Source code for mindspore.ops.function.linalg_func

# Copyright 2022 Huawei Technologies Co., Ltd
#
# Licensed under the Apache License, Version 2.0 (the "License");
# you may not use this file except in compliance with the License.
# You may obtain a copy of the License at
#
# http://www.apache.org/licenses/LICENSE-2.0
#
# Unless required by applicable law or agreed to in writing, software
# distributed under the License is distributed on an "AS IS" BASIS,
# WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
# See the License for the specific language governing permissions and
# limitations under the License.
# ============================================================================

"""Operators for linalg function."""
from __future__ import absolute_import

import mindspore.ops as ops
from mindspore.common import dtype as mstype
from mindspore.common.tensor import Tensor
from mindspore.ops import operations as P
from mindspore.ops import functional as F
from mindspore.ops.operations import _inner_ops as inner
from mindspore.ops.function.math_func import _check_input_dtype, _check_attr_dtype
from mindspore._c_expression import Tensor as Tensor_
from mindspore.ops.auto_generate import geqrf

from ..operations import linalg_ops
from .._primitive_cache import _get_cache_prim

__all__ = ['cond', 'eig', 'eigvals', 'geqrf', 'svd', 'pinv', 'qr']

dtype_ = P.DType()
geqrf_ = P.Geqrf()
slice_ = P.Slice()

