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3681 lines (2977 loc) · 113 KB
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"""Lite version of scipy.linalg.
Notes
-----
This module is a lite version of the linalg.py module in SciPy which
contains high-level Python interface to the LAPACK library. The lite
version only accesses the following LAPACK functions: dgesv, zgesv,
dgeev, zgeev, dgesdd, zgesdd, dgelsd, zgelsd, dsyevd, zheevd, dgetrf,
zgetrf, dpotrf, zpotrf, dgeqrf, zgeqrf, zungqr, dorgqr.
"""
__all__ = ['matrix_power', 'solve', 'tensorsolve', 'tensorinv', 'inv',
'cholesky', 'eigvals', 'eigvalsh', 'pinv', 'slogdet', 'det',
'svd', 'svdvals', 'eig', 'eigh', 'lstsq', 'norm', 'qr', 'cond',
'matrix_rank', 'LinAlgError', 'multi_dot', 'trace', 'diagonal',
'cross', 'outer', 'tensordot', 'matmul', 'matrix_transpose',
'matrix_norm', 'vector_norm', 'vecdot']
import functools
import operator
import warnings
from typing import NamedTuple, Any
from numpy._utils import set_module
from numpy._core import (
array, asarray, zeros, empty, empty_like, intc, single, double,
csingle, cdouble, inexact, complexfloating, newaxis, all, inf, dot,
add, multiply, sqrt, sum, isfinite, finfo, errstate, moveaxis, amin,
amax, prod, abs, atleast_2d, intp, asanyarray, object_, matmul,
swapaxes, divide, count_nonzero, isnan, sign, argsort, sort,
reciprocal, overrides, diagonal as _core_diagonal, trace as _core_trace,
cross as _core_cross, outer as _core_outer, tensordot as _core_tensordot,
matmul as _core_matmul, matrix_transpose as _core_matrix_transpose,
transpose as _core_transpose, vecdot as _core_vecdot,
)
from numpy._globals import _NoValue
from numpy.lib._twodim_base_impl import triu, eye
from numpy.lib.array_utils import normalize_axis_index, normalize_axis_tuple
from numpy.linalg import _umath_linalg
from numpy._typing import NDArray
class EigResult(NamedTuple):
eigenvalues: NDArray[Any]
eigenvectors: NDArray[Any]
class EighResult(NamedTuple):
eigenvalues: NDArray[Any]
eigenvectors: NDArray[Any]
class QRResult(NamedTuple):
Q: NDArray[Any]
R: NDArray[Any]
class SlogdetResult(NamedTuple):
sign: NDArray[Any]
logabsdet: NDArray[Any]
class SVDResult(NamedTuple):
U: NDArray[Any]
S: NDArray[Any]
Vh: NDArray[Any]
array_function_dispatch = functools.partial(
overrides.array_function_dispatch, module='numpy.linalg'
)
fortran_int = intc
@set_module('numpy.linalg')
class LinAlgError(ValueError):
"""
Generic Python-exception-derived object raised by linalg functions.
General purpose exception class, derived from Python's ValueError
class, programmatically raised in linalg functions when a Linear
Algebra-related condition would prevent further correct execution of the
function.
