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android / platform / external / pytorch / ecaa208fd6 / . / torch / linalg / __init__.py
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# -*- coding: utf-8 -*-
import sys
import torch
from torch._C import _add_docstr, _linalg # type: ignore[attr-defined]
Tensor = torch.Tensor
# Note: This not only adds doc strings for functions in the linalg namespace, but
# also connects the torch.linalg Python namespace to the torch._C._linalg builtins.
cholesky = _add_docstr(_linalg.linalg_cholesky, r"""
linalg.cholesky(A, *, out=None) -> Tensor
Computes the Cholesky decomposition of a complex Hermitian or real symmetric positive-definite matrix.
Letting :math:`\mathbb{K}` be :math:`\mathbb{R}` or :math:`\mathbb{C}`,
the **Cholesky decomposition** of a complex Hermitian or real symmetric positive-definite matrix
:math:`A \in \mathbb{K}^{n \times n}` is defined as
.. math::
A = LL^{\text{H}}\mathrlap{\qquad L \in \mathbb{K}^{n \times n}}
where :math:`L` is a lower triangular matrix and
:math:`L^{\text{H}}` is the conjugate transpose when :math:`L` is complex, and the transpose when :math:`L` is real-valued.
Supports inputs of float, double, cfloat and cdouble dtypes.
Also supports batched inputs, and, if the input is batched, the output is batched with the same dimensions.
.. note:: For CUDA inputs, this function synchronizes that device with the CPU.
.. seealso::
:func:`torch.linalg.eigh` for a different decomposition of a Hermitian matrix.
The eigenvalue decomposition gives more information about the matrix but it
slower to compute than the Cholesky decomposition.
Args:
A (Tensor): tensor of shape `(*, n, n)` where `*` is zero or more batch dimensions
consisting of symmetric or Hermitian positive-definite matrices.
Keyword args:
out (Tensor, optional): output tensor. Ignored if `None`. Default: `None`.
Raises:
RuntimeError: if the :attr:`A` matrix or any matrix in a batched :attr:`A` is not Hermitian
(resp. symmetric) positive-definite. If :attr:`A` is a batch of matrices,
the error message will include the batch index of the first matrix that fails
to meet this condition.
Examples::
>>> a = torch.randn(2, 2, dtype=torch.complex128)
>>> a = a @ a.t().conj() + torch.eye(2) # creates a Hermitian positive-definite matrix
>>> l = torch.linalg.cholesky(a)
>>> a
tensor([[2.5266+0.0000j, 1.9586-2.0626j],
[1.9586+2.0626j, 9.4160+0.0000j]], dtype=torch.complex128)
>>> l
tensor([[1.5895+0.0000j, 0.0000+0.0000j],
[1.2322+1.2976j, 2.4928+0.0000j]], dtype=torch.complex128)
>>> l @ l.t().conj()
tensor([[2.5266+0.0000j, 1.9586-2.0626j],
[1.9586+2.0626j, 9.4160+0.0000j]], dtype=torch.complex128)
>>> a = torch.randn(3, 2, 2, dtype=torch.float64)
>>> a = a @ a.transpose(-2, -1) + torch.eye(2).squeeze(0) # symmetric positive definite matrices
>>> l = torch.linalg.cholesky(a)
>>> a
tensor([[[ 1.1629, 2.0237],
[ 2.0237, 6.6593]],
[[ 0.4187, 0.1830],
[ 0.1830, 0.1018]],
[[ 1.9348, -2.5744],
[-2.5744, 4.6386]]], dtype=torch.float64)
>>> l
tensor([[[ 1.0784, 0.0000],
[ 1.8766, 1.7713]],
[[ 0.6471, 0.0000],
[ 0.2829, 0.1477]],
[[ 1.3910, 0.0000],
[-1.8509, 1.1014]]], dtype=torch.float64)
>>> torch.allclose(l @ l.transpose(-2, -1), a)
True
""")
inv = _add_docstr(_linalg.linalg_inv, r"""
linalg.inv(A, *, out=None) -> Tensor
Computes the inverse of a square matrix if it exists.
Throws a `RuntimeError` if the matrix is not invertible.
Letting :math:`\mathbb{K}` be :math:`\mathbb{R}` or :math:`\mathbb{C}`,
for a matrix :math:`A \in \mathbb{K}^{n \times n}`,
its **inverse matrix** :math:`A^{-1} \in \mathbb{K}^{n \times n}` (if it exists) is defined as
.. math::
A^{-1}A = AA^{-1} = \mathrm{I}_n
where :math:`\mathrm{I}_n` is the `n`-dimensional identity matrix.
The inverse matrix exists if and only if :math:`A` is `invertible`_. In this case,
the inverse is unique.
Supports inputs of float, double, cfloat and cdouble dtypes.
Also supports batched inputs, and, if the input is batched, the output is batched with the same dimensions.
.. note:: For CUDA inputs, this function synchronizes that device with the CPU.
.. seealso::
:func:`torch.linalg.pinv` computes the pseudoinverse (Moore-Penrose inverse) of matrices
of any shape.
:func:`torch.linalg.solve` computes :attr:`A`\ `.inv() @ \ `:attr:`B` with a stable algorithm.
It is always prefered to use :func:`torch.linalg.solve` when possible, as it is
faster and more stable than computing the inverse and then multiplying.
Args:
A (Tensor): tensor of shape `(*, n, n)` where `*` is zero or more batch dimensions
consisting of invertible matrices.
Keyword args:
out (Tensor, optional): output tensor. Ignored if `None`. Default: `None`.
Raises:
RuntimeError: if the matrix :attr:`A` or any matrix in the batch of matrices :attr:`A` is not invertible.
Examples::
>>> x = torch.rand(4, 4)
>>> y = torch.linalg.inv(x)
>>> z = x @ y
>>> z
tensor([[ 1.0000, -0.0000, -0.0000, 0.0000],
[ 0.0000, 1.0000, 0.0000, 0.0000],
[ 0.0000, 0.0000, 1.0000, 0.0000],
[ 0.0000, -0.0000, -0.0000, 1.0000]])
>>> torch.dist(z, torch.eye(4))
tensor(1.1921e-07)
>>> # Batched inverse example
>>> x = torch.randn(2, 3, 4, 4)
>>> y = torch.linalg.inv(x)
>>> z = x @ y
>>> torch.dist(z, torch.eye(4).expand_as(x))
tensor(1.9073e-06)
>>> x = torch.rand(4, 4, dtype=torch.cdouble)
>>> y = torch.linalg.inv(x)
>>> z = x @ y
>>> torch.dist(z, torch.eye(4, dtype=torch.cdouble))
tensor(7.5107e-16, dtype=torch.float64)
.. _invertible:
https://en.wikipedia.org/wiki/Invertible_matrix#The_invertible_matrix_theorem
""")
det = _add_docstr(_linalg.linalg_det, r"""
linalg.det(A, *, out=None) -> Tensor
Computes the determinant of a square matrix.
Supports inputs of float, double, cfloat and cdouble dtypes.
Also supports batched inputs, and, if the input is batched, the output is batched with the same dimensions.
.. note:: This function is computed using :func:`torch.lu`.
For CUDA inputs, this function synchronizes that device with the CPU.
.. seealso::
:func:`torch.linalg.slogdet` computes the sign (resp. angle) and natural logarithm of the
absolute value (resp. modulus) of the determinant of real-valued (resp. complex)
square matrices.
Args:
A (Tensor): tensor of shape `(*, n, n)` where `*` is zero or more batch dimensions.
Keyword args:
out (Tensor, optional): output tensor. Ignored if `None`. Default: `None`.
Examples::
>>> a = torch.randn(3, 3)
>>> a
tensor([[ 0.9478, 0.9158, -1.1295],
[ 0.9701, 0.7346, -1.8044],
[-0.2337, 0.0557, 0.6929]])
>>> torch.linalg.det(a)
tensor(0.0934)
>>> out = torch.empty(0)
>>> torch.linalg.det(a, out=out)
tensor(0.0934)
>>> out
tensor(0.0934)
>>> a = torch.randn(3, 2, 2)
>>> a
tensor([[[ 0.9254, -0.6213],
[-0.5787, 1.6843]],
[[ 0.3242, -0.9665],
[ 0.4539, -0.0887]],
[[ 1.1336, -0.4025],
[-0.7089, 0.9032]]])
>>> torch.linalg.det(a)
tensor([1.1990, 0.4099, 0.7386])
""")
slogdet = _add_docstr(_linalg.linalg_slogdet, r"""
linalg.slogdet(A, *, out=None) -> (Tensor, Tensor)
Computes the sign and natural logarithm of the absolute value of the determinant of a square matrix.
For complex :attr:`A`, it returns the angle and the natural logarithm of the modulus of the
determinant, that is, a logarithmic polar decomposition of the determinant.
Supports inputs of float, double, cfloat and cdouble dtypes.
Also supports batched inputs, and, if the input is batched, the output is batched with the same dimensions.
.. note:: This function is computed using :func:`torch.lu`.
For CUDA inputs, this function synchronizes that device with the CPU.
.. note:: The determinant can be recovered as `sign * exp(logabsdet)`.
.. note:: When a matrix has a determinant of zero, it returns `(0, -inf)`.
.. seealso::
:func:`torch.linalg.det` computes the determinant of square matrices.
Args:
A (Tensor): tensor of shape `(*, n, n)` where `*` is zero or more batch dimensions.
Keyword args:
out (tuple, optional): output tuple of two tensors. Ignored if `None`. Default: `None`.
Returns:
A named tuple `(sign, logabsdet)`.
`logabsdet` will always be real-valued, even when :attr:`A` is complex.
`sign` will have the same dtype as :attr:`A`.
Examples::
>>> A = torch.randn(3, 3)
>>> A
tensor([[ 0.0032, -0.2239, -1.1219],
[-0.6690, 0.1161, 0.4053],
[-1.6218, -0.9273, -0.0082]])
>>> torch.linalg.det(A)
tensor(-0.7576)
>>> torch.linalg.logdet(A)
tensor(nan)
>>> torch.linalg.slogdet(A)
torch.return_types.linalg_slogdet(sign=tensor(-1.), logabsdet=tensor(-0.2776))
""")
eig = _add_docstr(_linalg.linalg_eig, r"""
linalg.eig(A, *, out=None) -> (Tensor, Tensor)
Computes the eigenvalue decomposition of a square matrix if it exists.
