/spec/API_specification/array_object.md @@ -288,7 +288,7 @@ For floating-point operands, let `self` equal `x`. - **self**: _<array>_ - - array instance. + - array instance. Should have a numeric data type. #### Returns @@ -337,11 +337,11 @@ Floating-point addition is a commutative operation, but not always associative. - **self**: _<array>_ - - array instance (augend array). + - array instance (augend array). Should have a numeric data type. -- **other**: _<array>_ +- **other**: _Union\[ int, float, <array> ]_ - - addend array. Must be compatible with `self` (see {ref}`broadcasting`). + - addend array. Must be compatible with `self` (see {ref}`broadcasting`). Should have a numeric data type. #### Returns @@ -363,11 +363,11 @@ Evaluates `self_i & other_i` for each element of an array instance with the resp - **self**: _<array>_ - - array instance. Must have an integer or boolean data type. + - array instance. Should have an integer or boolean data type. -- **other**: _<array>_ +- **other**: _Union\[ int, bool, <array> ]_ - - other array. Must be compatible with `self` (see {ref}`broadcasting`). Must have an integer or boolean data type. + - other array. Must be compatible with `self` (see {ref}`broadcasting`). Should have an integer or boolean data type. #### Returns @@ -432,9 +432,15 @@ Exports the array for consumption by {ref}`function-from_dlpack` as a DLPack cap - array instance. -- **stream**: _Optional\[ int ]_ +- **stream**: _Optional\[ Union\[ int, Any ]]_ + + - for CUDA and ROCm, a Python integer representing a pointer to a stream, on devices that support streams. `stream` is provided by the consumer to the producer to instruct the producer to ensure that operations can safely be performed on the array (e.g., by inserting a dependency between streams via "wait for event"). The pointer must be a positive integer or `-1`. If `stream` is `-1`, the value may be used by the consumer to signal "producer must not perform any synchronization". The ownership of the stream stays with the consumer. + + On CPU and other device types without streams, only `None` is accepted. + + For other device types which do have a stream, queue or similar synchronization mechanism, the most appropriate type to use for `stream` is not yet determined. E.g., for SYCL one may want to use an object containing an in-order `cl::sycl::queue`. This is allowed when libraries agree on such a convention, and may be standardized in a future version of this API standard. - - a Python integer representing a pointer to a stream. `stream` is provided by the consumer to the producer to instruct the producer to ensure that operations can safely be performed on the array. The pointer must be a positive integer or `-1`. If `stream` is `-1`, the value may be used by the consumer to signal "producer must not perform any synchronization". Device-specific notes: + Device-specific notes: :::{admonition} CUDA - `None`: producer must assume the legacy default stream (default). @@ -505,11 +511,11 @@ Computes the truth value of `self_i == other_i` for each element of an array ins - **self**: _<array>_ - - array instance. + - array instance. May have any data type. -- **other**: _<array>_ +- **other**: _Union\[ int, float, bool, <array> ]_ - - other array. Must be compatible with `self` (see {ref}`broadcasting`). + - other array. Must be compatible with `self` (see {ref}`broadcasting`). May have any data type. #### Returns @@ -548,11 +554,11 @@ Evaluates `self_i // other_i` for each element of an array instance with the res - **self**: _<array>_ - - array instance. + - array instance. Should have a numeric data type. -- **other**: _<array>_ +- **other**: _Union\[ int, float, <array> ]_ - - other array. Must be compatible with `self` (see {ref}`broadcasting`). + - other array. Must be compatible with `self` (see {ref}`broadcasting`). Should have a numeric data type. #### Returns @@ -574,11 +580,11 @@ Computes the truth value of `self_i >= other_i` for each element of an array ins - **self**: _<array>_ - - array instance. + - array instance. Should have a numeric data type. -- **other**: _<array>_ +- **other**: _Union\[ int, float, <array> ]_ - - other array. Must be compatible with `self` (see {ref}`broadcasting`). + - other array. Must be compatible with `self` (see {ref}`broadcasting`). Should have a numeric data type. #### Returns @@ -598,7 +604,7 @@ Returns `self[key]`. #### Parameters -- **self**: _<array;>_ +- **self**: _<array>_ - array instance. @@ -621,11 +627,11 @@ Computes the truth value of `self_i > other_i` for each element of an array inst - **self**: _<array>_ - - array instance. + - array instance. Should have a numeric data type. -- **other**: _<array>_ +- **other**: _Union\[ int, float, <array> ]_ - - other array. Must be compatible with `self` (see {ref}`broadcasting`). + - other array. Must be compatible with `self` (see {ref}`broadcasting`). Should have a numeric data type. #### Returns @@ -664,7 +670,7 @@ Evaluates `~self_i` for each element of an array instance. - **self**: _<array>_ - - array instance. Must have an integer or boolean data type. + - array instance. Should have an integer or boolean data type. #### Returns @@ -686,11 +692,11 @@ Computes the truth value of `self_i <= other_i` for each element of an array ins - **self**: _<array>_ - - array instance. + - array instance. Should have a numeric data type. -- **other**: _<array>_ +- **other**: _Union\[ int, float, <array> ]_ - - other array. Must be compatible with `self` (see {ref}`broadcasting`). + - other array. Must be compatible with `self` (see {ref}`broadcasting`). Should have a numeric data type. #### Returns @@ -717,11 +723,11 @@ Evaluates `self_i << other_i` for each element of an array instance with the res - **self**: _<array>_ - - array instance. Must have an integer data type. + - array instance. Should have an integer data type. -- **other**: _<array>_ +- **other**: _Union\[ int, <array> ]_ - - other array. Must be compatible with `self` (see {ref}`broadcasting`). Must have an integer data type. Each element must be greater than or equal to `0`. + - other array. Must be compatible with `self` (see {ref}`broadcasting`). Should have an integer data type. Each element must be greater than or equal to `0`. #### Returns @@ -731,7 +737,7 @@ Evaluates `self_i << other_i` for each element of an array instance with the res ```{note} -Element-wise results must equal the results returned by the equivalent element-wise function [`less_equal(x1, x2)`](elementwise_functions.md#bitwise_left_shiftx1-x2-). +Element-wise results must equal the results returned by the equivalent element-wise function [`bitwise_left_shift(x1, x2)`](elementwise_functions.md#bitwise_left_shiftx1-x2-). ``` (method-__lt__)= @@ -743,11 +749,11 @@ Computes the truth value of `self_i < other_i` for each element of an array inst - **self**: _<array>_ - - array instance. + - array instance. Should have a numeric data type. -- **other**: _<array>_ +- **other**: _Union\[ int, float, <array> ]_ - - other array. Must be compatible with `self` (see {ref}`broadcasting`). + - other array. Must be compatible with `self` (see {ref}`broadcasting`). Should have a numeric data type. #### Returns @@ -763,23 +769,47 @@ Element-wise results must equal the results returned by the equivalent element-w (method-__matmul__)= ### \_\_matmul\_\_(self, other, /) -_TODO: awaiting `matmul` functional equivalent._ +Computes the matrix product. + +```{note} + +The `matmul` function must implement the same semantics as the built-in `@` operator (see [PEP 465](https://www.python.org/dev/peps/pep-0465)). +``` #### Parameters - **self**: _<array>_ - - array instance. + - array instance. Should have a numeric data type. Must have at least one dimension. If `self` is one-dimensional having shape `(M)` and `other` has more than one dimension, `self` 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 `self` has more than one dimension (including after vector-to-matrix promotion), `self` must be compatible with `other` (see {ref}`broadcasting`). If `self` has shape `(..., M, K)`, the innermost two dimensions form matrices on which to perform matrix multiplication. - **other**: _<array>_ - - other array. Must be compatible with `self` (see {ref}`broadcasting`). + - other array. Should have a numeric data type. Must have at least one dimension. If `other` is one-dimensional having shape `(N)` and `self` has more than one dimension, `other` 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 `other` has more than one dimension (including after vector-to-matrix promotion), `other` must be compatible with `self` (see {ref}`broadcasting`). If `other` has shape `(..., K, N)`, the innermost two dimensions form matrices on which to perform matrix multiplication. #### Returns - **out**: _<array>_ - - _TODO_ + - if both `self` and `other` are one-dimensional arrays having shape `(N)`, a zero-dimensional array containing the inner product as its only element. + - if `self` is a two-dimensional array having shape `(M, K)` and `other` is a two-dimensional array having shape `(K, N)`, a two-dimensional array containing the [conventional matrix product](https://en.wikipedia.org/wiki/Matrix_multiplication) and having shape `(M, N)`. + - if `self` is a one-dimensional array having shape `(K)` and `other` 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](https://en.wikipedia.org/wiki/Matrix_multiplication). + - if `self` is an array having shape `(..., M, K)` and `other` 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](https://en.wikipedia.org/wiki/Matrix_multiplication). + - if `self` is a two-dimensional array having