Parameters
----------
None
Examples
--------
>>> from numpy import linalg as LA
>>> LA.inv(np.zeros((2,2)))
Traceback (most recent call last):
File "<stdin>", line 1, in <module>
File "...linalg.py", line 350,
in inv return wrap(solve(a, identity(a.shape[0], dtype=a.dtype)))
File "...linalg.py", line 249,
in solve
raise LinAlgError('Singular matrix')
numpy.linalg.LinAlgError: Singular matrix
"""
def _raise_linalgerror_singular(err, flag):
raise LinAlgError("Singular matrix")
def _raise_linalgerror_nonposdef(err, flag):
raise LinAlgError("Matrix is not positive definite")
def _raise_linalgerror_eigenvalues_nonconvergence(err, flag):
raise LinAlgError("Eigenvalues did not converge")
def _raise_linalgerror_svd_nonconvergence(err, flag):
raise LinAlgError("SVD did not converge")
def _raise_linalgerror_lstsq(err, flag):
raise LinAlgError("SVD did not converge in Linear Least Squares")
def _raise_linalgerror_qr(err, flag):
raise LinAlgError("Incorrect argument found while performing "
"QR factorization")
def _makearray(a):
new = asarray(a)
wrap = getattr(a, "__array_wrap__", new.__array_wrap__)
return new, wrap
def isComplexType(t):
return issubclass(t, complexfloating)
_real_types_map = {single: single,
double: double,
csingle: single,
cdouble: double}
_complex_types_map = {single: csingle,
double: cdouble,
csingle: csingle,
cdouble: cdouble}
def _realType(t, default=double):
return _real_types_map.get(t, default)
def _complexType(t, default=cdouble):
return _complex_types_map.get(t, default)
def _commonType(*arrays):
# in lite version, use higher precision (always double or cdouble)
result_type = single
is_complex = False
for a in arrays:
type_ = a.dtype.type
if issubclass(type_, inexact):
if isComplexType(type_):
is_complex = True
rt = _realType(type_, default=None)
if rt is double:
result_type = double
elif rt is None:
# unsupported inexact scalar
raise TypeError("array type %s is unsupported in linalg" %
(a.dtype.name,))
else:
result_type = double
if is_complex:
result_type = _complex_types_map[result_type]
return cdouble, result_type
else:
return double, result_type
def _to_native_byte_order(*arrays):
ret = []
for arr in arrays:
if arr.dtype.byteorder not in ('=', '|'):
ret.append(asarray(arr, dtype=arr.dtype.newbyteorder('=')))
else:
ret.append(arr)
if len(ret) == 1:
return ret[0]
else:
return ret
def _assert_2d(*arrays):
for a in arrays:
if a.ndim != 2:
raise LinAlgError('%d-dimensional array given. Array must be '
'two-dimensional' % a.ndim)
def _assert_stacked_2d(*arrays):
for a in arrays:
if a.ndim < 2:
raise LinAlgError('%d-dimensional array given. Array must be '
'at least two-dimensional' % a.ndim)
def _assert_stacked_square(*arrays):
for a in arrays:
m, n = a.shape[-2:]
if m != n:
raise LinAlgError('Last 2 dimensions of the array must be square')
def _assert_finite(*arrays):
for a in arrays:
if not isfinite(a).all():
raise LinAlgError("Array must not contain infs or NaNs")
def _is_empty_2d(arr):
# check size first for efficiency
return arr.size == 0 and prod(arr.shape[-2:]) == 0
def transpose(a):
"""
Transpose each matrix in a stack of matrices.
Unlike np.transpose, this only swaps the last two axes, rather than all of
them
Parameters
----------
a : (...,M,N) array_like
Returns
-------
aT : (...,N,M) ndarray
"""
return swapaxes(a, -1, -2)
# Linear equations
def _tensorsolve_dispatcher(a, b, axes=None):
return (a, b)
@array_function_dispatch(_tensorsolve_dispatcher)
def tensorsolve(a, b, axes=None):
"""
Solve the tensor equation ``a x = b`` for x.
It is assumed that all indices of `x` are summed over in the product,
together with the rightmost indices of `a`, as is done in, for example,
``tensordot(a, x, axes=x.ndim)``.
Parameters
----------
a : array_like
Coefficient tensor, of shape ``b.shape + Q``. `Q`, a tuple, equals
the shape of that sub-tensor of `a` consisting of the appropriate
number of its rightmost indices, and must be such that
``prod(Q) == prod(b.shape)`` (in which sense `a` is said to be
'square').
b : array_like
Right-hand tensor, which can be of any shape.
axes : tuple of ints, optional
Axes in `a` to reorder to the right, before inversion.
If None (default), no reordering is done.
Returns
-------
x : ndarray, shape Q
Raises
------
LinAlgError
If `a` is singular or not 'square' (in the above sense).