Letting :math:`\mathbb{K}` be :math:`\mathbb{R}` or :math:`\mathbb{C}`,
the **eigenvalue decomposition** of a square matrix
:math:`A \in \mathbb{K}^{n \times n}` (if it exists) is defined as
.. math::
A = V \operatorname{diag}(\Lambda) V^{-1}\mathrlap{\qquad V \in \mathbb{C}^{n \times n}, \Lambda \in \mathbb{C}^n}
This decomposition exists if and only if :math:`A` is `diagonalizable`_.
This is the case when all its eigenvalues are different.
Supports inputs of float, double, cfloat and cdouble dtypes.
Also supports batched inputs, and, if the input is batched, the output is batched with the same dimensions.
.. note:: The eigenvalues and eigenvectors of a real matrix may be complex.
.. note:: For CUDA inputs, this function synchronizes that device with the CPU.
.. warning:: The eigenvectors of a matrix are not unique, nor are they continuous with respect to
:attr:`A`. Due to this lack of uniqueness, different hardware and software may compute
different eigenvectors.
This non-uniqueness is caused by the fact that multiplying an eigenvector by a
non-zero number produces another set of valid eigenvectors of the matrix.
In this implmentation, the returned eigenvectors are normalized to have norm
`1` and largest real component.
.. warning:: This function assumes that :attr:`A` is `diagonalizable`_ (e.g. when all the
eigenvalues are different). If it is not diagonalizable, the returned
eigenvalues will be correct but :math:`A \neq V \operatorname{diag}(\Lambda)V^{-1}`.
.. warning:: This function does not yet support autograd.
.. seealso::
:func:`torch.linalg.eigvals` computes only the eigenvalues.
However, that function is not differentiable.
:func:`torch.linalg.eigh` for a (faster) function that computes the eigenvalue decomposition
for Hermitian and symmetric matrices.
:func:`torch.linalg.svd` for a function that computes another type of spectral
decomposition that works on matrices of any shape.
:func:`torch.linalg.qr` for another (much faster) decomposition that works on matrices of
any shape.
Args:
A (Tensor): tensor of shape `(*, n, n)` where `*` is zero or more batch dimensions
consisting of diagonalizable matrices.
Keyword args:
out (tuple, optional): output tuple of two tensors. Ignored if `None`. Default: `None`.
Returns:
A named tuple `(eigenvalues, eigenvectors)` which corresponds to :math:`\Lambda` and :math:`V` above.
`eigenvalues` and `eigenvectors` will always be complex-valued, even when :attr:`A` is real.
Examples::
>>> a = torch.randn(2, 2, dtype=torch.complex128)
>>> a
tensor([[ 0.9828+0.3889j, -0.4617+0.3010j],
[ 0.1662-0.7435j, -0.6139+0.0562j]], dtype=torch.complex128)
>>> w, v = torch.linalg.eig(a)
>>> w
tensor([ 1.1226+0.5738j, -0.7537-0.1286j], dtype=torch.complex128)
>>> v
tensor([[ 0.9218+0.0000j, 0.1882-0.2220j],
[-0.0270-0.3867j, 0.9567+0.0000j]], dtype=torch.complex128)
>>> torch.allclose(torch.matmul(v, torch.matmul(w.diag_embed(), v.inverse())), a)
True
>>> a = torch.randn(3, 2, 2, dtype=torch.float64)
>>> w, v = torch.linalg.eig(a)
>>> torch.allclose(torch.matmul(v, torch.matmul(w.diag_embed(), v.inverse())).real, a)
True
.. _diagonalizable:
https://en.wikipedia.org/wiki/Diagonalizable_matrix#Definition
""")
eigvals = _add_docstr(_linalg.linalg_eigvals, r"""
linalg.eigvals(A, *, out=None) -> Tensor
Computes the eigenvalues of a square matrix.
Letting :math:`\mathbb{K}` be :math:`\mathbb{R}` or :math:`\mathbb{C}`,
the **eigenvalues** of a square matrix :math:`A \in \mathbb{K}^{n \times n}` are defined
as the roots (counted with multiplicity) of the polynomial `p` of degree `n` given by
.. math::
p(\lambda) = \operatorname{det}(A - \lambda \mathrm{I}_n)\mathrlap{\qquad \lambda \in \mathbb{C}}
where :math:`\mathrm{I}_n` is the `n`-dimensional identity matrix.
Supports inputs of float, double, cfloat and cdouble dtypes.
Also supports batched inputs, and, if the input is batched, the output is batched with the same dimensions.
.. note:: The eigenvalues of a real matrix may be complex, as the roots of a real polynomial may be complex.
The eigenvalues of a matrix are always well-defined, even when the matrix is not diagonalizable.
.. note:: For CUDA inputs, this function synchronizes that device with the CPU.
.. note:: This function is not differentiable. If you need differentiability use
:func:`torch.linalg.eig` instead, which also computes the eigenvectors.
Args:
A (Tensor): tensor of shape `(*, n, n)` where `*` is zero or more batch dimensions.
Keyword args:
out (Tensor, optional): output tensor. Ignored if `None`. Default: `None`.
Returns:
A complex-valued tensor cointaining the eigenvalues even when :attr:`A` is real.
Examples::
>>> a = torch.randn(2, 2, dtype=torch.complex128)
>>> a
tensor([[ 0.9828+0.3889j, -0.4617+0.3010j],
[ 0.1662-0.7435j, -0.6139+0.0562j]], dtype=torch.complex128)
>>> w = torch.linalg.eigvals(a)
>>> w
tensor([ 1.1226+0.5738j, -0.7537-0.1286j], dtype=torch.complex128)
""")
eigh = _add_docstr(_linalg.linalg_eigh, r"""
linalg.eigh(A, UPLO='L', *, out=None) -> (Tensor, Tensor)
Computes the eigenvalue decomposition of a complex Hermitian or real symmetric matrix.
Letting :math:`\mathbb{K}` be :math:`\mathbb{R}` or :math:`\mathbb{C}`,
the **eigenvalue decomposition** of a complex Hermitian or real symmetric matrix
:math:`A \in \mathbb{K}^{n \times n}` is defined as
.. math::
A = Q \operatorname{diag}(\Lambda) Q^{\text{H}}\mathrlap{\qquad Q \in \mathbb{K}^{n \times n}, \Lambda \in \mathbb{R}^n}
where :math:`Q^{\text{H}}` is the conjugate transpose when :math:`Q` is complex, and the transpose when :math:`Q` is real-valued.
:math:`Q` is orthogonal in the real case and unitary in the complex case.
Supports inputs of float, double, cfloat and cdouble dtypes.
Also supports batched inputs, and, if the input is batched, the output is batched with the same dimensions.
:attr:`A` is assumed to be Hermitian (resp. symmetric), but this is not checked internally, instead:
- If :attr:`UPLO`\ `= 'L'` (default), only the lower triangular part of the matrix is used in the computation.
- If :attr:`UPLO`\ `= 'U'`, only the upper triangular part of the matrix is used.
The eigenvalues are returned in ascending order.
.. note:: For CUDA inputs, this function synchronizes that device with the CPU.
.. note:: The eigenvalues of real symmetric or complex Hermitian matrices are always real.
.. warning:: The eigenvectors of a symmetric matrix are not unique, nor are they continuous with
respect to :attr:`A`. Due to this lack of uniqueness, different hardware and
software may compute different eigenvectors.
This non-uniqueness is caused by the fact that multiplying an eigenvector by
`-1` in the real case or by :math:`e^{i \phi}, \phi \in \mathbb{R}` in the complex
case produces another set of valid eigenvectors of the matrix.
This non-uniqueness problem is even worse when the matrix has repeated eigenvalues.
In this case, one may multiply the associated eigenvectors spanning
the subspace by a rotation matrix and the resulting eigenvectors will be valid
eigenvectors.
.. warning:: Gradients computed using the `eigenvectors` tensor will only be finite when
:attr:`A` has unique eigenvalues.
Furthermore, if the distance between any two eigvalues is close to zero,
the gradient will be numerically unstable, as it depends on the eigenvalues
:math:`\lambda_i` through the computation of
:math:`\frac{1}{\min_{i \neq j} \lambda_i - \lambda_j}`.
.. seealso::
:func:`torch.linalg.eigvalsh` computes only the eigenvalues.
However, that function is not differentiable.
:func:`torch.linalg.cholesky` for a different decomposition of a Hermitian matrix.
The Cholesky decomposition gives less information about the matrix but is much faster
to compute than the eigenvalue decomposition.
:func:`torch.linalg.eig` for a (slower) function that computes the eigenvalue decomposition
of a not necessarily Hermitian square matrix.
:func:`torch.linalg.svd` for a (slower) function that computes the more general SVD
decomposition of matrices of any shape.
:func:`torch.linalg.qr` for another (much faster) decomposition that works on general
matrices.
Args:
A (Tensor): tensor of shape `(*, n, n)` where `*` is zero or more batch dimensions
consisting of symmetric or Hermitian matrices.
UPLO ('L', 'U', optional): controls whether to use the upper or lower triangular part
of :attr:`A` in the computations. Default: `'L'`.
Keyword args:
out (tuple, optional): output tuple of two tensors. Ignored if `None`. Default: `None`.
Returns:
A named tuple `(eigenvalues, eigenvectors)` which corresponds to :math:`\Lambda` and :math:`V` above.
`eigenvalues` will always be real-valued, even when :attr:`A` is complex.
It will also be ordered in ascending order.
`eigenvectors` will have the same dtype as :attr:`A`.