shape `(M, K)` and `other` is an array having shape `(..., K, N)`, an array having shape `(..., M, N)` and containing the [conventional matrix product](https://en.wikipedia.org/wiki/Matrix_multiplication) for each stacked matrix. + - if `self` is an array having shape `(..., M, K)` and `other` is a two-dimensional array having shape `(K, N)`, an array having shape `(..., M, N)` and containing the [conventional matrix product](https://en.wikipedia.org/wiki/Matrix_multiplication) for each stacked matrix. + - if either `self` or `other` has more than two dimensions, an array having a shape determined by {ref}`broadcasting` `self` against `other` and containing the [conventional matrix product](https://en.wikipedia.org/wiki/Matrix_multiplication) for each stacked matrix. + + The returned array must have a data type determined by {ref}`type-promotion`. + + ```{note} + + Results must equal the results returned by the equivalent function [`matmul(x1, x2)`](linear_algebra_functions.md#matmulx1-x2-). + ``` + +#### Raises + +- if either `self` or `other` is a zero-dimensional array. +- if `self` is a one-dimensional array having shape `(N)`, `other` is a one-dimensional array having shape `(M)`, and `N != M`. +- if `self` is an array having shape `(..., M, K)`, `other` is an array having shape `(..., L, N)`, and `K != L`. (method-__mod__)= ### \_\_mod\_\_(self, other, /) @@ -790,11 +820,11 @@ Evaluates `self_i % other_i` for each element of an array instance with the resp - **self**: _<array>_ - - array instance. + - array instance. Should have a numeric data type. -- **other**: _<array>_ +- **other**: _Union\[ int, float, <array> ]_ - - other array. Must be compatible with `self` (see {ref}`broadcasting`). + - other array. Must be compatible with `self` (see {ref}`broadcasting`). Should have a numeric data type. #### Returns @@ -835,11 +865,11 @@ Floating-point multiplication is not always associative due to finite precision. - **self**: _<array>_ - - array instance. + - array instance. Should have a numeric data type. -- **other**: _<array>_ +- **other**: _Union\[ int, float, <array> ]_ - - other array. Must be compatible with `self` (see {ref}`broadcasting`). + - other array. Must be compatible with `self` (see {ref}`broadcasting`). Should have a numeric data type. #### Returns @@ -861,11 +891,11 @@ Computes the truth value of `self_i != other_i` for each element of an array ins - **self**: _<array>_ - - array instance. + - array instance. May have any data type. -- **other**: _<array>_ +- **other**: _Union\[ int, float, bool, <array> ]_ - - other array. Must be compatible with `self` (see {ref}`broadcasting`). + - other array. Must be compatible with `self` (see {ref}`broadcasting`). May have any data type. #### Returns @@ -887,7 +917,7 @@ Evaluates `-self_i` for each element of an array instance. - **self**: _<array>_ - - array instance. + - array instance. Should have a numeric data type. #### Returns @@ -909,11 +939,11 @@ Evaluates `self_i | other_i` for each element of an array instance with the resp - **self**: _<array>_ - - array instance. Must have an integer or boolean data type. + - array instance. Should have an integer or boolean data type. -- **other**: _<array>_ +- **other**: _Union\[ int, bool, <array> ]_ - - other array. Must be compatible with `self` (see {ref}`broadcasting`). Must have an integer or boolean data type. + - other array. Must be compatible with `self` (see {ref}`broadcasting`). Should have an integer or boolean data type. #### Returns @@ -935,7 +965,7 @@ Evaluates `+self_i` for each element of an array instance. - **self**: _<array>_ - - array instance. + - array instance. Should have a numeric data type. #### Returns @@ -986,11 +1016,11 @@ For floating-point operands, let `self` equal `x1` and `other` equal `x2`. - **self**: _<array>_ - - array instance whose elements correspond to the exponentiation base. + - array instance whose elements correspond to the exponentiation base. Should have a numeric data type. -- **other**: _<array>_ +- **other**: _Union\[ int, float, <array> ]_ - - other array whose elements correspond to the exponentiation exponent. Must be compatible with `self` (see {ref}`broadcasting`). + - other array whose elements correspond to the exponentiation exponent. Must be compatible with `self` (see {ref}`broadcasting`). Should have a numeric data type. #### Returns @@ -1012,11 +1042,11 @@ Evaluates `self_i >> other_i` for each element of an array instance with the res - **self**: _<array>_ - - array instance. Must have an integer data type. + - array instance. Should have an integer data type. -- **other**: _<array>_ +- **other**: _Union\[ int, <array> ]_ - - other array. Must be compatible with `self` (see {ref}`broadcasting`). Must have an integer data type. Each element must be greater than or equal to `0`. + - other array. Must be compatible with `self` (see {ref}`broadcasting`). Should have an integer data type. Each element must be greater than or equal to `0`. #### Returns @@ -1043,11 +1073,11 @@ Calculates the difference for each element of an array instance with the respect - **self**: _<array>_ - - array instance (minuend array). + - array instance (minuend array). Should have a numeric data type. -- **other**: _<array>_ +- **other**: _Union\[ int, float, <array> ]_ - - subtrahend array. Must be compatible with `self` (see {ref}`broadcasting`). + - subtrahend array. Must be compatible with `self` (see {ref}`broadcasting`). Should have a numeric data type. #### Returns @@ -1096,11 +1126,11 @@ For floating-point operands, let `self` equal `x1` and `other` equal `x2`. - **self**: _<array>_ - - array instance. + - array instance. Should have a numeric data type. -- **other**: _<array>_ +- **other**: _Union\[ int, float, <array> ]_ - - other array. Must be compatible with `self` (see {ref}`broadcasting`). + - other array. Must be compatible with `self` (see {ref}`broadcasting`). Should have a numeric data type. #### Returns @@ -1122,11 +1152,11 @@ Evaluates `self_i ^ other_i` for each element of an array instance with the resp - **self**: _<array>_ - - array instance. Must have an integer or boolean data type. + - array instance. Should have an integer or boolean data type. -- **other**: _<array>_ +- **other**: _Union\[ int, bool, <array> ]_ - - other array. Must be compatible with `self` (see {ref}`broadcasting`). Must have an integer or boolean data type. + - other array. Must be compatible with `self` (see {ref}`broadcasting`). Should have an integer or boolean data type. #### Returns diff --git a/spec/API_specification/constants.md b/spec/API_specification/constants.md index 4a9cd040f..78cdc8de7 100644 --- a/spec/API_specification/constants.md +++ b/spec/API_specification/constants.md @@ -2,7 +2,9 @@ > Array API specification for constants. -A conforming implementation of the array API standard must provide and support the following constants. +A conforming implementation of the array API standard must provide and support the following constants adhering to the following conventions. + +- Each constant must have a Python floating-point data type (i.e., `float`) and be provided as a Python scalar value. @@ -11,7 +13,7 @@ A conforming implementation of the array API standard must provide and support t (constant-e)= ### e -Euler's constant. +IEEE 754 floating-point representation of Euler's constant. ```text e = 2.71828182845904523536028747135266249775724709369995... @@ -30,7 +32,7 @@ IEEE 754 floating-point representation of Not a Number (`NaN`). (constant-pi)= ### pi -The mathematical constant `π`. +IEEE 754 floating-point representation of the mathematical constant `π`. ```text pi = 3.1415926535897932384626433... diff --git a/spec/API_specification/creation_functions.md b/spec/API_specification/creation_functions.md index 7ce12f5d9..8b1bec16e 100644 --- a/spec/API_specification/creation_functions.md +++ b/spec/API_specification/creation_functions.md @@ -12,7 +12,7 @@ A conforming implementation of the array API standard must provide and support t (function-arange)= -### arange(start, /, *, stop=None, step=1, dtype=None, device=None) +### arange(start, /, stop=None, step=1, *, dtype=None, device=None) Returns evenly spaced values within the half-open interval `[start, stop)` as a one-dimensional array. @@ -33,11 +33,11 @@ This function cannot guarantee that the interval does not include the `stop` val - **step**: _Union\[ int, float ]_ - - the distance between two adjacent elements (`out[i+1] - out[i]`). Default: `1`. + - 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 the default floating-point data type. Default: `None`. + - 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 floating-point data type. Default: `None`. - **device**: _Optional\[ <device> ]_ @@ -47,7 +47,7 @@ This function cannot guarantee that the interval does not include the `stop` val - **out**: _<array>_ - - a one-dimensional array containing evenly spaced values. The length of the output array must be `ceil((stop-start)/step)`. + - 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. (function-asarray)= @@ -68,7 +68,7 @@ Convert the input to an array. - **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`. Default: `None`. + - 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 if they're all `bool` the output dtype will be `bool`; if they're a mix of `bool`s and `int` the output dtype will be the default integer data type; if they contain `float`s the output dtype will be the default floating-point data type. Default: `None`. - **device**: _Optional\[ <device> ]_ @@ -86,7 +86,7 @@ Convert the input to an