[docs]def cond(A, p=None): r""" Returns the matrix norm or vector norm of a given tensor. `p` is the calculation mode of norm. The following norm modes are supported. ========================== ================================ ========================================== `p` norm for matrices norm for vectors ========================== ================================ ========================================== ``None`` (default) `2`-norm (see below) `2`-norm (see below) ``'fro'`` Frobenius norm -- not supported -- ``'nuc'`` nuclear norm -- not supported -- ``inf`` :math:`max(sum(abs(x), dim=1))` :math:`max(abs(x))` ``-inf`` :math:`min(sum(abs(x), dim=1))` :math:`min(abs(x))` ``0`` -- not supported -- :math:`sum(x != 0)` ``1`` :math:`max(sum(abs(x), dim=0))` as below ``-1`` :math:`min(sum(abs(x), dim=0))` as below ``2`` largest singular value as below ``-2`` smallest singular value as below other ``int`` or ``float`` -- not supported -- :math:`sum(abs(x)^{p})^{(1 / p)}` ========================== ================================ ========================================== Note: Currently, complex numbers are not supported. Args: A (Tensor): Tensor of shape :math:`(*, n)` or :math:`(*, m, n)` where :math:`*` is zero or more batch dimensions. p (Union[int, float, inf, -inf, 'fro', 'nuc'], optional): norm's mode. Refer to the table above for behavior. Default: ``None``. Returns: Tensor, the result of norm calculation on the specified dimension, `dim`, has the same dtype as `A`. Raises: TypeError: If `A` is a vector and `p` is a str. ValueError: If `A` is a matrices and `p` is not in valid mode. ValueError: If `A` is a matrix and `p` is an integer that is not in [1, -1, 2, -2]. Supported Platforms: ``GPU`` ``CPU`` Examples: >>> import mindspore as ms >>> x = ms.Tensor([[1.0, 0.0, -1.0], [0.0, 1.0, 0.0], [1.0, 0.0, 1.0]]) >>> print(ms.ops.cond(x)) 1.4142 >>> print(ms.ops.cond(x, 'fro')) 3.1622777 """ matrix_inverse = _get_cache_prim(P.MatrixInverse)(adjoint=False) if p is None: p = 2 if A.dim() >= 3: shape_ori = A.shape[0:-2] A_flatten = ops.flatten(A, start_dim=0, end_dim=-3) out = [] for i in range(A_flatten.shape[0]): norm_a = F.norm(A_flatten[i], p) norm_inv_a = F.norm(matrix_inverse(A_flatten[i]), p) cond_i = ops.fill(mstype.float32, (1, 1), norm_a * norm_inv_a) out.append(cond_i) out_stacked = ops.hstack(out) output = ops.reshape(out_stacked, shape_ori) return output norm_a = F.norm(A, p) norm_inv_a = F.norm(matrix_inverse(A), p) return norm_a * norm_inv_a
def eig(A): """ Computes the eigenvalues and eigenvectors of a square matrix(batch square matrices). .. warning:: This is an experimental API that is subject to change or deletion. Args: A (Tensor): Square matrices of shape :math:`(*, N, N)`, with float32, float64, complex64 or complex128 data type. Returns: - **eigen_values** (Tensor) - Shape :math:`(*, N)`. eigenvalues of the corresponding matrix. The eigenvalues may not have an order. - **eigen_vectors** (Tensor) - Shape :math:`(*, N, N)`,columns of eigen vectors represent - **normalized** (unit length) eigenvectors of corresponding eigenvalues. Raises: TypeError: If dtype of `A` is not one of: float64, float32, complex64 or complex128. TypeError: If `A` is not a Tensor. ValueError: If `A` is not a square(batch squares). Supported Platforms: ``Ascend`` ``CPU`` Examples: >>> import mindspore >>> import numpy as np >>> from mindspore import Tensor, ops >>> input_x = Tensor(np.array([[1.0, 0.0], [0.0, 2.0]]), mindspore.float32) >>> u, v = ops.eig(input_x) >>> print(u) [1.+0.j 2.+0.j] >>> print(v) [[1.+0.j 0.+0.j] [0.+0.j 1.+0.j]] """ return _get_cache_prim(P.Eig)(compute_v=True)(A)
[docs]def eigvals(A): """ Computes the eigenvalues of a square matrix(batch square matrices). .. warning:: This is an experimental API that is subject to change or deletion. Args: A (Tensor): Square matrices of shape :math:`(*, N, N)`, with float32, float64, complex64 or complex128 data type. Returns: Tensor, with shape :math:`(*, N)`. Returns the eigenvalues of the corresponding matrix, which may not have an order. Raises: TypeError: If dtype of `A` is not one of: float64, float32, complex64 or complex128. TypeError: If `A` is not a Tensor. ValueError: If `A` is not a square(batch squares). Supported Platforms: ``Ascend`` ``CPU`` Examples: >>> import mindspore >>> from mindspore import Tensor, ops >>> import numpy as np >>> input_x = Tensor(np.array([[1.0, 0.0], [0.0, 2.0]]), mindspore.float32) >>> u = ops.eigvals(input_x) >>> print(u) [1.+0.j 2.+0.j] """ u, _ = _get_cache_prim(P.Eig)(compute_v=False)(A) return u