Examples::
>>> a = torch.randn(2, 2, dtype=torch.complex128)
>>> a = a + a.t().conj() # creates a Hermitian matrix
>>> a
tensor([[2.9228+0.0000j, 0.2029-0.0862j],
[0.2029+0.0862j, 0.3464+0.0000j]], dtype=torch.complex128)
>>> w, v = torch.linalg.eigh(a)
>>> w
tensor([0.3277, 2.9415], dtype=torch.float64)
>>> v
tensor([[-0.0846+-0.0000j, -0.9964+0.0000j],
[ 0.9170+0.3898j, -0.0779-0.0331j]], dtype=torch.complex128)
>>> torch.allclose(torch.matmul(v, torch.matmul(w.to(v.dtype).diag_embed(), v.t().conj())), a)
True
>>> a = torch.randn(3, 2, 2, dtype=torch.float64)
>>> a = a + a.transpose(-2, -1) # creates a symmetric matrix
>>> w, v = torch.linalg.eigh(a)
>>> torch.allclose(torch.matmul(v, torch.matmul(w.diag_embed(), v.transpose(-2, -1))), a)
True
""")
eigvalsh = _add_docstr(_linalg.linalg_eigvalsh, r"""
linalg.eigvalsh(A, UPLO='L', *, out=None) -> Tensor
Computes the eigenvalues of a complex Hermitian or real symmetric matrix.
Letting :math:`\mathbb{K}` be :math:`\mathbb{R}` or :math:`\mathbb{C}`,
the **eigenvalues** of a complex Hermitian or real symmetric matrix :math:`A \in \mathbb{K}^{n \times n}`
are defined as the roots (counted with multiplicity) of the polynomial `p` of degree `n` given by
.. math::
p(\lambda) = \operatorname{det}(A - \lambda \mathrm{I}_n)\mathrlap{\qquad \lambda \in \mathbb{R}}
where :math:`\mathrm{I}_n` is the `n`-dimensional identity matrix.
The eigenvalues of a real symmetric or complex Hermitian matrix are always real.
Supports inputs of float, double, cfloat and cdouble dtypes.
Also supports batched inputs, and, if the input is batched, the output is batched with the same dimensions.
The eigenvalues are returned in ascending order.
:attr:`A` is assumed to be Hermitian (resp. symmetric), but this is not checked internally, instead:
- If :attr:`UPLO`\ `= 'L'` (default), only the lower triangular part of the matrix is used in the computation.
- If :attr:`UPLO`\ `= 'U'`, only the upper triangular part of the matrix is used.
.. note:: For CUDA inputs, this function synchronizes that device with the CPU.
.. note:: This function is not differentiable. If you need differentiability use
:func:`torch.linalg.eigh` instead, which also computes the eigenvectors.
Args:
A (Tensor): tensor of shape `(*, n, n)` where `*` is zero or more batch dimensions
consisting of symmetric or Hermitian matrices.
UPLO ('L', 'U', optional): controls whether to use the upper or lower triangular part
of :attr:`A` in the computations. Default: `'L'`.
Keyword args:
out (Tensor, optional): output tensor. Ignored if `None`. Default: `None`.
Returns:
A real-valued tensor cointaining the eigenvalues even when :attr:`A` is complex.
The eigenvalues are returned in ascending order.
Examples::
>>> a = torch.randn(2, 2, dtype=torch.complex128)
>>> a = a + a.t().conj() # creates a Hermitian matrix
>>> a
tensor([[2.9228+0.0000j, 0.2029-0.0862j],
[0.2029+0.0862j, 0.3464+0.0000j]], dtype=torch.complex128)
>>> w = torch.linalg.eigvalsh(a)
>>> w
tensor([0.3277, 2.9415], dtype=torch.float64)
>>> a = torch.randn(3, 2, 2, dtype=torch.float64)
>>> a = a + a.transpose(-2, -1) # creates a symmetric matrix
>>> a
tensor([[[ 2.8050, -0.3850],
[-0.3850, 3.2376]],
[[-1.0307, -2.7457],
[-2.7457, -1.7517]],
[[ 1.7166, 2.2207],
[ 2.2207, -2.0898]]], dtype=torch.float64)
>>> w = torch.linalg.eigvalsh(a)
>>> w
tensor([[ 2.5797, 3.4629],
[-4.1605, 1.3780],
[-3.1113, 2.7381]], dtype=torch.float64)
""")
householder_product = _add_docstr(_linalg.linalg_householder_product, r"""
householder_product(A, tau, *, out=None) -> Tensor
Computes the first `n` columns of a product of Householder matrices.
Letting :math:`\mathbb{K}` be :math:`\mathbb{R}` or :math:`\mathbb{C}`,
for a matrix :math:`V \in \mathbb{K}^{m \times n}` with columns :math:`v_i \in \mathbb{K}^m`
with :math:`m \geq n` and a vector :math:`\tau \in \mathbb{K}^k` with :math:`k \leq n`,
this function computes the first :math:`n` columns of the matrix
.. math::
H_1H_2 ... H_k \qquad\text{with}\qquad H_i = \mathrm{I}_m - \tau_i v_i v_i^{\text{H}}
where :math:`\mathrm{I}_m` is the `m`-dimensional identity matrix and
:math:`v^{\text{H}}` is the conjugate transpose when :math:`v` is complex, and the transpose when :math:`v` is real-valued.
See `Representation of Orthogonal or Unitary Matrices`_ for further details.
Supports inputs of float, double, cfloat and cdouble dtypes.
Also supports batched inputs, and, if the input is batched, the output is batched with the same dimensions.
.. note:: This function only uses the values strictly below the main diagonal of :attr:`A`.
The other values are ignored.
.. seealso::
:func:`torch.geqrf` can be used together with this function to form the `Q` from the
:func:`~qr` decomposition.
Args:
A (Tensor): tensor of shape `(*, m, n)` where `*` is zero or more batch dimensions.
tau (Tensor): tensor of shape `(*, k)` where `*` is zero or more batch dimensions.
Keyword args:
out (Tensor, optional): output tensor. Ignored if `None`. Default: `None`.
Raises:
RuntimeError: if :attr:`A` doesn't satisfy the requirement `m >= n`,
or :attr:`tau` doesn't satisfy the requirement `n >= k`.
Examples::
>>> a = torch.randn(2, 2)
>>> h, tau = torch.geqrf(a)
>>> q = torch.linalg.householder_product(h, tau)
>>> torch.allclose(q, torch.linalg.qr(a)[0])
True
>>> h = torch.randn(3, 2, 2, dtype=torch.complex128)
>>> tau = torch.randn(3, 1, dtype=torch.complex128)
>>> q = torch.linalg.householder_product(h, tau)
>>> q
tensor([[[ 1.8034+0.4184j, 0.2588-1.0174j],
[-0.6853+0.7953j, 2.0790+0.5620j]],
[[ 1.4581+1.6989j, -1.5360+0.1193j],
[ 1.3877-0.6691j, 1.3512+1.3024j]],
[[ 1.4766+0.5783j, 0.0361+0.6587j],
[ 0.6396+0.1612j, 1.3693+0.4481j]]], dtype=torch.complex128)
.. _Representation of Orthogonal or Unitary Matrices:
https://www.netlib.org/lapack/lug/node128.html
""")
lstsq = _add_docstr(_linalg.linalg_lstsq, r"""
torch.linalg.lstsq(A, B, cond=None, *, driver=None) -> (Tensor, Tensor, Tensor, Tensor)
Computes a solution to the least squares problem of a system of linear equations.
Letting :math:`\mathbb{K}` be :math:`\mathbb{R}` or :math:`\mathbb{C}`,
the **least squares problem** for a linear system :math:`AX = B` with
:math:`A \in \mathbb{K}^{m \times n}, B \in \mathbb{K}^{m \times k}` is defined as
.. math::
\min_{X \in \mathbb{K}^{n \times k}} \|AX - B\|_F
where :math:`\|-\|_F` denotes the Frobenius norm.
Supports inputs of float, double, cfloat and cdouble dtypes.
Also supports batched inputs, and, if the input is batched, the output is batched with the same dimensions.
:attr:`driver` chooses the LAPACK/MAGMA function that will be used.
For CPU inputs the valid values are `'gels'`, `'gelsy'`, `'gelsd`, `'gelss'`.
For CUDA input, the only valid driver is `'gels'`, which assumes that :attr:`A` is full-rank and `m < n`.
To choose the best driver on CPU consider:
- If :attr:`A` is well-conditioned (its `condition number`_ is not too large), or you do not mind some precision loss.
- For a general matrix: `'gelsy'` (QR with pivoting) (default)
- If :attr:`A` is full-rank: `'gels'` (QR)
- If :attr:`A` is not well-conditioned.
- `'gelsd'` (tridiagonal reduction and SVD)
- But if you run into memory issues: `'gelss'` (full SVD).
See also the `full description of these drivers`_
:attr:`cond` is used to determine the effective rank of the matrices in :attr:`A`
when :attr:`driver` is one of (`'gelsy'`, `'gelsd'`, `'gelss'`).
In this case, if :math:`\sigma_i` are the singular values of `A` in decreasing order,
:math:`\sigma_i` will be rounded down to zero if :math:`\sigma_i \leq \text{cond} \cdot \sigma_1`.
If :attr:`cond`\ `= None` (default), :attr:`cond` is set to the machine precision of the dtype of :attr:`A`.
This function returns the solution to the problem and some extra information in a named tuple of
four tensors `(solution, residuals, rank, singular_values)`. For inputs :attr:`A`, :attr:`B`
of shape `(*, m, n)`, `(*, m, k)` respectively, it cointains
- `solution`: the least squares solution. It has shape `(*, n, k)`.
- `residuals`: the squared residuals of the solutions, that is, :math:`\|AX - B\|_F^2`.
It has shape equal to the batch dimensions of :attr:`A`.
It is computed when `m > n` and every matrix in :attr:`A` is full-rank,
otherwise, it is an empty tensor.
- `rank`: tensor of ranks of the matrices in :attr:`A`.
It has shape equal to the batch dimensions of :attr:`A`.
It is computed when :attr:`driver` is one of (`'gelsy'`, `'gelsd'`, `'gelss'`),
otherwise it is an empty tensor.