array. (function-empty)= -### empty(shape, /, *, dtype=None, device=None) +### empty(shape, *, dtype=None, device=None) Returns an uninitialized array having a specified `shape`. @@ -136,19 +136,19 @@ Returns an uninitialized array with the same `shape` as an input array `x`. - an array having the same shape as `x` and containing uninitialized data. (function-eye)= -### eye(N, /, *, M=None, k=0, dtype=None, device=None) +### eye(n_rows, n_cols=None, /, *, k=0, dtype=None, device=None) Returns a two-dimensional array with ones on the `k`th diagonal and zeros elsewhere. #### Parameters -- **N**: _int_ +- **n_rows**: _int_ - number of rows in the output array. -- **M**: _Optional\[ int ]_ +- **n_cols**: _Optional\[ int ]_ - - number of columns in the output array. If `None`, the default number of columns in the output array is `N`. Default: `None`. + - 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**: _Optional\[ int ]_ @@ -190,7 +190,7 @@ Returns a new array containing the data from another (array) object with a `__dl ``` (function-full)= -### full(shape, fill_value, /, *, dtype=None, device=None) +### full(shape, fill_value, *, dtype=None, device=None) Returns a new array having a specified `shape` and filled with `fill_value`. @@ -206,7 +206,7 @@ Returns a new array having a specified `shape` and filled with `fill_value`. - **dtype**: _Optional\[ <dtype> ]_ - - output array data type. If `dtype` is `None`, the output array data type must be the default floating-point data type. Default: `None`. + - output array data type. If `dtype` is `None`, the output array data type must be inferred from `fill_value`. If it's an `int`, the output array dtype must be the default integer dtype; if it's a `float`, then the output array dtype must be the default floating-point data type; if it's a `bool` then the output array must have boolean dtype. Default: `None`. - **device**: _Optional\[ <device> ]_ @@ -219,7 +219,7 @@ Returns a new array having a specified `shape` and filled with `fill_value`. - an array where every element is equal to `fill_value`. (function-full_like)= -### full_like(x, fill_value, /, *, dtype=None, device=None) +### full_like(x, /, fill_value, *, dtype=None, device=None) Returns a new array filled with `fill_value` and having the same `shape` as an input array `x`. @@ -235,7 +235,7 @@ Returns a new array filled with `fill_value` and having the same `shape` as an i - **dtype**: _Optional\[ <dtype> ]_ - - output array data type. If `dtype` is `None`, the output array data type must be inferred from `x`. Default: `None`. + - output array data type. If `dtype` is `None`, the output array data type must be inferred from `fill_value` (see {ref}`function-full`). Default: `None`. - **device**: _Optional\[ <device> ]_ @@ -248,7 +248,7 @@ Returns a new array filled with `fill_value` and having the same `shape` as an i - an array having the same shape as `x` and where every element is equal to `fill_value`. (function-linspace)= -### linspace(start, stop, num, /, *, dtype=None, device=None, endpoint=True) +### linspace(start, stop, /, num, *, dtype=None, device=None, endpoint=True) Returns evenly spaced numbers over a specified interval. @@ -290,7 +290,7 @@ Returns evenly spaced numbers over a specified interval. - a one-dimensional array containing evenly spaced values. (function-meshgrid)= -### meshgrid(*arrays, /, *, indexing='xy') +### meshgrid(*arrays, indexing='xy') Returns coordinate matrices from coordinate vectors. @@ -302,26 +302,26 @@ Returns coordinate matrices from coordinate vectors. - **indexing**: _str_ - - 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'`. + - 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\[ <array>, ... ]_ - - 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)`, + - 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 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)`. + - 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)`. + 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)`. + 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)`. - The returned arrays must have a numeric data type determined by {ref}`type-promotion`. + The returned arrays must have a numeric data type determined by {ref}`type-promotion`. (function-ones)= -### ones(shape, /, *, dtype=None, device=None) +### ones(shape, *, dtype=None, device=None) Returns a new array having a specified `shape` and filled with ones. @@ -371,7 +371,7 @@ Returns a new array filled with ones and having the same `shape` as an input arr - an array having the same shape as `x` and filled with ones. (function-zeros)= -### zeros(shape, /, *, dtype=None, device=None) +### zeros(shape, *, dtype=None, device=None) Returns a new array having a specified `shape` and filled with zeros. diff --git a/spec/API_specification/data_type_functions.md b/spec/API_specification/data_type_functions.md index 8df12d409..0b76b386b 100644 --- a/spec/API_specification/data_type_functions.md +++ b/spec/API_specification/data_type_functions.md @@ -7,14 +7,14 @@ A conforming implementation of the array API standard must provide and support t ## Objects in API -(function-broadcast-arrays)= -### broadcast_arrays(\*args, /) +(function-broadcast_arrays)= +### broadcast_arrays(*arrays) Broadcasts one or more arrays against one another. #### Parameters -- **\*args**: _Sequence\[ <array> ]_ +- **arrays**: _Sequence\[ <array> ]_ - arrays to broadcast. @@ -24,8 +24,8 @@ Broadcasts one or more arrays against one another. - a list of broadcasted arrays. Each array must have the same shape. Each array must have the same dtype as its corresponding input array. -(function-broadcast-to)= -### broadcast_to(x, shape, /) +(function-broadcast_to)= +### broadcast_to(x, /, shape) Broadcasts an array to a specified shape. @@ -49,14 +49,14 @@ Broadcasts an array to a specified shape. - if the array is incompatible with the specified shape (see {ref}`broadcasting`). -(function-can-cast)= -### can_cast(from, to, /) +(function-can_cast)= +### can_cast(from_, to, /) Determines if one data type can be cast to another data type according {ref}`type-promotion` rules. #### Parameters -- **from**: _Union\[ <dtype>, <array>]_ +- **from_**: _Union\[ <dtype>, <array>]_ - input data type or array from which to cast. diff --git a/spec/API_specification/data_types.md b/spec/API_specification/data_types.md index b1c83b4d5..3788d8494 100644 --- a/spec/API_specification/data_types.md +++ b/spec/API_specification/data_types.md @@ -6,10 +6,9 @@ A conforming implementation of the array API standard must provide and support the following data types. -A conforming implementation of the array API standard must define a default floating-point data type (either `float32` or `float64`), as well as a default data type for an array index (either `int32` or `int64`). +A conforming implementation of the array API standard must define a default floating-point data type (either `float32` or `float64`), as well as a default integer data type (`int32` or `int64`). These default data types must be the same across platforms. The default integer data type may vary depending on whether Python is 32-bit or 64-bit. ```{note} - The default floating-point and array index integer data types should be clearly defined in a conforming library's documentation. ``` diff --git a/spec/API_specification/elementwise_functions.md b/spec/API_specification/elementwise_functions.md index 3dcf0dcf4..aa69c9a6a 100644 --- a/spec/API_specification/elementwise_functions.md +++ b/spec/API_specification/elementwise_functions.md @@ -570,7 +570,7 @@ Computes the truth value of `x1_i == x2_i` for each element `x1_i` of the input - **x2**: _<array>_ - - second input array. Must be compatible with `x1` (see {ref}`broadcasting`). + - second input array. Must be compatible with `x1` (see {ref}`broadcasting`). May have any data type. #### Returns @@ -946,11 +946,11 @@ For floating-point operands, - **x1**: _<array>_ - - first input array. + - first input array. Should have a floating-point data type. - **x2**: _<array>_ - - second input array. Must be compatible with `x1` (see {ref}`broadcasting`). + - second input array. Must be compatible with `x1` (see {ref}`broadcasting`). Should have a floating-point data type. #### Returns diff --git a/spec/API_specification/indexing.md b/spec/API_specification/indexing.md index 57689d489..98db203a3 100644 --- a/spec/API_specification/indexing.md +++ b/spec/API_specification/indexing.md @@ -160,6 +160,12 @@ _Rationale: this is consistent with bounds-checking for single-axis indexing. An ## Boolean Array Indexing +:::{admonition} Data-dependent output shape +:class: important + +For common boolean array use cases (e.g., using a dynamically-sized boolean array mask to filter the values of another array), the shape of the output array is data-dependent; hence, libraries that build array computation graphs (e.g., JAX, Dask, etc.) may find boolean array indexing difficult to implement. Accordingly, such libraries may choose to omit boolean array indexing. See {ref}`data-dependent-output-shapes` section for more details. +::: + An array must support indexing where the **sole index** is an `M`-dimensional boolean array `B` with shape `S1 = (s1, ..., sM)` according to the following rules. Let `A` be an `N`-dimensional array with shape `S2 = (s1, ..., sM, ..., sN)`. - If `N >= M`, then `A[B]` must replace the first `M` dimensions of `A` with a single dimension having a size equal to the number of `True` elements in `B`. The values in the resulting array must be in row-major (C-style order); this is equivalent to `A[nonzero(B)]`. diff --git a/spec/API_specification/linear_algebra_functions.md