See Also
--------
numpy.tensordot, tensorinv, numpy.einsum
Examples
--------
>>> import numpy as np
>>> a = np.eye(2*3*4)
>>> a.shape = (2*3, 4, 2, 3, 4)
>>> rng = np.random.default_rng()
>>> b = rng.normal(size=(2*3, 4))
>>> x = np.linalg.tensorsolve(a, b)
>>> x.shape
(2, 3, 4)
>>> np.allclose(np.tensordot(a, x, axes=3), b)
True
"""
a, wrap = _makearray(a)
b = asarray(b)
an = a.ndim
if axes is not None:
allaxes = list(range(0, an))
for k in axes:
allaxes.remove(k)
allaxes.insert(an, k)
a = a.transpose(allaxes)
oldshape = a.shape[-(an-b.ndim):]
prod = 1
for k in oldshape:
prod *= k
if a.size != prod ** 2:
raise LinAlgError(
"Input arrays must satisfy the requirement \
prod(a.shape[b.ndim:]) == prod(a.shape[:b.ndim])"
)
a = a.reshape(prod, prod)
b = b.ravel()
res = wrap(solve(a, b))
res.shape = oldshape
return res
def _solve_dispatcher(a, b):
return (a, b)
@array_function_dispatch(_solve_dispatcher)
def solve(a, b):
"""
Solve a linear matrix equation, or system of linear scalar equations.
Computes the "exact" solution, `x`, of the well-determined, i.e., full
rank, linear matrix equation `ax = b`.
Parameters
----------
a : (..., M, M) array_like
Coefficient matrix.
b : {(M,), (..., M, K)}, array_like
Ordinate or "dependent variable" values.
Returns
-------
x : {(..., M,), (..., M, K)} ndarray
Solution to the system a x = b. Returned shape is (..., M) if b is
shape (M,) and (..., M, K) if b is (..., M, K), where the "..." part is
broadcasted between a and b.
Raises
------
LinAlgError
If `a` is singular or not square.
See Also
--------
scipy.linalg.solve : Similar function in SciPy.
Notes
-----
.. versionadded:: 1.8.0
Broadcasting rules apply, see the `numpy.linalg` documentation for
details.
The solutions are computed using LAPACK routine ``_gesv``.
`a` must be square and of full-rank, i.e., all rows (or, equivalently,
columns) must be linearly independent; if either is not true, use
`lstsq` for the least-squares best "solution" of the
system/equation.
.. versionchanged:: 2.0
The b array is only treated as a shape (M,) column vector if it is
exactly 1-dimensional. In all other instances it is treated as a stack
of (M, K) matrices. Previously b would be treated as a stack of (M,)
vectors if b.ndim was equal to a.ndim - 1.
References
----------
.. [1] G. Strang, *Linear Algebra and Its Applications*, 2nd Ed., Orlando,
FL, Academic Press, Inc., 1980, pg. 22.
Examples
--------
Solve the system of equations:
``x0 + 2 * x1 = 1`` and
``3 * x0 + 5 * x1 = 2``:
>>> import numpy as np
>>> a = np.array([[1, 2], [3, 5]])
>>> b = np.array([1, 2])
>>> x = np.linalg.solve(a, b)
>>> x
array([-1., 1.])
Check that the solution is correct:
>>> np.allclose(np.dot(a, x), b)
True
"""
a, _ = _makearray(a)
_assert_stacked_2d(a)
_assert_stacked_square(a)
b, wrap = _makearray(b)
t, result_t = _commonType(a, b)
# We use the b = (..., M,) logic, only if the number of extra dimensions
# match exactly
if b.ndim == 1:
gufunc = _umath_linalg.solve1
else:
gufunc = _umath_linalg.solve
signature = 'DD->D' if isComplexType(t) else 'dd->d'
with errstate(call=_raise_linalgerror_singular, invalid='call',
over='ignore', divide='ignore', under='ignore'):
r = gufunc(a, b, signature=signature)
return wrap(r.astype(result_t, copy=False))
def _tensorinv_dispatcher(a, ind=None):
return (a,)
@array_function_dispatch(_tensorinv_dispatcher)
def tensorinv(a, ind=2):
"""
Compute the 'inverse' of an N-dimensional array.
The result is an inverse for `a` relative to the tensordot operation
``tensordot(a, b, ind)``, i. e., up to floating-point accuracy,
``tensordot(tensorinv(a), a, ind)`` is the "identity" tensor for the
tensordot operation.
Parameters
----------
a : array_like
Tensor to 'invert'. Its shape must be 'square', i. e.,
``prod(a.shape[:ind]) == prod(a.shape[ind:])``.
ind : int, optional
Number of first indices that are involved in the inverse sum.