- `singular_values`: tensor of singular values of the matrices in :attr:`A`.
It has shape `(*, min(m, n))`.
It is computed when :attr:`driver` is one of (`'gelsd'`, `'gelss'`),
otherwise it is an empty tensor.
.. note::
While `X = \ `:attr:`A`\ `.pinv() @ \ `:attr:`B`, this function computes the
solution in a faster and more numerically stable way that performing the
computations separately.
.. warning::
The default value of :attr:`cond` may change in the future.
It is therefore recommended to use a fixed value to avoid potential
breaking changes.
Args:
A (Tensor): lhs tensor of shape `(*, m, n)` where `*` is zero or more batch dimensions.
B (Tensor): rhs tensor of shape `(*, m, k)` where `*` is zero or more batch dimensions.
cond (float, optional): used to determine the effective rank of :attr:`A`.
If :attr:`cond`\ `= None`, :attr:`cond` is set to the machine
precision of the dtype of :attr:`A`. Default: `None`.
Keyword args:
driver (str, optional): name of the LAPACK/MAGMA method to be used.
If `None`, `'gelsy'` is used for CPU inputs and `'gels'` for CUDA inputs.
Default: `None`.
Returns:
A named tuple `(solution, residuals, rank, singular_values)`.
Examples::
>>> a = torch.tensor([[10, 2, 3], [3, 10, 5], [5, 6, 12]], dtype=torch.float)
>>> a.unsqueeze_(0)
>>> b = torch.tensor([[[2, 5, 1], [3, 2, 1], [5, 1, 9]],
[[4, 2, 9], [2, 0, 3], [2, 5, 3]]], dtype=torch.float)
>>> x = torch.linalg.lstsq(a, b).solution
>>> torch.dist(x, a.pinverse() @ b)
tensor(2.0862e-07)
>>> sv = torch.linalg.lstsq(a, driver='gelsd').singular_values
>>> torch.dist(sv, a.svd().S)
tensor(5.7220e-06)
>>> a[:, 0].zero_()
>>> xx, rank, _ = torch.linalg.lstsq(a, b)
>>> rank
tensor([2])
.. _condition number:
https://pytorch.org/docs/master/linalg.html#torch.linalg.cond
.. _full description of these drivers:
https://www.netlib.org/lapack/lug/node27.html
""")
matrix_power = _add_docstr(_linalg.linalg_matrix_power, r"""
matrix_power(A, n, *, out=None) -> Tensor
Computes the `n`-th power of a square matrix for an integer `n`.
Supports inputs of float, double, cfloat and cdouble dtypes.
Also supports batched inputs, and, if the input is batched, the output is batched with the same dimensions.
If :attr:`n`\ `= 0`, it returns the identity matrix (or batch) of the same shape
as :attr:`A`. If :attr:`n` is negative, it returns the inverse of each matrix
(if invertible) raised to the power of `abs(n)`.
.. seealso::
:func:`torch.linalg.solve` computes :attr:`A`\ `.inv() @ \ `:attr:`B` with a stable algorithm.
It is always prefered to use :func:`~matrix_power` with :attr:`n`\ `> 0` followed
by :func:`torch.linalg.solve` when possible, rather than
:func:`~matrix_power` with :attr:`n`\ `< 0`. The former method is
faster and more stable than computing :math:`A^{-n}` explicitly.
Args:
A (Tensor): tensor of shape `(*, m, m)` where `*` is zero or more batch dimensions.
n (int): the exponent.
Keyword args:
out (Tensor, optional): output tensor. Ignored if `None`. Default: `None`.
Raises:
RuntimeError: if :attr:`n`\ `< 0` and the matrix :attr:`A` or any matrix in the
batch of matrices :attr:`A` is not invertible.
Examples::
>>> a = torch.randn(3, 3)
>>> a
tensor([[-0.2270, 0.6663, -1.3515],
[-0.9838, -0.4002, -1.9313],
[-0.7886, -0.0450, 0.0528]])
>>> torch.linalg.matrix_power(a, 0)
tensor([[1., 0., 0.],
[0., 1., 0.],
[0., 0., 1.]])
>>> torch.linalg.matrix_power(a, 3)
tensor([[ 1.0756, 0.4980, 0.0100],
[-1.6617, 1.4994, -1.9980],
[-0.4509, 0.2731, 0.8001]])
>>> torch.linalg.matrix_power(a.expand(2, -1, -1), -2)
tensor([[[ 0.2640, 0.4571, -0.5511],
[-1.0163, 0.3491, -1.5292],
[-0.4899, 0.0822, 0.2773]],
[[ 0.2640, 0.4571, -0.5511],
[-1.0163, 0.3491, -1.5292],
[-0.4899, 0.0822, 0.2773]]])
""")
matrix_rank = _add_docstr(_linalg.linalg_matrix_rank, r"""
matrix_rank(A, tol=None, hermitian=False, *, out=None) -> Tensor
Computes the numerical rank of a matrix.
The matrix rank is computed as the number of singular values
(or eigenvalues in absolute value when :attr:`hermitian`\ `= True`)
that are greater than the specified :attr:`tol` threshold.
Supports inputs of float, double, cfloat and cdouble dtypes.
Also supports batched inputs, and, if the input is batched, the output is batched with the same dimensions.
If :attr:`hermitian`\ `= True`, :attr:`A` is assumed to be Hermitian if complex or
symmetric if real, but this is not checked internally. Instead, just the lower
triangular part of the matrix is used in the computations.
If :attr:`tol` is not specified and :attr:`A` is a matrix of dimensions `(m, n)`,
the tolerance is set to be
.. math::
\text{tol} = \sigma_1 \max(m, n) \varepsilon
where :math:`\sigma_1` is the largest singular value
(or eigenvalue in absolute value when :attr:`hermitian`\ `= True`), and
:math:`\varepsilon` is the epsilon value for the dtype of :attr:`A` (see :class:`torch.finfo`).
If :attr:`A` is a batch of matrices, :attr:`tol` is computed this way for every element of
the batch.
.. note:: The matrix rank is computed using singular value decomposition
:func:`torch.linalg.svd` if :attr:`hermitian`\ `= False` (default) and the eigenvalue
decomposition :func:`torch.linalg.eigvalsh` when :attr:`hermitian`\ `= True`.
For CUDA inputs, this function synchronizes that device with the CPU.
Args:
A (Tensor): tensor of shape `(*, m, n)` where `*` is zero or more batch dimensions.
tol (float, Tensor, optional): the tolerance value. See above for the value it takes when `None`.
Default: `None`.
hermitian(bool, optional): indicates whether :attr:`A` is Hermitian if complex
or symmetric if real. Default: `False`.
Keyword args:
out (Tensor, optional): output tensor. Ignored if `None`. Default: `None`.
Examples::
>>> a = torch.eye(10)
>>> torch.linalg.matrix_rank(a)
tensor(10)
>>> b = torch.eye(10)
>>> b[0, 0] = 0
>>> torch.linalg.matrix_rank(b)
tensor(9)
>>> a = torch.randn(4, 3, 2)
>>> torch.linalg.matrix_rank(a)
tensor([2, 2, 2, 2])
>>> a = torch.randn(2, 4, 2, 3)
>>> torch.linalg.matrix_rank(a)
tensor([[2, 2, 2, 2],
[2, 2, 2, 2]])
>>> a = torch.randn(2, 4, 3, 3, dtype=torch.complex64)
>>> torch.linalg.matrix_rank(a)
tensor([[3, 3, 3, 3],
[3, 3, 3, 3]])
>>> torch.linalg.matrix_rank(a, hermitian=True)
tensor([[3, 3, 3, 3],
[3, 3, 3, 3]])
>>> torch.linalg.matrix_rank(a, tol=1.0)
tensor([[3, 2, 2, 2],
[1, 2, 1, 2]])
>>> torch.linalg.matrix_rank(a, tol=1.0, hermitian=True)
tensor([[2, 2, 2, 1],
[1, 2, 2, 2]])
""")
vector_norm = _add_docstr(_linalg.linalg_vector_norm, r"""
linalg.vector_norm(A, ord=None, dim=None, keepdim=False, *, dtype=None, out=None) -> Tensor
Computes a vector norm.
If :attr:`A` is complex valued, it computes the norm of :attr:`A`\ `.abs()`
Supports inputs of float, double, cfloat and cdouble dtypes.
Also supports batched inputs, and, if the input is batched, the output is batched with the same dimensions.
- If :attr:`dim`\ `= None`, :attr:`A` will be flattened before the norm is computed.
- If :attr:`dim` is an `int` or a `tuple`, the norm will be computed over these dimensions
and the other dimensions will be treated as batch dimensions.
:attr:`ord` defines the vector norm that is computed. The following norms are supported:
====================== ========================================================
:attr:`ord` vector norm
====================== ========================================================
`None` (default) `2`-norm
`inf` `max(abs(x))`
`-inf` `min(abs(x))`
`0` `sum(x != 0)`
other `int` or `float` `sum(abs(x)**\ `:attr:`ord`\ `)**(1./\ `:attr:`ord`\ `)`
====================== ========================================================
where `inf` refers to `float('inf')`, NumPy's `inf` object, or any equivalent object.
Args:
A (Tensor): tensor of shape `(*, n)` where `*` is zero or more batch dimensions.
ord (int, float, inf, -inf, 'fro', 'nuc', optional): order of norm. Default: `None`
dim (int, Tuple[int], optional): dimensions over which to compute
the norm. See above for the behavior when :attr:`dim`\ `= None`.
Default: `None`
keepdim (bool, optional): If set to `True`, the reduced dimensions are retained
in the result as dimensions with shape one. Default: `False`
Keyword args:
out (Tensor, optional): output tensor. Ignored if `None`. Default: `None`.
dtype (:class:`torch.dtype`, optional): If specified, the input tensor is cast to
:attr:`dtype` before performing the operation, and the returned tensor's type
will be :attr:`dtype`. Default: `None`
Returns:
A real-valued tensor, even when :attr:`A` is complex.