b/spec/API_specification/linear_algebra_functions.md index 100efccf2..3724c2c24 100644 --- a/spec/API_specification/linear_algebra_functions.md +++ b/spec/API_specification/linear_algebra_functions.md @@ -15,330 +15,131 @@ A conforming implementation of the array API standard must provide and support t -(function-cholesky)= -### cholesky() +(function-einsum)= +### einsum() TODO -(function-cross)= -### cross(x1, x2, /, *, axis=-1) +(function-matmul)= +### matmul(x1, x2, /) + +Computes the matrix product. + +```{note} -Returns the cross product of 3-element vectors. If `x1` and `x2` are multi-dimensional arrays (i.e., both have a rank greater than `1`), then the cross-product of each pair of corresponding 3-element vectors is independently computed. +The `matmul` function must implement the same semantics as the built-in `@` operator (see [PEP 465](https://www.python.org/dev/peps/pep-0465)). +``` #### Parameters - **x1**: _<array>_ - - first input array. Should have a numeric data type. + - 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), `x1` must be compatible with `x2` (see {ref}`broadcasting`). If `x1` has shape `(..., M, K)`, the innermost two dimensions form matrices on which to perform matrix multiplication. - **x2**: _<array>_ - - second input array. Must have the same shape as `x1`. Should have a numeric data type. - -- **axis**: _int_ - - - the axis (dimension) of `x1` and `x2` containing the vectors for which to compute the cross product. If set to `-1`, the function computes the cross product for vectors defined by the last axis (dimension). Default: `-1`. + - 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), `x2` must be compatible with `x1` (see {ref}`broadcasting`). If `x2` has shape `(..., K, N)`, the innermost two dimensions form matrices on which to perform matrix multiplication. #### Returns - **out**: _<array>_ - - an array containing the cross products. The returned array must have a data type determined by {ref}`type-promotion`. - -(function-det)= -### det(x, /) + - 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](https://en.wikipedia.org/wiki/Matrix_multiplication) 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](https://en.wikipedia.org/wiki/Matrix_multiplication). + - 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](https://en.wikipedia.org/wiki/Matrix_multiplication). + - 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](https://en.wikipedia.org/wiki/Matrix_multiplication) 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](https://en.wikipedia.org/wiki/Matrix_multiplication) for each stacked matrix. + - if either `x1` or `x2` has more than two dimensions, an array having a shape determined by {ref}`broadcasting` `x1` against `x2` and containing the [conventional matrix product](https://en.wikipedia.org/wiki/Matrix_multiplication) for each stacked matrix. -Returns the determinant of a square matrix (or stack of square matrices) `x`. + The returned array must have a data type determined by {ref}`type-promotion`. -#### Parameters +#### Raises -- **x**: _<array>_ +- if either `x1` or `x2` is a zero-dimensional array. +- if `x1` is a one-dimensional array having shape `(N)`, `x2` is a one-dimensional array having shape `(M)`, and `N != M`. +- if `x1` is an array having shape `(..., M, K)`, `x2` is an array having shape `(..., L, N)`, and `K != L`. - - input array having shape `(..., M, M)` and whose innermost two dimensions form square matrices. Should have a floating-point data type. +(function-tensordot)= +### tensordot(x1, x2, /, *, axes=2) -#### Returns - -- **out**: _<array>_ - - - 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`. - -(function-diagonal)= -### diagonal(x, /, *, axis1=0, axis2=1, offset=0) - -Returns the specified diagonals. If `x` has more than two dimensions, then the axes (dimensions) specified by `axis1` and `axis2` are used to determine the two-dimensional sub-arrays from which to return diagonals. +Returns a tensor contraction of `x1` and `x2` over specific axes. #### Parameters -- **x**: _<array>_ - - - input array. Must have at least `2` dimensions. - -- **axis1**: _int_ - - - first axis (dimension) with respect to which to take diagonal. Default: `0`. - -- **axis2**: _int_ - - - second axis (dimension) with respect to which to take diagonal. Default: `1`. - -- **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**: _<array>_ - - - if `x` is a two-dimensional array, a one-dimensional array containing the diagonal; otherwise, a multi-dimensional array containing the diagonals and whose shape is determined by removing `axis1` and `axis2` and appending a dimension equal to the size of the resulting diagonals. The returned array must have the same data type as `x`. - -(function-dot)= -### dot() - -TODO - -(function-eig)= -### eig() - -TODO - -(function-eigvalsh)= -### eigvalsh() +- **x1**: _<array>_ -TODO + - first input array. Should have a numeric data type. -(function-einsum)= -### einsum() +- **x2**: _<array>_ -TODO + - second input array. Must be compatible with `x1` (see {ref}`broadcasting`). Should have a numeric data type. -(function-inv)= -### inv(x, /) +- **axes**: _Union\[ int, Tuple\[ Sequence\[ int ], Sequence\[ int ] ] ]_ -Computes the multiplicative inverse of a square matrix (or a stack of square matrices) `x`. + - number of axes (dimensions) to contract or explicit sequences of axes (dimensions) for `x1` and `x2`, respectively. -#### Parameters + 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. -- **x**: _<array>_ + - 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). - - input array having shape `(..., M, M)` and whose innermost two dimensions form square matrices. Should have a floating-point data type. + If `axes` is a tuple of two sequences `(x1_axes, x2_axes)`, the first sequence must apply to `x` 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. #### Returns - **out**: _<array>_ - - an array containing the multiplicative inverses. The returned array must have a floating-point data type determined by {ref}`type-promotion` and must have the same shape as `x`. - -(function-lstsq)= -### lstsq() - -TODO - -(function-matmul)= -### matmul() - -TODO - -(function-matrix_power)= -### matrix_power() - -TODO - -(function-matrix_rank)= -### matrix_rank() + - 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 {ref}`type-promotion`. -TODO - -(function-norm)= -### norm(x, /, *, axis=None, keepdims=False, ord=None) +(function-transpose)= +### transpose(x, /, *, axes=None) -Computes the matrix or vector norm of `x`. +Transposes (or permutes the axes (dimensions)) of an array `x`. #### Parameters - **x**: _<array>_ - - input array. Should have a floating-point data type. - -- **axis**: _Optional\[ Union\[ int, Tuple\[ int, int ] ] ]_ - - - If an integer, `axis` specifies the axis (dimension) along which to compute vector norms. - - If a 2-tuple, `axis` specifies the axes (dimensions) defining two-dimensional matrices for which to compute matrix norms. - - If `None`, - - - if `x` is one-dimensional, the function must compute the vector norm. - - if `x` is two-dimensional, the function must compute the matrix norm. - - if `x` has more than two dimensions, the function must compute the vector norm 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 {ref}`broadcasting`). Otherwise, if `False`, the axes (dimensions) specified by `axis` 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 | matrix | vector | - | ---------------- | ------------------------------- | -------------------------- | - | 'fro' | 'fro' | - | - | 'nuc' | 'nuc' | - | - | 1 | max(sum(abs(x), axis=0)) | L1-norm (Manhattan) | - | 2 | largest singular value | L2-norm (Euclidean) | - | inf | max(sum(abs(x), axis=1)) | infinity norm | - | (int,float >= 1) | - | p-norm | - - The following non-mathematical "norms" must be supported: - - | ord | matrix | vector | - | ---------------- | ------------------------------- | ------------------------------ | - | 0 | - | sum(a != 0) | - | -1 | min(sum(abs(x), axis=0)) | 1./sum(1./abs(a)) | - | -2 | smallest singular value | 1./sqrt(sum(1./abs(a)\*\*2)) | - | -inf | min(sum(abs(x), axis=1)) | min(abs(a)) | - | (int,float < 1) | - | sum(abs(a)\*\*ord)\*\*(1./ord) | - - When `ord` is `None`, the following norms must be the default norms: - - | ord | matrix | vector | - | ---------------- | ------------------------------- | -------------------------- | - | None | 'fro' | L2-norm (Euclidean) | - - where `fro` corresponds to the **Frobenius norm**, `nuc` corresponds to the **nuclear norm**, and `-` indicates that the norm is **not** supported. - - For matrices, - - - 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). - - If `None`, + - input array. - - if matrix (or matrices), the function must compute the Frobenius norm. - - if vector (or vectors), the function must compute the L2-norm (Euclidean norm). +- **axes**: _Optional\[ Tuple\[ int, ... ] ]_ - Default: `None`. + - tuple containing a permutation of `(0, 1, ..., N-1)` where `N` is the number of axes (dimensions) of `x`. If `None`, the axes (dimensions) must be permuted in reverse order (i.e., equivalent to setting `axes=(N-1, ..., 1, 0)`). Default: `None`. #### Returns - **out**: _<array>_ - - an array containing the 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 2-tuple, the returned array must have a rank which is two less than the rank of `x`. The returned array must have a floating-point data type determined by {ref}`type-promotion`. + - an array containing the transpose. The returned array must have the same data type as `x`. -(function-outer)= -### outer(x1, x2, /) +(function-vecdot)= +### vecdot(x1, x2, /, *, axis=None) -Computes