[docs]def svd(input, full_matrices=False, compute_uv=True): """ Computes the singular value decompositions of one or more matrices. If :math:`A` is a matrix, the svd returns the singular values :math:`S`, the left singular vectors :math:`U` and the right singular vectors :math:`V`. It meets: .. math:: A=U*diag(S)*V^{T} Args: input (Tensor): Tensor of the matrices to be decomposed. The shape should be :math:`(*, M, N)`, the supported dtype are float32 and float64. full_matrices (bool, optional): If true, compute full-sized :math:`U` and :math:`V`. If false, compute only the leading P singular vectors, with P is the minimum of M and N. Default: ``False`` . compute_uv (bool, optional): If true, compute the left and right singular vectors. If false, compute only the singular values. Default: ``True`` . Returns: - **s** (Tensor) - Singular values. The shape is :math:`(*, P)`. - **u** (Tensor) - Left singular vectors. If `compute_uv` is False, u will not be returned. The shape is :math:`(*, M, P)`. If `full_matrices` is True, the shape will be :math:`(*, M, M)`. - **v** (Tensor) - Right singular vectors. If `compute_uv` is False, v will not be returned. The shape is :math:`(*, N, P)`. If `full_matrices` is True, the shape will be :math:`(*, N, N)`. Raises: TypeError: If `full_matrices` or `compute_uv` is not the type of bool. TypeError: If the rank of input less than 2. TypeError: If the type of input is not one of the following dtype: float32, float64. Supported Platforms: ``GPU`` ``CPU`` Examples: >>> import numpy as np >>> from mindspore import Tensor, set_context >>> from mindspore import ops >>> set_context(device_target="CPU") >>> input = Tensor(np.array([[1, 2], [-4, -5], [2, 1]]).astype(np.float32)) >>> s, u, v = ops.svd(input, full_matrices=True, compute_uv=True) >>> print(s) [7.0652843 1.040081 ] >>> print(u) [[ 0.30821905 -0.48819482 0.81649697] [-0.90613353 0.11070572 0.40824813] [ 0.2896955 0.8656849 0.4082479 ]] >>> print(v) [[ 0.63863593 0.769509 ] [ 0.769509 -0.63863593]] """ svd_ = _get_cache_prim(linalg_ops.Svd)(full_matrices=full_matrices, compute_uv=compute_uv) if compute_uv: return svd_(input) s, _, _ = svd_(input) return s
[docs]def pinv(x, *, atol=None, rtol=None, hermitian=False): r""" Computes the (Moore-Penrose) pseudo-inverse of a matrix. This function is computed using SVD. If :math:`x=U*S*V^{T}` ,Than the pseudo-inverse of x is: :math:`x^{+}=V*S^{+}*U^{T}` , :math:`S^{+}` is the reciprocal of each non-zero element on the diagonal of S, and zero remains in place. Batch matrices are supported. If x is a batch matrix, the output has the same batch dimension when atol or rtol is float. If atol or rtol is a Tensor, its shape must be broadcast to the singular value returned by `x.svd <https://www.mindspore.cn/docs/en/r2.3.0rc2/api_python/ops/mindspore.ops.svd.html>`_ . If x.shape is :math:`(B, M, N)`, and the shape of atol or rtol is :math:`(K, B)`, the output shape is :math:`(K, B, N, M)`. When the Hermitian is True, temporary support only real domain, x is treated as a real symmetric, so x is not checked internally, and only use the lower triangular part in the computations. When the singular value of x (or the norm of the eigenvalues when hermitian = True) that are below threshold (:math:`max(atol, \sigma \cdot rtol)`, :math:`\sigma` as the largest singular value or characteristic value), it is set to zero, and is not used in the computations. If rtol is not specified and x is a matrix of dimensions (M, N), then rtol is set to be :math:`rtol=max(M, N)*\varepsilon`, :math:`\varepsilon` is the `eps <https://www.mindspore.cn/docs/en/r2.3.0rc2/api_python/ops/mindspore.ops.Eps.html>`_ value of x.dtype. If rtol is not specified and atol specifies a value larger than zero, rtol is set to zero. .. note:: This function uses `svd <https://www.mindspore.cn/docs/en/r2.3.0rc2/api_python/ops/mindspore.ops.svd.html>`_ internally, (or `eigh <https://www.mindspore.cn/docs/en/r2.3.0rc2/api_python/scipy/mindspore.scipy.linalg.eigh.html>`_ , when `hermitian = True` ). So it has the same problem as these functions. For details, see the warnings in svd() and eigh(). Args: x (Tensor): A matrix to be calculated. Only `float32`, `float64` are supported Tensor dtypes. shape is :math:`(*, M, N)`, * is zero or more batch dimensions. - When `hermitian` is ``True``, batch dimensions are not supported temporarily. Keyword args: atol (float, Tensor): absolute tolerance value. Default: ``None`` . rtol (float, Tensor): relative tolerance value. Default: ``None`` . hermitian (bool): An optional bool. x is assumed to be symmetric if real. Default: ``False`` . Outputs: - **output** (Tensor) - same type as input. Shape is :math:`(*, N, M)`, * is zero or more batch dimensions. Raises: TypeError: If `hermitian` is not a bool. TypeError: If `x` is not a Tensor. ValueError: If the dimension of `x` is less than 2. Supported Platforms: ``CPU`` Examples: >>> import mindspore >>> from mindspore import Tensor, ops >>> x = Tensor([[4., 0.], [0., 5.]], mindspore.float32) >>> output = ops.pinv(x) >>> print(output) [[0.25 0. ] [0. 