Examples::
>>> from torch import linalg as LA
>>> a = torch.arange(9, dtype=torch.float) - 4
>>> a
tensor([-4., -3., -2., -1., 0., 1., 2., 3., 4.])
>>> b = a.reshape((3, 3))
>>> b
tensor([[-4., -3., -2.],
[-1., 0., 1.],
[ 2., 3., 4.]])
>>> LA.vector_norm(a, ord=3.5)
tensor(5.4345)
>>> LA.vector_norm(b, ord=3.5)
tensor(5.4345)
""")
multi_dot = _add_docstr(_linalg.linalg_multi_dot, r"""
linalg.multi_dot(tensors, *, out=None)
Efficiently multiplies two or more matrices by reordering the multiplications so that
the fewest arithmetic operations are performed.
Supports inputs of float, double, cfloat and cdouble dtypes.
This function does not support batched inputs.
Every tensor in :attr:`tensors` must be 2D, except for the first and last which
may be 1D. If the first tensor is a 1D vector of shape `(n,)` it is treated as a row vector
of shape `(1, n)`, similarly if the last tensor is a 1D vector of shape `(n,)` it is treated
as a column vector of shape `(n, 1)`.
If the first and last tensors are matrices, the output will be a matrix.
However, if either is a 1D vector, then the output will be a 1D vector.
Differences with `numpy.linalg.multi_dot`:
- Unlike `numpy.linalg.multi_dot`, the first and last tensors must either be 1D or 2D
whereas NumPy allows them to be nD
.. warning:: This function does not broadcast.
.. note:: This function is implemented by chaining :func:`torch.mm` calls after
computing the optimal matrix multiplication order.
.. note:: The cost of multiplying two matrices with shapes `(a, b)` and `(b, c)` is
`a * b * c`. Given matrices `A`, `B`, `C` with shapes `(10, 100)`,
`(100, 5)`, `(5, 50)` respectively, we can calculate the cost of different
multiplication orders as follows:
.. math::
\begin{align*}
\operatorname{cost}((AB)C) &= 10 \times 100 \times 5 + 10 \times 5 \times 50 = 7500 \\
\operatorname{cost}(A(BC)) &= 10 \times 100 \times 50 + 100 \times 5 \times 50 = 75000
\end{align*}
In this case, multiplying `A` and `B` first followed by `C` is 10 times faster.
Args:
tensors (Sequence[Tensor]): two or more tensors to multiply. The first and last
tensors may be 1D or 2D. Every other tensor must be 2D.
Keyword args:
out (Tensor, optional): output tensor. Ignored if `None`. Default: `None`.
Examples::
>>> from torch.linalg import multi_dot
>>> multi_dot([torch.tensor([1, 2]), torch.tensor([2, 3])])
tensor(8)
>>> multi_dot([torch.tensor([[1, 2]]), torch.tensor([2, 3])])
tensor([8])
>>> multi_dot([torch.tensor([[1, 2]]), torch.tensor([[2], [3]])])
tensor([[8]])
>>> a = torch.arange(2 * 3).view(2, 3)
>>> b = torch.arange(3 * 2).view(3, 2)
>>> c = torch.arange(2 * 2).view(2, 2)
>>> multi_dot((a, b, c))
tensor([[ 26, 49],
[ 80, 148]])
>>> multi_dot((a.to(torch.float), torch.empty(3, 0), torch.empty(0, 2)))
tensor([[0., 0.],
[0., 0.]])
""")
norm = _add_docstr(_linalg.linalg_norm, r"""
linalg.norm(A, ord=None, dim=None, keepdim=False, *, out=None, dtype=None) -> Tensor
Computes a vector or a matrix norm.
If :attr:`A` is complex valued, it computes the norm of :attr:`A`\ `.abs()`
Supports inputs of float, double, cfloat and cdouble dtypes.
Also supports batched inputs, and, if the input is batched, the output is batched with the same dimensions.
- If :attr:`dim` is an `int`, the vector norm will be computed.
- If :attr:`dim` is a `2`-`tuple`, the matrix norm will be computed.
- If :attr:`dim`\ `= None` and :attr:`ord`\ `= None`,
:attr:`A` will be flattened to 1D and the `2`-norm of the resulting vector will be computed.
- If :attr:`dim`\ `= None` and :attr:`ord` `!= None`, :attr:`A` must be 1D or 2D.
:attr:`ord` defines the norm that is computed. The following norms are supported:
====================== ========================= ========================================================
:attr:`ord` norm for matrices norm for vectors
====================== ========================= ========================================================
`None` (default) Frobenius norm `2`-norm
`'fro'` Frobenius norm -- not supported --
`'nuc'` nuclear norm -- not supported --
`inf` `max(sum(abs(x), dim=1))` `max(abs(x))`
`-inf` `min(sum(abs(x), dim=1))` `min(abs(x))`
`0` -- not supported -- `sum(x != 0)`
`1` `max(sum(abs(x), dim=0))` as below
`-1` `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 -- `sum(abs(x)**\ `:attr:`ord`\ `)**(1./\ `:attr:`ord`\ `)`
====================== ========================= ========================================================
where `inf` refers to `float('inf')`, NumPy's `inf` object, or any equivalent object.
Args:
A (Tensor): tensor of shape `(*, n)` or `(*, m, n)` where `*` is zero or more batch dimensions
ord (int, float, inf, -inf, 'fro', 'nuc', optional): order of norm. Default: `None`
dim (int, Tuple[int], optional): dimensions over which to compute
the vector or matrix norm. See above for the behavior when :attr:`dim`\ `= None`.
Default: `None`
keepdim (bool, optional): If set to `True`, the reduced dimensions are retained
in the result as dimensions with size one. Default: `False`
Keyword args:
out (Tensor, optional): output tensor. Ignored if `None`. Default: `None`.
dtype (:class:`torch.dtype`, optional): If specified, the input tensor is cast to
:attr:`dtype` before performing the operation, and the returned tensor's type
will be :attr:`dtype`. Default: `None`
Returns:
A real-valued tensor, even when :attr:`A` is complex.
Examples::
>>> from torch import linalg as LA
>>> a = torch.arange(9, dtype=torch.float) - 4
>>> a
tensor([-4., -3., -2., -1., 0., 1., 2., 3., 4.])
>>> b = a.reshape((3, 3))
>>> b
tensor([[-4., -3., -2.],
[-1., 0., 1.],
[ 2., 3., 4.]])
>>> LA.norm(a)
tensor(7.7460)
>>> LA.norm(b)
tensor(7.7460)
>>> LA.norm(b, 'fro')
tensor(7.7460)
>>> LA.norm(a, float('inf'))
tensor(4.)
>>> LA.norm(b, float('inf'))
tensor(9.)
>>> LA.norm(a, -float('inf'))
tensor(0.)
>>> LA.norm(b, -float('inf'))
tensor(2.)
>>> LA.norm(a, 1)
tensor(20.)
>>> LA.norm(b, 1)
tensor(7.)
>>> LA.norm(a, -1)
tensor(0.)
>>> LA.norm(b, -1)
tensor(6.)
>>> LA.norm(a, 2)
tensor(7.7460)
>>> LA.norm(b, 2)
tensor(7.3485)
>>> LA.norm(a, -2)
tensor(0.)
>>> LA.norm(b.double(), -2)
tensor(1.8570e-16, dtype=torch.float64)
>>> LA.norm(a, 3)
tensor(5.8480)
>>> LA.norm(a, -3)
tensor(0.)
Using the :attr:`dim` argument to compute vector norms::
>>> c = torch.tensor([[1., 2., 3.],
... [-1, 1, 4]])
>>> LA.norm(c, dim=0)
tensor([1.4142, 2.2361, 5.0000])
>>> LA.norm(c, dim=1)
tensor([3.7417, 4.2426])
>>> LA.norm(c, ord=1, dim=1)
tensor([6., 6.])
Using the :attr:`dim` argument to compute matrix norms::
>>> m = torch.arange(8, dtype=torch.float).reshape(2, 2, 2)
>>> LA.norm(m, dim=(1,2))
tensor([ 3.7417, 11.2250])
>>> LA.norm(m[0, :, :]), LA.norm(m[1, :, :])
(tensor(3.7417), tensor(11.2250))
""")
svd = _add_docstr(_linalg.linalg_svd, r"""
linalg.svd(A, full_matrices=True, compute_uv=True, *, out=None) -> (Tensor, Tensor, Tensor)
Computes the singular value decomposition (SVD) of a matrix.
Letting :math:`\mathbb{K}` be :math:`\mathbb{R}` or :math:`\mathbb{C}`,
the **full SVD** of a matrix
:math:`A \in \mathbb{K}^{m \times n}`, if `k = min(m,n)`, is defined as
.. math::
A = U \operatorname{diag}(S) V^{\text{H}}
\mathrlap{\qquad U \in \mathbb{K}^{m \times m}, S \in \mathbb{R}^k, V \in \mathbb{K}^{n \times n}}
where :math:`\operatorname{diag}(S) \in \mathbb{K}^{m \times n}`,
:math:`V^{\text{H}}` is the conjugate transpose when :math:`V` is complex, and the transpose when :math:`V` is real-valued.
The matrices :math:`U`, :math:`V` (and thus :math:`V^{\text{H}}`) are orthogonal in the real case, and unitary in the complex case.
When `m > n` (resp. `m < n`) we can drop the last `m - n` (resp. `n - m`) columns of `U` (resp. `V`) to form the **reduced SVD**:
.. math::
A = U \operatorname{diag}(S) V^{\text{H}}
\mathrlap{\qquad U \in \mathbb{K}^{m \times k}, S \in \mathbb{R}^k, V \in \mathbb{K}^{k \times n}}
where :math:`\operatorname{diag}(S) \in \mathbb{K}^{k \times k}`.
In this case, :math:`U` and :math:`V` also have orthonormal columns.
Supports inputs of float, double, cfloat and cdouble dtypes.
Also supports batched inputs, and, if the input is batched, the output is batched with the same dimensions.
The returned decomposition is a named tuple `(U, S, Vh)`
which corresponds to :math:`U`, :math:`S`, :math:`V^{\text{H}}` above.