the outer product of two vectors `x1` and `x2`. +Computes the (vector) dot product of two arrays. #### Parameters - **x1**: _<array>_ - - first one-dimensional input array of size `N`. Should have a numeric data type. + - first input array. Should have a numeric data type. - **x2**: _<array>_ - - second one-dimensional input array of size `M`. Should have a numeric data type. - -#### Returns - -- **out**: _<array>_ - - - a two-dimensional array containing the outer product and whose shape is `(N, M)`. The returned array must have a data type determined by {ref}`type-promotion`. - -(function-pinv)= -### pinv() - -TODO - -(function-qr)= -### qr() - -TODO - -(function-slogdet)= -### slogdet() - -TODO - -(function-solve)= -### solve() - -TODO - -(function-svd)= -### svd() - -TODO + - second input array. Must be compatible with `x1` (see {ref}`broadcasting`). Should have a numeric data type. -(function-trace)= -### trace(x, /, *, axis1=0, axis2=1, offset=0) +- **axis**: _Optional\[ int ]_ -Returns the sum along the specified diagonals. If `x` has more than two dimensions, then the axes (dimensions) specified by `axis1` and `axis2` are used to determine the two-dimensional sub-arrays for which to compute the trace. - -#### Parameters - -- **x**: _<array>_ - - - input array. Must have at least `2` dimensions. Should have a numeric data type. - -- **axis1**: _int_ - - - first axis (dimension) with respect to which to compute the trace. Default: `0`. - -- **axis2**: _int_ - - - second axis (dimension) with respect to which to compute the trace. Default: `1`. - -- **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`. + - 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 {ref}`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). If `None`, the function must compute the dot product over the last axis. Default: `None`. #### Returns - **out**: _<array>_ - - if `x` is a two-dimensional array, the returned array must be a zero-dimensional array containing the trace; otherwise, the returned array must be a multi-dimensional array containing the traces. - - The shape of a multi-dimensional output array is determined by removing `axis1` and `axis2` and storing the traces in the last array dimension. For example, if `x` has rank `k` and shape `(I, J, K, ..., L, M, N)` and `axis1=-2` and `axis1=-1`, then a multi-dimensional output array has rank `k-2` and shape `(I, J, K, ..., L)` where + - 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 {ref}`broadcasting`. The returned array must have a data type determined by {ref}`type-promotion`. - ```text - out[i, j, k, ..., l] = trace(a[i, j, k, ..., l, :, :]) - ``` - - The returned array must have the same data type as `x`. - -(function-transpose)= -### transpose(x, /, *, axes=None) +#### Raises -Transposes (or permutes the axes (dimensions)) of an array `x`. - -#### Parameters - -- **x**: _<array>_ - - - input array. - -- **axes**: _Optional\[ Tuple\[ int, ... ] ]_ - - - tuple containing a permutation of `(0, 1, ..., N-1)` where `N` is the number of axes (dimensions) of `x`. If `None`, the axes (dimensions) must be permuted in reverse order (i.e., equivalent to setting `axes=(N-1, ..., 1, 0)`). Default: `None`. - -#### Returns - -- **out**: _<array>_ - - - an array containing the transpose. The returned array must have the same data type as `x`. +- if provided an invalid `axis`. +- if the size of the axis over which to compute the dot product is not the same for both `x1` and `x2`. diff --git a/spec/API_specification/manipulation_functions.md b/spec/API_specification/manipulation_functions.md index 29fca5a29..ae8499370 100644 --- a/spec/API_specification/manipulation_functions.md +++ b/spec/API_specification/manipulation_functions.md @@ -39,7 +39,7 @@ Joins a sequence of arrays along an existing axis. ``` (function-expand_dims)= -### expand_dims(x, axis, /) +### expand_dims(x, /, *, axis) Expands the shape of an array by inserting a new axis (dimension) of size one at the position specified by `axis`. @@ -81,7 +81,7 @@ Reverses the order of elements in an array along the given axis. The shape of th - an output array having the same data type and shape as `x` and whose elements, relative to `x`, are reordered. (function-reshape)= -### reshape(x, shape, /) +### reshape(x, /, shape) Reshapes an array without changing its data. @@ -102,7 +102,7 @@ Reshapes an array without changing its data. - an output array having the same data type, elements, and underlying element order as `x`. (function-roll)= -### roll(x, shift, /, *, axis=None) +### roll(x, /, shift, *, axis=None) 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. @@ -127,7 +127,7 @@ Rolls array elements along a specified axis. Array elements that roll beyond the - an output array having the same data type as `x` and whose elements, relative to `x`, are shifted. (function-squeeze)= -### squeeze(x, /, *, axis=None) +### squeeze(x, /, axis) Removes singleton dimensions (axes) from `x`. @@ -137,9 +137,9 @@ Removes singleton dimensions (axes) from `x`. - input array. -- **axis**: _Optional\[ Union\[ int, Tuple\[ int, ... ] ] ]_ +- **axis**: _Union\[ int, Tuple\[ int, ... ] ]_ - - axis (or axes) to squeeze. If provided, only the specified axes must be squeezed. If `axis` is `None`, all singleton dimensions (axes) must be removed. If a specified axis has a size greater than one, the specified axis must be left unchanged. Default: `None`. + - axis (or axes) to squeeze. If a specified axis has a size greater than one, a `ValueError` must be raised. #### Returns diff --git a/spec/API_specification/searching_functions.md b/spec/API_specification/searching_functions.md index d3d06d7a6..a521f98aa 100644 --- a/spec/API_specification/searching_functions.md +++ b/spec/API_specification/searching_functions.md @@ -69,6 +69,12 @@ Returns the indices of the minimum values along a specified axis. When the minim (function-nonzero)= ### nonzero(x, /) +:::{admonition} Data-dependent output shape +:class: important + +The shape of the output array for this function depends on the data values in the input array; hence, libraries that build array 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 {ref}`data-dependent-output-shapes` section for more details. +::: + Returns the indices of the array elements which are non-zero. #### Parameters diff --git a/spec/API_specification/set_functions.md b/spec/API_specification/set_functions.md index 73d8338cc..e6b949cb7 100644 --- a/spec/API_specification/set_functions.md +++ b/spec/API_specification/set_functions.md @@ -15,6 +15,12 @@ A conforming implementation of the array API standard must provide and support t (function-unique)= ### unique(x, /, *, return_counts=False, return_index=False, return_inverse=False) +:::{admonition} Data-dependent output shape +:class: important + +The shapes of one or more of output arrays for this function depend on the data values in the input array; hence, libraries that build array 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 {ref}`data-dependent-output-shapes` section for more details. +::: + Returns the unique elements of an input array `x`. #### Parameters @@ -51,11 +57,11 @@ Returns the unique elements of an input array `x`. - **indices**: _<array>_ - - an array containing the indices (first occurrences) of `x` that result in `unique`. The returned array must have the default array index data type. + - an array containing the indices (first occurrences) of `x` that result in `unique`. The returned array must have the default integer data type. - **inverse**: _<array>_ - - an array containing the indices of `unique` that reconstruct `x`. The returned array must have the default array index data type. + - an array containing the indices of `unique` that reconstruct `x`. The returned array must have the default integer data type. - **counts**: _<array>_ diff --git a/spec/design_topics/data_dependent_output_shapes.md b/spec/design_topics/data_dependent_output_shapes.md new file mode 100644 index 000000000..b110d7880 --- /dev/null +++ b/spec/design_topics/data_dependent_output_shapes.md @@ -0,0 +1,15 @@ +(data-dependent-output-shapes)= + +# Data-dependent output shapes + +Array libraries which build array computation graphs commonly employ static memory allocation techniques for performance optimization. In order to efficiently perform static memory allocation, such libraries must be able to infer array sizes during ahead-of-time (AOT) compilation. Functions and operations which are value-dependent present difficulties for such libraries, as array sizes cannot be inferred AOT without also knowing the contents of the respective arrays. + +While value-dependent functions and operations are not impossible for array computation graph libraries to implement, this specification does not want to impose and undue burden on such libraries and permits omission of value-dependent operations. All other array libraries are expected, however, to implement the value-dependent operations included in this specification in order to be array specification compliant. + +Value-dependent operations are demarcated in this specification using an admonition similar to the following: + +:::{admonition} Data-dependent output shape +:class: important + +The shape of the output array for this function/operation depends on the data values in the input array; hence, libraries that build array computation graphs (e.g., JAX, Dask, etc.) may find this function/operation difficult to implement without knowing array values. Accordingly, such libraries may choose to omit this function. See {ref}`data-dependent-output-shapes` section for