Must be a positive integer, default is 2.
Returns
-------
b : ndarray
`a`'s tensordot inverse, shape ``a.shape[ind:] + a.shape[:ind]``.
Raises
------
LinAlgError
If `a` is singular or not 'square' (in the above sense).
See Also
--------
numpy.tensordot, tensorsolve
Examples
--------
>>> import numpy as np
>>> a = np.eye(4*6)
>>> a.shape = (4, 6, 8, 3)
>>> ainv = np.linalg.tensorinv(a, ind=2)
>>> ainv.shape
(8, 3, 4, 6)
>>> rng = np.random.default_rng()
>>> b = rng.normal(size=(4, 6))
>>> np.allclose(np.tensordot(ainv, b), np.linalg.tensorsolve(a, b))
True
>>> a = np.eye(4*6)
>>> a.shape = (24, 8, 3)
>>> ainv = np.linalg.tensorinv(a, ind=1)
>>> ainv.shape
(8, 3, 24)
>>> rng = np.random.default_rng()
>>> b = rng.normal(size=24)
>>> np.allclose(np.tensordot(ainv, b, 1), np.linalg.tensorsolve(a, b))
True
"""
a = asarray(a)
oldshape = a.shape
prod = 1
if ind > 0:
invshape = oldshape[ind:] + oldshape[:ind]
for k in oldshape[ind:]:
prod *= k
else:
raise ValueError("Invalid ind argument.")
a = a.reshape(prod, -1)
ia = inv(a)
return ia.reshape(*invshape)
# Matrix inversion
def _unary_dispatcher(a):
return (a,)
@array_function_dispatch(_unary_dispatcher)
def inv(a):
"""
Compute the inverse of a matrix.
Given a square matrix `a`, return the matrix `ainv` satisfying
``a @ ainv = ainv @ a = eye(a.shape[0])``.
Parameters
----------
a : (..., M, M) array_like
Matrix to be inverted.
Returns
-------
ainv : (..., M, M) ndarray or matrix
Inverse of the matrix `a`.
Raises
------
LinAlgError
If `a` is not square or inversion fails.
See Also
--------
scipy.linalg.inv : Similar function in SciPy.
numpy.linalg.cond : Compute the condition number of a matrix.
numpy.linalg.svd : Compute the singular value decomposition of a matrix.
Notes
-----
.. versionadded:: 1.8.0
Broadcasting rules apply, see the `numpy.linalg` documentation for
details.
If `a` is detected to be singular, a `LinAlgError` is raised. If `a` is
ill-conditioned, a `LinAlgError` may or may not be raised, and results may
be inaccurate due to floating-point errors.
References
----------
.. [1] Wikipedia, "Condition number",
https://en.wikipedia.org/wiki/Condition_number
Examples
--------
>>> import numpy as np
>>> from numpy.linalg import inv
>>> a = np.array([[1., 2.], [3., 4.]])
>>> ainv = inv(a)
>>> np.allclose(a @ ainv, np.eye(2))
True
>>> np.allclose(ainv @ a, np.eye(2))
True
If a is a matrix object, then the return value is a matrix as well:
>>> ainv = inv(np.matrix(a))
>>> ainv
matrix([[-2. , 1. ],
[ 1.5, -0.5]])
Inverses of several matrices can be computed at once:
>>> a = np.array([[[1., 2.], [3., 4.]], [[1, 3], [3, 5]]])
>>> inv(a)
array([[[-2. , 1. ],
[ 1.5 , -0.5 ]],
[[-1.25, 0.75],
[ 0.75, -0.25]]])
If a matrix is close to singular, the computed inverse may not satisfy
``a @ ainv = ainv @ a = eye(a.shape[0])`` even if a `LinAlgError`
is not raised:
>>> a = np.array([[2,4,6],[2,0,2],[6,8,14]])
>>> inv(a) # No errors raised
array([[-1.12589991e+15, -5.62949953e+14, 5.62949953e+14],
[-1.12589991e+15, -5.62949953e+14, 5.62949953e+14],
[ 1.12589991e+15, 5.62949953e+14, -5.62949953e+14]])
>>> a @ inv(a)
array([[ 0. , -0.5 , 0. ], # may vary
[-0.5 , 0.625, 0.25 ],
[ 0. , 0. , 1. ]])
To detect ill-conditioned matrices, you can use `numpy.linalg.cond` to
compute its *condition number* [1]_. The larger the condition number, the
more ill-conditioned the matrix is. As a rule of thumb, if the condition
number ``cond(a) = 10**k``, then you may lose up to ``k`` digits of
accuracy on top of what would be lost to the numerical method due to loss
of precision from arithmetic methods.