The singular values are returned in descending order.
The parameter :attr:`full_matrices` chooses between the full (default) and reduced SVD.
Differences with `numpy.linalg.svd`:
- Unlike `numpy.linalg.svd`, this function always returns a tuple of three tensors.
When :attr:`compute_uv`\ `= False`, `U`, `Vh` are empty tensors.
This behavior may change in a future PyTorch release.
Please use :func:`torch.linalg.svdvals`, which computes only the singular values,
instead of :attr:`compute_uv`\ `= False`.
.. note:: When :attr:`full_matrices`\ `= True`, the gradients with respect to `U[..., :, min(m, n):]`
and `Vh[..., min(m, n):, :]` will be ignored, as those vectors can be arbitrary bases
of the corresponding subspaces.
.. warning:: The returned tensors `U` and `V` are not unique, nor are they continuous with
respect to :attr:`A`.
Due to this lack of uniqueness, different hardware and software may compute
different singular vectors.
This non-uniqueness is caused by the fact that multiplying any pair of singular
vectors :math:`u_k, v_k` by `-1` in the real case or by
:math:`e^{i \phi}, \phi \in \mathbb{R}` in the complex case produces another two
valid singular vectors of the matrix.
This non-uniqueness problem is even worse when the matrix has repeated singular values.
In this case, one may multiply the associated singular vectors of `U` and `V` spanning
the subspace by a rotation matrix and `the resulting vectors will span the same subspace`_.
.. warning:: Gradients computed using `U` or `Vh` will only be finite when
:attr:`A` does not have zero as a singular value or repeated singular values.
Furthermore, if the distance between any two singular values is close to zero,
the gradient will be numerically unstable, as it depends on the singular values
:math:`\sigma_i` through the computation of
:math:`\frac{1}{\min_{i \neq j} \sigma_i^2 - \sigma_j^2}`.
The gradient will also be numerically unstable when :attr:`A` has small singular
values, as it also depends on the computaiton of :math:`\frac{1}{\sigma_i}`.
.. seealso::
:func:`torch.linalg.svdvals` computes only the singular values.
However, that function is not differentiable.
:func:`torch.linalg.eig` for a function that computes another type of spectral
decomposition of a matrix. The eigendecomposition works just on on square matrices.
:func:`torch.linalg.eigh` for a (faster) function that computes the eigenvalue decomposition
for Hermitian and symmetric matrices.
:func:`torch.linalg.qr` for another (much faster) decomposition that works on general
matrices.
Args:
A (Tensor): tensor of shape `(*, m, n)` where `*` is zero or more batch dimensions.
full_matrices (bool, optional): controls whether to compute the full or reduced
SVD, and consequently,
the shape of the returned tensors
`U` and `Vh`. Default: `True`.
compute_uv (bool, optional): controls whether to compute `U` and `Vh`. Default: `True`.
Keyword args:
out (tuple, optional): output tuple of three tensors. Ignored if `None`.
If :attr:`compute_uv`\ `= False`, the 1st and 3rd arguments must be
tensors, but they are ignored. Default: `None`
Returns:
A named tuple `(U, S, Vh)` which corresponds to :math:`U`, :math:`S`, :math:`V^{\text{H}}` above.
`S` will always be real-valued, even when :attr:`A` is complex.
It will also be ordered in descending order.
`U` and `Vh` will have the same dtype as :attr:`A`.
Examples::
>>> a = torch.randn(5, 3)
>>> a
tensor([[-0.3357, -0.2987, -1.1096],
[ 1.4894, 1.0016, -0.4572],
[-1.9401, 0.7437, 2.0968],
[ 0.1515, 1.3812, 1.5491],
[-1.8489, -0.5907, -2.5673]])
>>>
>>> # reconstruction in the full_matrices=False case
>>> u, s, vh = torch.linalg.svd(a, full_matrices=False)
>>> u.shape, s.shape, vh.shape
(torch.Size([5, 3]), torch.Size([3]), torch.Size([3, 3]))
>>> torch.dist(a, u @ torch.diag(s) @ vh)
tensor(1.0486e-06)
>>>
>>> # reconstruction in the full_matrices=True case
>>> u, s, vh = torch.linalg.svd(a)
>>> u.shape, s.shape, vh.shape
(torch.Size([5, 5]), torch.Size([3]), torch.Size([3, 3]))
>>> torch.dist(a, u[:, :3] @ torch.diag(s) @ vh)
>>> torch.dist(a, u[:, :3] @ torch.diag(s) @ vh)
tensor(1.0486e-06)
>>>
>>> # extra dimensions
>>> a_big = torch.randn(7, 5, 3)
>>> u, s, vh = torch.linalg.svd(a_big, full_matrices=False)
>>> torch.dist(a_big, u @ torch.diag_embed(s) @ vh)
tensor(3.0957e-06)
.. _the resulting vectors will span the same subspace:
https://en.wikipedia.org/wiki/Singular_value_decomposition#Singular_values,_singular_vectors,_and_their_relation_to_the_SVD
""")
svdvals = _add_docstr(_linalg.linalg_svdvals, r"""
linalg.svdvals(A, *, out=None) -> Tensor
Computes the singular values of a matrix.
Supports inputs of float, double, cfloat and cdouble dtypes.
Also supports batched inputs, and, if the input is batched, the output is batched with the same dimensions.
The singular values are returned in descending order.
.. note:: This function is not differentiable. If you need differentiability use
:func:`torch.linalg.svd` instead, which also computes the singular vectors.
.. note:: This function is equivalent to NumPy's `linalg.svd(A, compute_uv=False)`.
.. note:: For CUDA inputs, this function synchronizes that device with the CPU.
Args:
A (Tensor): tensor of shape `(*, m, n)` where `*` is zero or more batch dimensions.
Keyword args:
out (Tensor, optional): output tensor. Ignored if `None`. Default: `None`.
Returns:
A real-valued tensor, even when :attr:`A` is complex.
Examples::
>>> import torch
>>> a = torch.randn(5, 3)
>>> a
tensor([[-1.3490, -0.1723, 0.7730],
[-1.6118, -0.3385, -0.6490],
[ 0.0908, 2.0704, 0.5647],
[-0.6451, 0.1911, 0.7353],
[ 0.5247, 0.5160, 0.5110]])
>>> s = torch.linalg.svdvals(a)
>>> s
tensor([2.5139, 2.1087, 1.1066])
""")
cond = _add_docstr(_linalg.linalg_cond, r"""
linalg.cond(A, p=None, *, out=None) -> Tensor
Computes the condition number of a matrix with respect to a matrix norm.
Letting :math:`\mathbb{K}` be :math:`\mathbb{R}` or :math:`\mathbb{C}`,
the **condition number** :math:`\kappa` of a matrix
:math:`A \in \mathbb{K}^{n \times n}` is defined as
.. math::
\kappa(A) = \|A\|_p\|A^{-1}\|_p
The condition number of :attr:`A` measures the numerical stability of the linear system `AX = B`
with respect to a matrix norm.
Supports inputs of float, double, cfloat and cdouble dtypes.
Also supports batched inputs, and, if the input is batched, the output is batched with the same dimensions.
:attr:`p` defines the matrix norm that is computed. The following norms are supported:
========= =================================
:attr:`p` matrix norm
========= =================================
`None` `2`-norm (largest singular value)
`'fro'` Frobenius norm
`'nuc'` nuclear norm
`inf` `max(sum(abs(x), dim=1))`
`-inf` `min(sum(abs(x), dim=1))`
`1` `max(sum(abs(x), dim=0))`
`-1` `min(sum(abs(x), dim=0))`
`2` largest singular value
`-2` smallest singular value
========= =================================
where `inf` refers to `float('inf')`, NumPy's `inf` object, or any equivalent object.
For :attr:`p` is one of `('fro', 'nuc', inf, -inf, 1, -1)`, this function uses
:func:`torch.linalg.norm` and :func:`torch.linalg.inv`.
As such, in this case, the matrix (or every matrix in the batch) :attr:`A` has to be square
and invertible.
For :attr:`p` in `(2, -2)`, this function can be computed in terms of the singular values
:math:`\sigma_1 \geq \ldots \geq \sigma_n`
.. math::
\kappa_2(A) = \frac{\sigma_1}{\sigma_n}\qquad \kappa_{-2}(A) = \frac{\sigma_n}{\sigma_1}
In these cases, it is computed using :func:`torch.linalg.svd`. For these norms, the matrix
(or every matrix in the batch) :attr:`A` may have any shape.
.. note :: For CUDA inputs, this function synchronizes that device with the CPU if
if :attr:`p` is one of `('fro', 'nuc', inf, -inf, 1, -1)`.
.. seealso::
:func:`torch.linalg.solve` for a function that solves linear systems of square matrices.
:func:`torch.linalg.lstsq` for a function that solves linear systems of general matrices.
Args:
A (Tensor): tensor of shape `(*, m, n)` where `*` is zero or more batch dimensions
for :attr:`p` in `(2, -2)`, and of shape `(*, n, n)` where every matrix
is invertible for :attr:`p` in `('fro', 'nuc', inf, -inf, 1, -1)`.
p (int, inf, -inf, 'fro', 'nuc', optional):
the type of the matrix norm to use in the computations (see above). Default: `None`
Keyword args:
out (Tensor, optional): output tensor. Ignored if `None`. Default: `None`.
Returns:
A real-valued tensor, even when :attr:`A` is complex.
Raises:
RuntimeError:
if :attr:`p` is one of `('fro', 'nuc', inf, -inf, 1, -1)`
and the :attr:`A` matrix or any matrix in the batch :attr:`A` is not square
or invertible.
Examples::
>>> a = torch.randn(3, 4, 4, dtype=torch.complex64)
>>> torch.linalg.cond(a)
>>> a = torch.tensor([[1., 0, -1], [0, 1, 0], [1, 0, 1]])
>>> torch.linalg.cond(a)
tensor([1.4142])
>>> torch.linalg.cond(a, 'fro')
tensor(3.1623)
>>> torch.linalg.cond(a, 'nuc')
tensor(9.2426)
>>> torch.linalg.cond(a, float('inf'))
tensor(2.)