more details. +::: \ No newline at end of file diff --git a/spec/design_topics/data_interchange.md b/spec/design_topics/data_interchange.md index 0cfa0c784..a834db3ae 100644 --- a/spec/design_topics/data_interchange.md +++ b/spec/design_topics/data_interchange.md @@ -111,6 +111,8 @@ The `__dlpack__` method will produce a `PyCapsule` containing a `from_dlpack` - therefore it is consumed exactly once, and it will not be visible to users of the Python API. +The producer must set the PyCapsule name to ``"dltensor"`` so that it can +be inspected by name. The consumer must set the PyCapsule name to `"used_dltensor"`, and call the `deleter` of the `DLPackManagedTensor` when it no longer needs the data. diff --git a/spec/design_topics/index.rst b/spec/design_topics/index.rst index 6e41899ca..2729cdbe4 100644 --- a/spec/design_topics/index.rst +++ b/spec/design_topics/index.rst @@ -6,6 +6,7 @@ Design topics & constraints :maxdepth: 1 copies_views_and_mutation + data_dependent_output_shapes data_interchange device_support static_typing diff --git a/spec/extensions/index.rst b/spec/extensions/index.rst new file mode 100644 index 000000000..1b3b7470f --- /dev/null +++ b/spec/extensions/index.rst @@ -0,0 +1,10 @@ +.. _extensions: + +Extensions +========== + +.. toctree:: + :caption: Extensions + :maxdepth: 3 + + linear_algebra_functions diff --git a/spec/extensions/linear_algebra_functions.md b/spec/extensions/linear_algebra_functions.md new file mode 100644 index 000000000..5f9eaf0a7 --- /dev/null +++ b/spec/extensions/linear_algebra_functions.md @@ -0,0 +1,651 @@ +# Linear Algebra Functions + +> Array API specification for linear algebra functions. + +A conforming implementation of the array API standard must provide and support the following functions adhering to the following conventions. + +- Positional parameters must be [positional-only](https://www.python.org/dev/peps/pep-0570/) parameters. Positional-only parameters have no externally-usable name. When a function accepting positional-only parameters is called, positional arguments are mapped to these parameters based solely on their order. +- Optional parameters must be [keyword-only](https://www.python.org/dev/peps/pep-3102/) arguments. +- Broadcasting semantics must follow the semantics defined in {ref}`broadcasting`. +- Unless stated otherwise, functions must support the data types defined in {ref}`data-types`. +- Unless stated otherwise, functions must adhere to the type promotion rules defined in {ref}`type-promotion`. +- Unless stated otherwise, floating-point operations must adhere to IEEE 754-2019. + +## Design Principles + +A principal goal of this specification is to standardize commonly implemented interfaces among array libraries. While this specification endeavors to avoid straying too far from common practice, this specification does, with due restraint, seek to address design decisions arising more from historical accident than first principles. This is especially true for linear algebra APIs, which have arisen and evolved organically over time and have often been tied to particular underlying implementations (e.g., to BLAS and LAPACK). + +Accordingly, the standardization process affords the opportunity to reduce interface complexity among linear algebra APIs by inferring and subsequently codifying common design themes, thus allowing more consistent APIs. What follows is the set of design principles governing the APIs which follow: + +1. **Batching**: if an operation is explicitly defined in terms of matrices (i.e., two-dimensional arrays), then the associated interface should support "batching" (i.e., the ability to perform the operation over a "stack" of matrices). Example operations include: + + - `inv`: computing the multiplicative inverse of a square matrix. + - `cholesky`: performing Cholesky decomposition. + - `matmul`: performing matrix multiplication. + +2. **Data types**: if an operation requires decimal operations and {ref}`type-promotion` semantics are undefined (e.g., as is the case for mixed-kind promotions), then the associated interface should be specified as being restricted to floating-point data types. While the specification uses the term "SHOULD" rather than "MUST", a conforming implementation of the array API standard should only ignore the restriction provided overly compelling reasons for doing so. Example operations which should be limited to floating-point data types include: + + - `inv`: computing the multiplicative inverse. + - `slogdet`: computing the natural logarithm of the absolute value of the determinant. + - `norm`: computing the matrix or vector norm. + + Certain operations are solely comprised of multiplications and additions. Accordingly, associated interfaces need not be restricted to floating-point data types. However, careful consideration should be given to overflow, and use of floating-point data types may be more prudent in practice. Example operations include: + + - `matmul`: performing matrix multiplication. + - `trace`: computing the sum along the diagonal. + - `cross`: computing the vector cross product. + + Lastly, certain operations may be performed independent of data type, and, thus, the associated interfaces should support all data types specified in this standard. Example operations include: + + - `transpose`: computing the transpose. + - `diagonal`: returning the diagonal. + +3. **Return values**: if an interface has more than one return value, the interface should return a namedtuple consisting of each value. + + In general, interfaces should avoid polymorphic return values (e.g., returning an array **or** a namedtuple, dependent on, e.g., an optional keyword argument). Dedicated interfaces for each return value type are preferred, as dedicated interfaces are easier to reason about at both the implementation level and user level. Example interfaces which could be combined into a single overloaded interface, but are not, include: + + - `eig`: computing both eigenvalues and eignvectors. + - `eigvals`: computing only eigenvalues. + +4. **Implementation agnosticism**: a standardized interface should eschew parameterization (including keyword arguments) biased toward particular implementations. + + Historically, at a time when all array computing happened on CPUs, BLAS and LAPACK underpinned most numerical computing libraries and environments. Naturally, language and library abstractions catered to the parameterization of those libraries, often exposing low-level implementation details verbatim in their higher-level interfaces, even if such choices would be considered poor or ill-advised by today's standards (e.g., NumPy's use of `UPLO` in `eigh`). However, the present day is considerably different. While still important, BLAS and LAPACK no longer hold a monopoly over linear algebra operations, especially given the proliferation of devices and hardware on which such operations must be performed. Accordingly, interfaces must be conservative in the parameterization they support in order to best ensure universality. Such conservatism applies even to performance optimization parameters afforded by certain hardware. + +5. **Orthogonality**: an interface should have clearly defined and delineated functionality which, ideally, has no overlap with the functionality of other interfaces in the specification. Providing multiple interfaces which can all perform the same operation creates unnecessary confusion regarding interface applicability (i.e., which interface is best at which time) and decreases readability of both library and user code. Where overlap is possible, the specification must be parsimonious in the number of interfaces, ensuring that each interface provides a unique and compelling abstraction. Examples of related interfaces which provide distinct levels of abstraction (and generality) include: + + - `vecdot`: computing the dot product of two vectors. + - `matmul`: performing matrix multiplication (including between two vectors and thus the dot product). + - `tensordot`: computing tensor contractions (generalized sum-products). + - `einsum`: expressing operations in terms of Einstein summation convention, including dot products and tensor contractions. + + The above can be contrasted with, e.g., NumPy, which provides the following interfaces for computing the dot product or related operations: + + - `dot`: dot product, matrix multiplication, and tensor contraction. + - `inner`: dot product. + - `vdot`: dot product with flattening and complex conjugation. + - `multi_dot`: chained dot product. + - `tensordot`: tensor contraction. + - `matmul`: matrix multiplication (dot product for two vectors). + - `einsum`: Einstein summation convention. + + where `dot` is overloaded based on input array dimensionality and `vdot` and `inner` exhibit a high degree of overlap with other interfaces. By consolidating interfaces and more clearly delineating behavior, this specification aims to ensure that each interface has a unique purpose and defined use case. + +## Objects in API + + + +(function-linalg-cholesky)= +### linalg.cholesky(x, /, *, upper=False) + +Returns the Cholesky decomposition of a symmetric positive-definite matrix (or a stack of symmetric positive-definite 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. + +- **upper**: _bool_ + + - If `True`, the result must be the upper-triangular Cholesky factor. If `False`, the result must be the lower-triangular Cholesky factor. Default: `False`. + +#### Returns + +- **out**: _<array>_ + + - an array containing the Cholesky factors for each square matrix. The returned array must have a floating-point data type determined by {ref}`type-promotion` and shape as `x`. + +(function-linalg-cross)= +### linalg.cross(x1, x2, /, *, axis=-1) + +Returns the cross product of 3-element vectors. If `x1` and `x2` are multi-dimensional arrays (i.e., both have 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. Should have a numeric data type. + +- **x2**: _<array>_ + + - second input array. Must have the same shape as `x1`. Should have a numeric data type. + +- **axis**: _int_ + + - the axis (dimension) of `x1` and `x2` containing the vectors for which to compute the cross product. If set to `-1`, the function computes the cross product for vectors defined by the last axis (dimension). Default: `-1`. + +#### Returns + +- **out**: _<array>_ + + - an array containing the cross products. The returned array must have a data type determined by {ref}`type-promotion`. + +(function-linalg-det)= +### linalg.det(x, /) + +Returns the determinant of a square matrix (or 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**: _<array>_ + + - 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`. + +(function-linalg-diagonal)= +### linalg.diagonal(x, /, *, axis1=0, axis2=1, offset=0) + +Returns the specified diagonals. If `x` has more than two dimensions, then the axes (dimensions) specified by `axis1` and `axis2` are used to determine the two-dimensional sub-arrays from which to return diagonals. + +#### Parameters + +- **x**: _<array>_ + + - input array. Must have at least `2` dimensions. + +- **axis1**: _int_ + + - first axis (dimension) with respect to which to take diagonal. Default: `0`. + +- **axis2**: _int_ + + - second axis (dimension) with respect to which to take diagonal. Default: `1`. + +- **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**: _<array>_ + + - if `x` is a two-dimensional array, a one-dimensional array containing the diagonal; otherwise, a multi-dimensional array containing the diagonals and whose shape is determined by removing `axis1` and `axis2` and appending a dimension equal to the size of the resulting diagonals. The returned array must have the same data type as `x`. + +(function-linalg-eig)= +### linalg.eig() + +TODO + +(function-linalg-eigh)= +### linalg.eigh(x, /, *, upper=False) + +Returns the eigenvalues and eigenvectors of a symmetric matrix (or a stack of symmetric matrices) `x`. + + + +#### Parameters + +- **x**: _<array>_ + + - input array having shape `(..., M, M)` and whose innermost two dimensions form square matrices. Must have a floating-point data type. + +- **upper**: _bool_ + + - If `True`, use the upper-triangular part to compute the eigenvalues and eigenvectors. If `False`, use the lower-triangular part to compute the eigenvalues and eigenvectors. Default: `False`. + +#### Returns + +- **out**: _Tuple\[ <array> ]_ + + - a namedtuple (`e`, `v`) whose + + - first element must have shape `(..., M)` and consist of computed eigenvalues. + - second element must have shape `(..., M, M)`and have the columns of the inner most matrices contain the computed eigenvectors. + + Each returned array must have the same floating-point data type as `x`. + +```{note} + +Eigenvalue sort order is left unspecified. +``` + +(function-linalg-eigvals)= +### linalg.eigvals() + +TODO + +(function-linalg-eigvalsh)= +### linalg.eigvalsh(x, /, *, upper=False) + +Computes the eigenvalues of a symmetric matrix (or a stack of symmetric matrices) `x`. + + + +#### Parameters + +- **x**: _<array>_ + + - input array having shape `(..., M, M)` and whose innermost two dimensions form square matrices. Must have a floating-point data type. + +- **upper**: _bool_ + + - If `True`, use the upper-triangular part to compute the eigenvalues. If `False`, use the lower-triangular part to compute the eigenvalues. Default: `False`. + +#### Returns + +- **out**: _<array>_ + + - an array containing the computed eigenvalues. The returned array must have shape `(..., M)` and have the same data type as `x`. + +```{note} + +Eigenvalue sort order is left unspecified. +``` + +(function-linalg-einsum)= +### linalg.einsum() + +Alias for {ref}`function-einsum`. + +(function-linalg-inv)= +### linalg.inv(x, /) + +Computes the multiplicative inverse 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**: _<array>_ + + - an array containing the multiplicative inverses. The returned array must have a floating-point data type determined by {ref}`type-promotion` and must have the same shape as `x`. + +(function-linalg-lstsq)= +### linalg.lstsq(x1, x2, /, *, rtol=None) + +Returns the least-squares solution to a linear matrix equation `Ax = b`. + +#### Parameters + +- **x1**: _<array>_ + + - coefficient array `A` having shape `(..., M, N)` and whose innermost two dimensions form `MxN` matrices. 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)`, and `shape(x2)` must be compatible with `shape(x1)[:-1]` (see {ref}`broadcasting`). If `x2` has shape `(..., M, K)`, each column `k` defines a set of ordinate values for which to compute a solution, and `shape(x2)[:-1]` must be compatible with `shape(x1)[:-1]` (see {ref}`broadcasting`). Should have a floating-point data type. + +- **rtol**: _Optional\[ Union\[ float, <array> ] ]_ + + - relative tolerance for small singular values. Singular values 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 data type determined by {ref}`type-promotion` (as applied to `x1` and `x2`) and must be broadcast against each matrix. If an `array`, must have a floating-point data type and must be compatible with `shape(x1)[:-2]` (see {ref}`broadcasting`). If `None`, the default value is `max(M, N) * eps`, where `eps` must be the machine epsilon associated with the floating-point data type determined by {ref}`type-promotion` (as applied to `x1` and `x2`). Default: `None`. + +#### Returns + +- **out**: _Tuple\[ <array>, <array>, <array>, <array> ]_ + + - a namedtuple `(x, residuals, rank, s)` whose + + - first element must have the field name `x` and must be an array containing the least-squares solution for each `MxN` matrix in `x1`. The array containing the solutions must have shape `(N, K)` and must have a floating-point data type determined by {ref}`type-promotion`. + - second element must have the field name `residuals` and must be an array containing the sum of squares residuals (i.e., the squared Euclidean 2-norm for each column in `b - Ax`). The array containing the residuals must have shape `(K,)` and must have a floating-point data type determined by {ref}`type-promotion`. + - third element must have the field name `rank` and must be an array containing the effective rank of each `MxN` matrix. The array containing the ranks must have shape `shape(x1)[:-2]` and must have an integer data type. + - fourth element must have the field name `s` and must be an array containing the singular values for each `MxN` matrix in `x1`. The array containing the singular values must have shape `(..., min(M, N))` and must have a floating-point data type determined by {ref}`type-promotion`. + +(function-linalg-matmul)= +### linalg.matmul(x1, x2, /) + +Alias for {ref}`function-matmul`. + +(function-linalg-matrix_power)= +### linalg.matrix_power(x, n, /) + +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**: _<array>_ + + - 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 {ref}`type-promotion`. + +#### Raises + +- if the innermost two dimensions of `x` are not the same size (i.e., form square matrices). +- if `n` is less than zero and a square matrix is not invertible. + +(function-linalg-matrix_rank)= +### linalg.matrix_rank(x, /, *, rtol=None) + +Computes the rank (i.e., number of non-zero singular values) of a matrix (or a stack of matrices). + +#### 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 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 floating-point data type determined by {ref}`type-promotion` (as applied to `x`) and must be broadcast against each matrix. If an `array`, must have a floating-point data type and must be compatible with `shape(x)[:-2]` (see {ref}`broadcasting`). If `None`, the default value is `max(M, N) * eps`, where `eps` must be the machine epsilon associated with the floating-point data type determined by {ref}`type-promotion` (as applied to `x`). Default: `None`. + +#### Returns + +- **out**: _<array>_ + + - an array containing the ranks. The returned array must have a floating-point data type determined by {ref}`type-promotion` and must have shape `(...)