>>> from numpy.linalg import cond
>>> cond(a)
np.float64(8.659885634118668e+17) # may vary
It is also possible to detect ill-conditioning by inspecting the matrix's
singular values directly. The ratio between the largest and the smallest
singular value is the condition number:
>>> from numpy.linalg import svd
>>> sigma = svd(a, compute_uv=False) # Do not compute singular vectors
>>> sigma.max()/sigma.min()
8.659885634118668e+17 # may vary
"""
a, wrap = _makearray(a)
_assert_stacked_2d(a)
_assert_stacked_square(a)
t, result_t = _commonType(a)
signature = 'D->D' if isComplexType(t) else 'd->d'
with errstate(call=_raise_linalgerror_singular, invalid='call',
over='ignore', divide='ignore', under='ignore'):
ainv = _umath_linalg.inv(a, signature=signature)
return wrap(ainv.astype(result_t, copy=False))
def _matrix_power_dispatcher(a, n):
return (a,)
@array_function_dispatch(_matrix_power_dispatcher)
def matrix_power(a, n):
"""
Raise a square matrix to the (integer) power `n`.
For positive integers `n`, the power is computed by repeated matrix
squarings and matrix multiplications. If ``n == 0``, the identity matrix
of the same shape as M is returned. If ``n < 0``, the inverse
is computed and then raised to the ``abs(n)``.
.. note:: Stacks of object matrices are not currently supported.
Parameters
----------
a : (..., M, M) array_like
Matrix to be "powered".
n : int
The exponent can be any integer or long integer, positive,
negative, or zero.
Returns
-------
a**n : (..., M, M) ndarray or matrix object
The return value is the same shape and type as `M`;
if the exponent is positive or zero then the type of the
elements is the same as those of `M`. If the exponent is
negative the elements are floating-point.
Raises
------
LinAlgError
For matrices that are not square or that (for negative powers) cannot
be inverted numerically.
Examples
--------
>>> import numpy as np
>>> from numpy.linalg import matrix_power
>>> i = np.array([[0, 1], [-1, 0]]) # matrix equiv. of the imaginary unit
>>> matrix_power(i, 3) # should = -i
array([[ 0, -1],
[ 1, 0]])
>>> matrix_power(i, 0)
array([[1, 0],
[0, 1]])
>>> matrix_power(i, -3) # should = 1/(-i) = i, but w/ f.p. elements
array([[ 0., 1.],
[-1., 0.]])
Somewhat more sophisticated example
>>> q = np.zeros((4, 4))
>>> q[0:2, 0:2] = -i
>>> q[2:4, 2:4] = i
>>> q # one of the three quaternion units not equal to 1
array([[ 0., -1., 0., 0.],
[ 1., 0., 0., 0.],
[ 0., 0., 0., 1.],
[ 0., 0., -1., 0.]])
>>> matrix_power(q, 2) # = -np.eye(4)
array([[-1., 0., 0., 0.],
[ 0., -1., 0., 0.],
[ 0., 0., -1., 0.],
[ 0., 0., 0., -1.]])
"""
a = asanyarray(a)
_assert_stacked_2d(a)
_assert_stacked_square(a)
try:
n = operator.index(n)
except TypeError as e:
raise TypeError("exponent must be an integer") from e
# Fall back on dot for object arrays. Object arrays are not supported by
# the current implementation of matmul using einsum
if a.dtype != object:
fmatmul = matmul
elif a.ndim == 2:
fmatmul = dot
else:
raise NotImplementedError(
"matrix_power not supported for stacks of object arrays")
if n == 0:
a = empty_like(a)
a[...] = eye(a.shape[-2], dtype=a.dtype)
return a
elif n < 0:
a = inv(a)
n = abs(n)
# short-cuts.
if n == 1:
return a
elif n == 2:
return fmatmul(a, a)
elif n == 3:
return fmatmul(fmatmul(a, a), a)