>>> torch.linalg.cond(a, float('-inf'))
tensor(1.)
>>> torch.linalg.cond(a, 1)
tensor(2.)
>>> torch.linalg.cond(a, -1)
tensor(1.)
>>> torch.linalg.cond(a, 2)
tensor([1.4142])
>>> torch.linalg.cond(a, -2)
tensor([0.7071])
>>> a = torch.randn(2, 3, 3)
>>> a
tensor([[[-0.9204, 1.1140, 1.2055],
[ 0.3988, -0.2395, -0.7441],
[-0.5160, 0.3115, 0.2619]],
[[-2.2128, 0.9241, 2.1492],
[-1.1277, 2.7604, -0.8760],
[ 1.2159, 0.5960, 0.0498]]])
>>> torch.linalg.cond(a)
tensor([[9.5917],
[3.2538]])
>>> a = torch.randn(2, 3, 3, dtype=torch.complex64)
>>> a
tensor([[[-0.4671-0.2137j, -0.1334-0.9508j, 0.6252+0.1759j],
[-0.3486-0.2991j, -0.1317+0.1252j, 0.3025-0.1604j],
[-0.5634+0.8582j, 0.1118-0.4677j, -0.1121+0.7574j]],
[[ 0.3964+0.2533j, 0.9385-0.6417j, -0.0283-0.8673j],
[ 0.2635+0.2323j, -0.8929-1.1269j, 0.3332+0.0733j],
[ 0.1151+0.1644j, -1.1163+0.3471j, -0.5870+0.1629j]]])
>>> torch.linalg.cond(a)
tensor([[4.6245],
[4.5671]])
>>> torch.linalg.cond(a, 1)
tensor([9.2589, 9.3486])
""")
pinv = _add_docstr(_linalg.linalg_pinv, r"""
linalg.pinv(A, rcond=1e-15, hermitian=False, *, out=None) -> Tensor
Computes the pseudoinverse (Moore-Penrose inverse) of a matrix.
The pseudoinverse may be `defined algebraically`_
but it is more computationally convenient to understand it `through the SVD`_
Supports inputs of float, double, cfloat and cdouble dtypes.
Also supports batched inputs, and, if the input is batched, the output is batched with the same dimensions.
If :attr:`hermitian`\ `= True`, :attr:`A` is assumed to be Hermitian if complex or
symmetric if real, but this is not checked internally. Instead, just the lower
triangular part of the matrix is used in the computations.
The singular values (or the norm of the eigenvalues when :attr:`hermitian`\ `= True`)
that are below the specified :attr:`rcond` threshold are treated as zero and discarded in the computation.
.. note:: This function uses :func:`torch.linalg.svd` if :attr:`hermitian`\ `= False` and
:func:`torch.linalg.eigh` if :attr:`hermitian`\ `= True`.
For CUDA inputs, this function synchronizes that device with the CPU.
.. seealso::
:func:`torch.linalg.inv` computes the inverse of a square matrix.
:func:`torch.linalg.lstsq` computes :attr:`A`\ `.pinv() @ \ `:attr:`B` with a stable algorithm.
It is always prefered to use :func:`torch.linalg.lstsq` when possible, as it is
faster and more stable than computing the pseudoinverse and then multiplying.
Args:
A (Tensor): tensor of shape `(*, m, n)` where `*` is zero or more batch dimensions.
rcond (float or Tensor, optional): the tolerance value to determine when is a singular value zero
If it is a :class:`torch.Tensor`, its shape must be
broadcastable to that of the singular values of
:attr:`A` as returned by :func:`torch.svd`.
Default: `1e-15`.
hermitian(bool, optional): indicates whether :attr:`A` is Hermitian if complex
or symmetric if real. Default: `False`.
Keyword args:
out (Tensor, optional): output tensor. Ignored if `None`. Default: `None`.
Examples::
>>> A = torch.randn(3, 5)
>>> A
tensor([[ 0.5495, 0.0979, -1.4092, -0.1128, 0.4132],
[-1.1143, -0.3662, 0.3042, 1.6374, -0.9294],
[-0.3269, -0.5745, -0.0382, -0.5922, -0.6759]])
>>> torch.linalg.pinv(A)
tensor([[ 0.0600, -0.1933, -0.2090],
[-0.0903, -0.0817, -0.4752],
[-0.7124, -0.1631, -0.2272],
[ 0.1356, 0.3933, -0.5023],
[-0.0308, -0.1725, -0.5216]])
Batched linalg.pinv example
>>> A = torch.randn(2, 6, 3)
>>> B = torch.linalg.pinv(A)
>>> torch.matmul(B, A).round()
tensor([[[1., -0., 0.],
[0., 1., -0.],
[0., 0., 1.]],
[[1., -0., 0.],
[-0., 1., 0.],
[-0., -0., 1.]]])
Hermitian input example
>>> A = torch.randn(3, 3, dtype=torch.complex64)
>>> A = A + A.t().conj() # creates a Hermitian matrix
>>> B = torch.linalg.pinv(A, hermitian=True)
>>> torch.matmul(B, A)
tensor([[ 1.0000e+00+0.0000e+00j, -1.1921e-07-2.3842e-07j,
5.9605e-08-2.3842e-07j],
[ 5.9605e-08+2.3842e-07j, 1.0000e+00+2.3842e-07j,
-4.7684e-07+1.1921e-07j],
[-1.1921e-07+0.0000e+00j, -2.3842e-07-2.9802e-07j,
1.0000e+00-1.7897e-07j]])
Non-default rcond example
>>> rcond = 0.5
>>> A = torch.randn(3, 3)
>>> torch.linalg.pinv(A)
tensor([[ 0.2971, -0.4280, -2.0111],
[-0.0090, 0.6426, -0.1116],
[-0.7832, -0.2465, 1.0994]])
>>> torch.linalg.pinv(A, rcond)
tensor([[-0.2672, -0.2351, -0.0539],
[-0.0211, 0.6467, -0.0698],
[-0.4400, -0.3638, -0.0910]])
Matrix-wise rcond example
>>> A = torch.randn(5, 6, 2, 3, 3)
>>> rcond = torch.rand(2) # different rcond values for each matrix in a[:, :, 0] and a[:, :, 1]
>>> torch.linalg.pinv(A, rcond)
>>> rcond = torch.randn(5, 6, 2) # different rcond value for each matrix in 'a'
>>> torch.linalg.pinv(A, rcond)
.. _defined algebraically:
https://en.wikipedia.org/wiki/Moore%E2%80%93Penrose_inverse#Existence_and_uniqueness
.. _through the SVD:
https://en.wikipedia.org/wiki/Moore%E2%80%93Penrose_inverse#Singular_value_decomposition_(SVD)
""")
solve = _add_docstr(_linalg.linalg_solve, r"""
linalg.solve(A, B, *, out=None) -> Tensor
Computes the solution of a square system of linear equations with a unique solution.
Letting :math:`\mathbb{K}` be :math:`\mathbb{R}` or :math:`\mathbb{C}`,
this function computes the solution :math:`X \in \mathbb{K}^{n \times k}` of the **linear system** associated to
:math:`A \in \mathbb{K}^{n \times n}, B \in \mathbb{K}^{m \times k}`, which is defined as
.. math:: AX = B
This system of linear equations has one solution if and only if :math:`A` is `invertible`_.
This function assumes that :math:`A` is invertible.
Supports inputs of float, double, cfloat and cdouble dtypes.
Also supports batched inputs, and, if the input is batched, the output is batched with the same dimensions.
Letting `*` be zero or more batch dimensions.
- If :attr:`A` has shape `(*, n, n)` and :attr:`B` has shape `(*, n)` (a batch of vectors) or shape
`(*, n, k)` (a batch of matrices or "multiple right-hand sides"), this function returns `X` of shape
`(*, n)` or `(*, n, k)` respectively.
- Otherwise, if :attr:`A` has shape `(*, n, n)` and :attr:`B` has shape `(n,)` or `(n, k)`, :attr:`B`
is broadcasted to have shape `(*, n)` or `(*, n, k)` respectively.
This function then returns the solution of the resulting batch of systems of linear equations.
.. note::
While `X = \ `:attr:`A`\ `.inv() @ \ `:attr:`B`, this function computes the
solution in a faster and more numerically stable way that performing the
computations separately.
.. note:: For CUDA inputs, this function synchronizes that device with the CPU.
Args:
A (Tensor): tensor of shape `(*, n, n)` where `*` is zero or more batch dimensions.
B (Tensor): right-hand side tensor of shape `(*, n)` or `(*, n, k)` or `(n,)` or `(n, k)`
according to the rules described above
Keyword args:
out (Tensor, optional): output tensor. Ignored if `None`. Default: `None`.
Raises:
RuntimeError: if the :attr:`A` matrix is not invertible or any matrix in a batched :attr:`A`
is not invertible.
Examples::
>>> A = torch.rand(3, 3)
>>> b = torch.randn(3)
>>> x = torch.linalg.solve(A, b)
>>> torch.allclose(A @ x, b)
True
>>> A = torch.randn(2, 3, 3)
>>> B = torch.randn(2, 3, 4)
>>> X = torch.linalg.solve(A, B)
>>> X.shape
torch.Size([2, 3, 4])
>>> torch.allclose(A @ X, B)
True
Broadcasting::
>>> A = torch.randn(2, 3, 3)
>>> b = torch.randn(3, 1)
>>> x = torch.linalg.solve(A, b) # b is broadcasted to size (2, 3, 1)
>>> x.shape
torch.Size([2, 3, 1])
>>> torch.allclose(A @ x, b)
True
>>> b = torch.rand(3)
>>> x = torch.linalg.solve(A, b) # b is broadcasted to size (2, 3)
>>> x.shape
torch.Size([2, 3])
>>> Ax = A @ x.unsqueeze(-1)
>>> torch.allclose(Ax, b.unsqueeze(-1).expand_as(Ax))
True
.. _invertible:
https://en.wikipedia.org/wiki/Invertible_matrix#The_invertible_matrix_theorem
""")
tensorinv = _add_docstr(_linalg.linalg_tensorinv, r"""
linalg.tensorinv(A, ind=2, *, out=None) -> Tensor
Computes the multiplicative inverse of :func:`torch.tensordot`.