` (i.e., must have a shape equal to `shape(x)[:-2]`). + +(function-linalg-norm)= +### linalg.norm(x, /, *, axis=None, keepdims=False, ord=None) + +Computes the matrix or vector norm of `x`. + +#### Parameters + +- **x**: _<array>_ + + - input array. Should have a floating-point data type. + +- **axis**: _Optional\[ Union\[ int, Tuple\[ int, int ] ] ]_ + + - If an integer, `axis` specifies the axis (dimension) along which to compute vector norms. + + If a 2-tuple, `axis` specifies the axes (dimensions) defining two-dimensional matrices for which to compute matrix norms. + + If `None`, + + - if `x` is one-dimensional, the function must compute the vector norm. + - if `x` is two-dimensional, the function must compute the matrix norm. + - if `x` has more than two dimensions, the function must compute the vector norm 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 {ref}`broadcasting`). Otherwise, if `False`, the axes (dimensions) specified by `axis` 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 | matrix | vector | + | ---------------- | ------------------------------- | -------------------------- | + | 'fro' | 'fro' | - | + | 'nuc' | 'nuc' | - | + | 1 | max(sum(abs(x), axis=0)) | L1-norm (Manhattan) | + | 2 | largest singular value | L2-norm (Euclidean) | + | inf | max(sum(abs(x), axis=1)) | infinity norm | + | (int,float >= 1) | - | p-norm | + + The following non-mathematical "norms" must be supported: + | ord | matrix | vector | + | ---------------- | ------------------------------- | ------------------------------ | + | 0 | - | sum(a != 0) | + | -1 | min(sum(abs(x), axis=0)) | 1./sum(1./abs(a)) | + | -2 | smallest singular value | 1./sqrt(sum(1./abs(a)\*\*2)) | + | -inf | min(sum(abs(x), axis=1)) | min(abs(a)) | + | (int,float < 1) | - | sum(abs(a)\*\*ord)\*\*(1./ord) | + + When `ord` is `None`, the following norms must be the default norms: + | ord | matrix | vector | + | ---------------- | ------------------------------- | -------------------------- | + | None | 'fro' | L2-norm (Euclidean) | + + where `fro` corresponds to the **Frobenius norm**, `nuc` corresponds to the **nuclear norm**, and `-` indicates that the norm is **not** supported. + + For matrices, + + - 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). + + If `None`, + + - if matrix (or matrices), the function must compute the Frobenius norm. + - if vector (or vectors), the function must compute the L2-norm (Euclidean norm). + + Default: `None`. + +#### Returns + +- **out**: _<array>_ + + - an array containing the 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 2-tuple, the returned array must have a rank which is two less than the rank of `x`. The returned array must have a floating-point data type determined by {ref}`type-promotion`. + +(function-linalg-outer)= +### linalg.outer(x1, x2, /) + +Computes the outer product of two vectors `x1` and `x2`. + +#### Parameters + +- **x1**: _<array>_ + + - first one-dimensional input array of size `N`. Should have a numeric data type. + +- **x2**: _<array>_ + + - second one-dimensional input array of size `M`. Should have a numeric data type. + +#### Returns + +- **out**: _<array>_ + + - a two-dimensional array containing the outer product and whose shape is `(N, M)`. The returned array must have a data type determined by {ref}`type-promotion`. + +(function-linalg-pinv)= +### linalg.pinv(x, /, *, rtol=None) + +Computes the (Moore-Penrose) pseudo-inverse of a matrix (or a stack of square 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. + +- **rtol**: _Optional\[ Union\[ float, <array> ] ]_ + + - relative tolerance for small singular values. Singular values 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 floating-point data type determined by {ref}`type-promotion` (as applied to `x`) and must be broadcast against each matrix. If an `array`, must have a floating-point data type and must be compatible with `shape(x)[:-2]` (see {ref}`broadcasting`). If `None`, the default value is `max(M, N) * eps`, where `eps` must be the machine epsilon associated with the floating-point data type determined by {ref}`type-promotion` (as applied to `x`). Default: `None`. + +#### Returns + +- **out**: _<array>_ + + - an array containing the pseudo-inverses. The returned array must have a floating-point data type determined by {ref}`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). + +(function-linalg-qr)= +### linalg.qr(x, /, *, mode='reduced') + +Computes the qr factorization of a matrix (or a stack of matrices), where `q` is an orthonormal matrix (or a stack of matrices) and `r` is an upper-triangular matrix (or a stack of matrices). + +#### Parameters + +- **x**: _<array>_ + + - input array having shape `(..., M, N)` and whose innermost two dimensions form `MxN` matrices. Should have a floating-point data type. + +- **mode**: _str_ + + - factorization 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**: _Tuple\[ <array>, <array> ]_ + + - a namedtuple `(q, r)` whose + + - first element must be an array whose shape depends on the value of `mode` and contain orthonormal matrices. 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 `x`. + - second element 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, M)`. 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 {ref}`type-promotion`. + +(function-linalg-slogdet)= +### linalg.slogdet(x, /) + +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. +``` + +#### 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**: _Tuple\[ <array>, <array> ]_ + + - a namedtuple (`sign`, `logabsdet`) whose + + - first element must be an array containing a number representing the sign of the determinant for each square matrix. + - second element must be an array containing the determinant for each square matrix. + + For a real matrix, the sign of the determinant must be either `1`, `0`, or `-1`. If a determinant is zero, then the corresponding `sign` must be `0` and `logabsdet` must be `-infinity`. In all cases, the determinant must be equal to `sign * exp(logsabsdet)`. + + Each returned array must have shape `shape(x)[:-2]` and a floating-point data type determined by {ref}`type-promotion`. + +(function-linalg-solve)= +### linalg.solve(x1, x2, /) + +Returns the solution to the system of linear equations represented by the well-determined (i.e., full rank) linear matrix equation `AX = B`. + +#### 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)`, and `shape(x2)` must be compatible with `shape(x1)[:-1]` (see {ref}`broadcasting`). If `x2` has shape `(..., M, K)`, each column `k` defines a set of ordinate values for which to compute a solution, and `shape(x2)[:-1]` must be compatible with `shape(x1)[:-1]` (see {ref}`broadcasting`). Should have a floating-point data type. + +#### Returns + +- **out**: _<array>_ + + - an array containing the solution to the system `AX = B` for each square matrix. The returned array must have the same shape as `x2` (i.e., the array corresponding to `B`) and must have a floating-point data type determined by {ref}`type-promotion`. + +(function-linalg-svd)= +### linalg.svd(x, /, *, full_matrices=True) + +Computes the singular value decomposition `A = USV` of a matrix (or a stack of matrices) `x`. + +#### 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 `v`, such that `u` has shape `(..., M, M)` and `v` has shape `(..., N, N)`. If `False`, compute on the leading `K` singular vectors, such that `u` has shape `(..., M, K)` and `v` has shape `(..., K, N)` and where `K = min(M, N)`. Default: `True`. + +#### Returns + +- **out**: _Union\[ <array>, Tuple\[ <array>, ... ] ]_ + + - a namedtuple `(u, s, v)` whose + + - first element must be an array whose shape depends on the value of `full_matrices` and contain unitary array(s) (i.e., the left singular vectors). The left singular vectors must be stored as columns. 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`. + - second element must be an array with shape `(..., K)` that contains the vector(s) of singular values of length `K`. 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`. + - third element must be an array whose shape depends on the value of `full_matrices` and contain unitary array(s) (i.e., the right singular vectors). The right singular vectors must be stored as rows (i.e., 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`. + + Each returned array must have the same floating-point data type as `x`. + +(function-linalg-tensordot)= +### linalg.tensordot(x1, x2, /, *, axes=2) + +Alias for {ref}`function-tensordot`. + +(function-linalg-svdvals)= +### linalg.svdvals(x, /) + +Computes the singular values of a matrix (or a stack of matrices) `x`. + +#### 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**: _Union\[ <array>, Tuple\[ <array>, ... ] ]_ + + - an array with shape `(..., K)` that contains the vector(s) of singular values of length `K`. 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 the same floating-point data type as `x`. + +(function-linalg-trace)= +### linalg.trace(x, /, *, axis1=0, axis2=1, offset=0) + +Returns the sum along the specified diagonals. If `x` has more than two dimensions, then the axes (dimensions) specified by `axis1` and `axis2` are used to determine the two-dimensional sub-arrays for which to compute the trace. + +#### Parameters + +- **x**: _<array>_ + + - input array. Must have at least `2` dimensions. Should have a numeric data type. + +- **axis1**: _int_ + + - first axis (dimension) with respect to which to compute the trace. Default: `0`. + +- **axis2**: _int_ + + - second axis (dimension) with respect to which to compute the trace. Default: `1`. + +- **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**: _<array>_ + + - if `x` is a two-dimensional array, the returned array must be a zero-dimensional array containing the trace; otherwise, the returned array must be a multi-dimensional array containing the traces. + + The shape of a multi-dimensional output array is determined by removing `axis1` and `axis2` and storing the traces in the last array dimension. For example, if `x` has rank `k` and shape `(I, J, K, ..., L, M, N)` and `axis1=-2` and `axis1=-1`, then a multi-dimensional output array has rank `k-2` and shape `(I, J, K, ..., L)` where + + ```text + out[i, j, k, ..., l] = trace(a[i, j, k, ..., l, :, :]) + ``` + + The returned array must have the same data type as `x`. + +(function-linalg-transpose)= +### linalg.transpose(x, /, *, axes=None) + +Alias for {ref}`function-transpose`. + +(function-linalg-vecdot)= +### linalg.vecdot(x1, x2, /, *, axis=None) + +Alias for {ref}`function-vecdot`. \ No newline at end of file diff --git a/spec/index.rst b/spec/index.rst index e7a466413..706c2f5e6 100644 --- a/spec/index.rst +++ b/spec/index.rst @@ -19,6 +19,7 @@ Contents design_topics/index future_API_evolution API_specification/index + extensions/index .. toctree:: :caption: Methodology and Usage