# Use binary decomposition to reduce the number of matrix multiplications.
# Here, we iterate over the bits of n, from LSB to MSB, raise `a` to
# increasing powers of 2, and multiply into the result as needed.
z = result = None
while n > 0:
z = a if z is None else fmatmul(z, z)
n, bit = divmod(n, 2)
if bit:
result = z if result is None else fmatmul(result, z)
return result
# Cholesky decomposition
def _cholesky_dispatcher(a, /, *, upper=None):
return (a,)
@array_function_dispatch(_cholesky_dispatcher)
def cholesky(a, /, *, upper=False):
"""
Cholesky decomposition.
Return the lower or upper Cholesky decomposition, ``L * L.H`` or
``U.H * U``, of the square matrix ``a``, where ``L`` is lower-triangular,
``U`` is upper-triangular, and ``.H`` is the conjugate transpose operator
(which is the ordinary transpose if ``a`` is real-valued). ``a`` must be
Hermitian (symmetric if real-valued) and positive-definite. No checking is
performed to verify whether ``a`` is Hermitian or not. In addition, only
the lower or upper-triangular and diagonal elements of ``a`` are used.
Only ``L`` or ``U`` is actually returned.
Parameters
----------
a : (..., M, M) array_like
Hermitian (symmetric if all elements are real), positive-definite
input matrix.
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
-------
L : (..., M, M) array_like
Lower or upper-triangular Cholesky factor of `a`. Returns a matrix
object if `a` is a matrix object.
Raises
------
LinAlgError
If the decomposition fails, for example, if `a` is not
positive-definite.
See Also
--------
scipy.linalg.cholesky : Similar function in SciPy.
scipy.linalg.cholesky_banded : Cholesky decompose a banded Hermitian
positive-definite matrix.
scipy.linalg.cho_factor : Cholesky decomposition of a matrix, to use in
`scipy.linalg.cho_solve`.
Notes
-----
.. versionadded:: 1.8.0
Broadcasting rules apply, see the `numpy.linalg` documentation for
details.
The Cholesky decomposition is often used as a fast way of solving
.. math:: A \\mathbf{x} = \\mathbf{b}
(when `A` is both Hermitian/symmetric and positive-definite).
First, we solve for :math:`\\mathbf{y}` in
.. math:: L \\mathbf{y} = \\mathbf{b},
and then for :math:`\\mathbf{x}` in
.. math:: L^{H} \\mathbf{x} = \\mathbf{y}.
Examples
--------
>>> import numpy as np
>>> A = np.array([[1,-2j],[2j,5]])
>>> A
array([[ 1.+0.j, -0.-2.j],
[ 0.+2.j, 5.+0.j]])
>>> L = np.linalg.cholesky(A)
>>> L
array([[1.+0.j, 0.+0.j],
[0.+2.j, 1.+0.j]])
>>> np.dot(L, L.T.conj()) # verify that L * L.H = A
array([[1.+0.j, 0.-2.j],
[0.+2.j, 5.+0.j]])
>>> A = [[1,-2j],[2j,5]] # what happens if A is only array_like?
>>> np.linalg.cholesky(A) # an ndarray object is returned
array([[1.+0.j, 0.+0.j],
[0.+2.j, 1.+0.j]])
>>> # But a matrix object is returned if A is a matrix object
>>> np.linalg.cholesky(np.matrix(A))
matrix([[ 1.+0.j, 0.+0.j],
[ 0.+2.j, 1.+0.j]])
>>> # The upper-triangular Cholesky factor can also be obtained.
>>> np.linalg.cholesky(A, upper=True)
array([[1.-0.j, 0.-2.j],
[0.-0.j, 1.-0.j]])
"""
gufunc = _umath_linalg.cholesky_up if upper else _umath_linalg.cholesky_lo
a, wrap = _makearray(a)
_assert_stacked_2d(a)
_assert_stacked_square(a)
t, result_t = _commonType(a)
signature = 'D->D' if isComplexType(t) else 'd->d'
with errstate(call=_raise_linalgerror_nonposdef, invalid='call',
over='ignore', divide='ignore', under='ignore'):
r = gufunc(a, signature=signature)
return wrap(r.astype(result_t, copy=False))
# outer product
def _outer_dispatcher(x1, x2):
return (x1, x2)
@array_function_dispatch(_outer_dispatcher)
def outer(x1, x2, /):
"""
Compute the outer product of two vectors.