0.2 ]] """ if not isinstance(x, (Tensor, Tensor_)): raise TypeError("The input x must be tensor") if x.shape == (): raise TypeError("For pinv, the 0-D input is not supported") x_shape = F.shape(x) if len(x_shape) < 2: raise ValueError("input x should have 2 or more dimensions, " f"but got {len(x_shape)}.") x_dtype = dtype_(x) _check_input_dtype("x", x_dtype, [mstype.float32, mstype.float64], "pinv") _check_attr_dtype("hermitian", hermitian, [bool], "pinv") if atol is not None: if rtol is None: rtol = Tensor(0.0) else: atol = Tensor(0.0) if rtol is None: rtol = max(ops.Shape()(x)) * ops.Eps()(Tensor(1.0, x_dtype)) if not inner.IsInstance()(rtol, mstype.tensor_type): rtol = F.cast(rtol, mstype.float32) if not inner.IsInstance()(atol, mstype.tensor_type): atol = F.cast(atol, mstype.float32) if not hermitian: s, u, v = linalg_ops.Svd()(x) max_singular_val = _narrow(s, -1, 0, 1) threshold = ops.Maximum()(atol.expand_dims(-1), rtol.expand_dims(-1) * max_singular_val) condition = s > threshold reciprocal_s_before = ops.Reciprocal()(s).broadcast_to(condition.shape) zero = F.zeros(condition.shape, s.dtype) s_pseudoinv = ops.Select()(condition, reciprocal_s_before, zero) output = ops.matmul(v * s_pseudoinv.expand_dims(-2), _nd_transpose(ops.Conj()(u))) else: s, u = linalg_ops.Eigh()(x) s_abs = ops.Abs()(s) max_singular_val = ops.amax(s_abs, -1, True) threshold = ops.Maximum()(atol.expand_dims(-1), rtol.expand_dims(-1) * max_singular_val) condition = s_abs > threshold reciprocal_s_before = ops.Reciprocal()(s) zero = F.zeros(condition.shape, s.dtype) s_pseudoinv = ops.Select()(condition, reciprocal_s_before, zero) output = ops.matmul(u * s_pseudoinv.expand_dims(-2), _nd_transpose(ops.Conj()(u))) return output
def qr(input, mode='reduced'): """ Returns the QR decomposition of one or more matrices. If `mode` is 'reduced'(the default), compute the P columns of Q where P is minimum of the 2 innermost dimensions of input. If `mode` is 'complete', compute full-sized Q and R. Args: input (Tensor): A matrix to be calculated. The matrix must be at least two dimensions, the supported dtype are float16, float32, float64, complex64 and complex128. Define the shape of input as :math:`(..., m, n)`, p as the minimum values of m and n. mode (Union['reduced', 'complete'], optional): If `mode` is ``'reduced'`` , computing reduce-sized QR decomposition, otherwise, computing the full-sized QR decomposition. Default: ``'reduced'`` . Returns: - **Q** (Tensor) - The orthonormal matrices of input. If `mode` is 'complete', the shape is :math:`(m, m)`, else the shape is :math:`(m, p)`. The dtype of `Q` is same as `input`. - **R** (Tensor) - The upper triangular matrices of input. If `mode` is 'complete', the shape is :math:`(m, n)`, else the shape is :math:`(p, n)`. The dtype of `R` is same as `input`. Raises: TypeError: If `input` is not a Tensor. TypeError: If `mode` is neither 'reduced' nor 'complete'. ValueError: If the dimension of `input` is less than 2. Supported Platforms: ``Ascend`` ``GPU`` ``CPU`` Examples: >>> input = Tensor(np.array([[20., -31, 7], [4, 270, -90], [-8, 17, -32]]), mstype.float32) >>> Q, R = ops.qr(input) >>> print(Q) [[-0.912871 0.16366126 0.37400758] [-0.18257418 -0.9830709 -0.01544376] [ 0.36514837 -0.08238228 0.92729706]] >>> print(R) [[ -21.908903 -14.788506 -1.6431675] [ 0. -271.9031 92.25824 ] [ 0. 0. -25.665514 ]] """ if mode not in ('reduced', 'complete'): raise TypeError(f"For qr, the arg mode must be 'reduced' or 'complete', but got {mode}.") qr_ = _get_cache_prim(P.Qr)(mode == 'complete') return qr_(input) def _narrow(x, axis, start, length): begins = [0] * x.ndim begins[axis] = start sizes = list(x.shape) sizes[axis] = length return slice_(x, begins, sizes) def _nd_transpose(a): """ _nd_transpose """ dims = a.ndim if dims < 2: raise TypeError("to do _nd_transpose for input a's ndim is not greater or equal to 2d, which is invalid.") axes = ops.make_range(0, dims) axes = axes[:-2] + (axes[-1],) + (axes[-2],) return ops.transpose(a, axes) __all__.sort()