If `m` is the product of the first :attr:`ind` dimensions of :attr:`A` and `n` is the product of
the rest of the dimensions, this function expects `m` and `n` to be equal.
If this is the case, it computes a tensor `X` such that
`tensordot(\ `:attr:`A`\ `, X, \ `:attr:`ind`\ `)` is the identity matrix in dimension `m`.
`X` will have the shape of :attr:`A` but with the first :attr:`ind` dimensions pushed back to the end
.. code:: text
X.shape == A.shape[ind:] + A.shape[:ind]
Supports inputs of float, double, cfloat and cdouble dtypes.
.. note:: When :attr:`A` is a `2`-dimensional tensor and :attr:`ind`\ `= 1`,
this function computes the (multiplicative) inverse of :attr:`A`
(see :func:`torch.linalg.inv`).
.. seealso::
:func:`torch.linalg.tensorsolve` computes
`torch.tensordot(tensorinv(\ `:attr:`A`\ `), \ `:attr:`B`\ `)`.
It is always prefered to use :func:`~tensorsolve` when possible, as it is
faster and more stable than using
:func:`~tensorinv` followed by :func:`torch.tensordot`.
Args:
A (Tensor): tensor to invert. Its shape must satisfy
`prod(\ `:attr:`A`\ `.shape[:\ `:attr:`ind`\ `]) ==
prod(\ `:attr:`A`\ `.shape[\ `:attr:`ind`\ `:])`.
ind (int): index at which to compute the inverse of :func:`torch.tensordot`. Default: `2`.
Keyword args:
out (Tensor, optional): output tensor. Ignored if `None`. Default: `None`.
Raises:
RuntimeError: if the reshaped :attr:`A` is not invertible or the product of the first
:attr:`ind` dimensions is not equal to the product of the rest.
Examples::
>>> a = torch.eye(4 * 6).reshape((4, 6, 8, 3))
>>> ainv = torch.linalg.tensorinv(a, ind=2)
>>> ainv.shape
torch.Size([8, 3, 4, 6])
>>> b = torch.randn(4, 6)
>>> torch.allclose(torch.tensordot(ainv, b), torch.linalg.tensorsolve(a, b))
True
>>> a = torch.randn(4, 4)
>>> a_tensorinv = torch.linalg.tensorinv(a, ind=1)
>>> a_inv = torch.inverse(a)
>>> torch.allclose(a_tensorinv, a_inv)
True
""")
tensorsolve = _add_docstr(_linalg.linalg_tensorsolve, r"""
linalg.tensorsolve(A, B, dims=None, *, out=None) -> Tensor
Computes the solution `X` to the system `torch.tensordot(A, X) = B`.
If `m` is the product of the first :attr:`B`\ `.ndim` dimensions of :attr:`A` and
`n` is the product of the rest of the dimensions, this function expects `m` and `n` to be equal.
The returned tensor `x` satisfies
`tensordot(\ `:attr:`A`\ `, x, dims=x.ndim) == \ `:attr:`B`.
`x` has shape :attr:`A`\ `[B.ndim:]`.
If :attr:`dims` is specified, :attr:`A` will be reshaped as
.. code:: text
A = movedim(A, dims, range(len(dims) - A.ndim + 1, 0))
Supports inputs of float, double, cfloat and cdouble dtypes.
.. seealso::
:func:`torch.linalg.tensorinv` computes the multiplicative inverse of
:func:`torch.tensordot`.
Args:
A (Tensor): tensor to solve for. Its shape must satisfy
`prod(\ `:attr:`A`\ `.shape[:\ `:attr:`B`\ `.ndim]) ==
prod(\ `:attr:`A`\ `.shape[\ `:attr:`B`\ `.ndim:])`.
B (Tensor): tensor of shape :attr:`A`\ `.shape[\ `:attr:`B`\ `.ndim]`.
dims (Tuple[int], optional): dimensions of :attr:`A` to be moved.
If `None`, no dimensions are moved. Default: `None`.
Keyword args:
out (Tensor, optional): output tensor. Ignored if `None`. Default: `None`.
Raises:
RuntimeError: if the reshaped :attr:`A`\ `.view(m, m)` with `m` as above is not
invertible or the product of the first :attr:`ind` dimensions is not equal
to the product of the rest of the dimensions.
Examples::
>>> a = torch.eye(2 * 3 * 4).reshape((2 * 3, 4, 2, 3, 4))
>>> b = torch.randn(2 * 3, 4)
>>> x = torch.linalg.tensorsolve(a, b)
>>> x.shape
torch.Size([2, 3, 4])
>>> torch.allclose(torch.tensordot(a, x, dims=x.ndim), b)
True
>>> a = torch.randn(6, 4, 4, 3, 2)
>>> b = torch.randn(4, 3, 2)
>>> x = torch.linalg.tensorsolve(a, b, dims=(0, 2))
>>> x.shape
torch.Size([6, 4])
>>> a = a.permute(1, 3, 4, 0, 2)
>>> a.shape[b.ndim:]
torch.Size([6, 4])
>>> torch.allclose(torch.tensordot(a, x, dims=x.ndim), b, atol=1e-6)
True
""")
qr = _add_docstr(_linalg.linalg_qr, r"""
qr(A, mode='reduced', *, out=None) -> (Tensor, Tensor)
Computes the QR decomposition of a matrix.
Letting :math:`\mathbb{K}` be :math:`\mathbb{R}` or :math:`\mathbb{C}`,
the **full QR decomposition** of a matrix
:math:`A \in \mathbb{K}^{m \times n}` is defined as
.. math::
A = QR\mathrlap{\qquad Q \in \mathbb{K}^{m \times m}, R \in \mathbb{K}^{m \times n}}
where :math:`Q` is orthogonal in the real case and unitary in the complex case, and :math:`R` is upper triangular.
When `m > n` (tall matrix), as `R` is upper triangular, its last `m - n` rows are zero.
In this case, we can drop the last `m - n` columns of `Q` to form the
**reduced QR decomposition**:
.. math::
A = QR\mathrlap{\qquad Q \in \mathbb{K}^{m \times n}, R \in \mathbb{K}^{n \times n}}
The reduced QR decomposition agrees with the full QR decomposition when `n >= m` (wide matrix).
Supports inputs of float, double, cfloat and cdouble dtypes.
Also supports batched inputs, and, if the input is batched, the output is batched with the same dimensions.
The parameter :attr:`mode` chooses between the full and reduced QR decomposition.
If :attr:`A` has shape `(*, m, n)`, denoting `k = min(m, n)`
- :attr:`mode`\ `= 'reduced'` (default): Returns `(Q, R)` of shapes `(*, m, k)`, `(*, k, n)` respectively.
- :attr:`mode`\ `= 'complete'`: Returns `(Q, R)` of shapes `(*, m, m)`, `(*, m, n)` respectively.
- :attr:`mode`\ `= 'r'`: Computes only the reduced `R`. Returns `(Q, R)` with `Q` empty and `R` of shape `(*, k, n)`.
Differences with `numpy.linalg.qr`:
- :attr:`mode`\ `= 'raw'` is not implemented.
- Unlike `numpy.linalg.qr`, this function always returns a tuple of two tensors.
When :attr:`mode`\ `= 'r'`, the `Q` tensor is an empty tensor.
This behavior may change in a future PyTorch release.
.. note:: The elements in the diagonal of `R` are not necessarily positive.
.. note:: :attr:`mode`\ `= 'r'` does not support backpropagation. Use :attr:`mode`\ `= 'reduced'` instead.
.. warning:: The QR decomposition is only unique up to the sign of the diagonal of `R` when the
first `k = min(m, n)` columns of :attr:`A` are linearly independent.
If this is not the case, different platforms, like NumPy,
or inputs on different devices, may produce different valid decompositions.
.. warning:: Gradient computations are only supported if the first `k = min(m, n)` columns
of every matrix in :attr:`A` are linearly independent.
If this condition is not met, no error will be thrown, but the gradient produced
will be incorrect.
This is because the QR decomposition is not differentiable at these points.
Args:
A (Tensor): tensor of shape `(*, m, n)` where `*` is zero or more batch dimensions.
mode (str, optional): one of `'reduced'`, `'complete'`, `'r'`.
Controls the shape of the returned tensors. Default: `'reduced'`.
Keyword args:
out (tuple, optional): output tuple of two tensors. Ignored if `None`. Default: `None`.
Returns:
A named tuple `(Q, R)`.
Examples::
>>> a = torch.tensor([[12., -51, 4], [6, 167, -68], [-4, 24, -41]])
>>> q, r = torch.linalg.qr(a)
>>> q
tensor([[-0.8571, 0.3943, 0.3314],
[-0.4286, -0.9029, -0.0343],
[ 0.2857, -0.1714, 0.9429]])
>>> r
tensor([[ -14.0000, -21.0000, 14.0000],
[ 0.0000, -175.0000, 70.0000],
[ 0.0000, 0.0000, -35.0000]])
>>> torch.mm(q, r).round()
tensor([[ 12., -51., 4.],
[ 6., 167., -68.],
[ -4., 24., -41.]])
>>> torch.mm(q.t(), q).round()
tensor([[ 1., 0., 0.],
[ 0., 1., -0.],
[ 0., -0., 1.]])
>>> q2, r2 = torch.linalg.qr(a, mode='r')
>>> q2
tensor([])
>>> torch.equal(r, r2)
True
>>> a = torch.randn(3, 4, 5)
>>> q, r = torch.linalg.qr(a, mode='complete')
>>> torch.allclose(torch.matmul(q, r), a, atol=1e-5)
True
>>> torch.allclose(torch.matmul(q.transpose(-2, -1), q), torch.eye(4), atol=1e-5)
True
""")