This function is Array API compatible. Compared to ``np.outer``
it accepts 1-dimensional inputs only.
Parameters
----------
x1 : (M,) array_like
One-dimensional input array of size ``N``.
Must have a numeric data type.
x2 : (N,) array_like
One-dimensional input array of size ``M``.
Must have a numeric data type.
Returns
-------
out : (M, N) ndarray
``out[i, j] = a[i] * b[j]``
See also
--------
outer
Examples
--------
Make a (*very* coarse) grid for computing a Mandelbrot set:
>>> rl = np.linalg.outer(np.ones((5,)), np.linspace(-2, 2, 5))
>>> rl
array([[-2., -1., 0., 1., 2.],
[-2., -1., 0., 1., 2.],
[-2., -1., 0., 1., 2.],
[-2., -1., 0., 1., 2.],
[-2., -1., 0., 1., 2.]])
>>> im = np.linalg.outer(1j*np.linspace(2, -2, 5), np.ones((5,)))
>>> im
array([[0.+2.j, 0.+2.j, 0.+2.j, 0.+2.j, 0.+2.j],
[0.+1.j, 0.+1.j, 0.+1.j, 0.+1.j, 0.+1.j],
[0.+0.j, 0.+0.j, 0.+0.j, 0.+0.j, 0.+0.j],
[0.-1.j, 0.-1.j, 0.-1.j, 0.-1.j, 0.-1.j],
[0.-2.j, 0.-2.j, 0.-2.j, 0.-2.j, 0.-2.j]])
>>> grid = rl + im
>>> grid
array([[-2.+2.j, -1.+2.j, 0.+2.j, 1.+2.j, 2.+2.j],
[-2.+1.j, -1.+1.j, 0.+1.j, 1.+1.j, 2.+1.j],
[-2.+0.j, -1.+0.j, 0.+0.j, 1.+0.j, 2.+0.j],
[-2.-1.j, -1.-1.j, 0.-1.j, 1.-1.j, 2.-1.j],
[-2.-2.j, -1.-2.j, 0.-2.j, 1.-2.j, 2.-2.j]])
An example using a "vector" of letters:
>>> x = np.array(['a', 'b', 'c'], dtype=object)
>>> np.linalg.outer(x, [1, 2, 3])
array([['a', 'aa', 'aaa'],
['b', 'bb', 'bbb'],
['c', 'cc', 'ccc']], dtype=object)
"""
x1 = asanyarray(x1)
x2 = asanyarray(x2)
if x1.ndim != 1 or x2.ndim != 1:
raise ValueError(
"Input arrays must be one-dimensional, but they are "
f"{x1.ndim=} and {x2.ndim=}."
)
return _core_outer(x1, x2, out=None)
# QR decomposition
def _qr_dispatcher(a, mode=None):
return (a,)
@array_function_dispatch(_qr_dispatcher)
def qr(a, mode='reduced'):
"""
Compute the qr factorization of a matrix.
Factor the matrix `a` as *qr*, where `q` is orthonormal and `r` is
upper-triangular.
Parameters
----------
a : array_like, shape (..., M, N)
An array-like object with the dimensionality of at least 2.
mode : {'reduced', 'complete', 'r', 'raw'}, optional, default: 'reduced'
If K = min(M, N), then
* 'reduced' : returns Q, R with dimensions (..., M, K), (..., K, N)
* 'complete' : returns Q, R with dimensions (..., M, M), (..., M, N)
* 'r' : returns R only with dimensions (..., K, N)
* 'raw' : returns h, tau with dimensions (..., N, M), (..., K,)
The options 'reduced', 'complete, and 'raw' are new in numpy 1.8,
see the notes for more information. The default is 'reduced', and to
maintain backward compatibility with earlier versions of numpy both
it and the old default 'full' can be omitted. Note that array h
returned in 'raw' mode is transposed for calling Fortran. The
'economic' mode is deprecated. The modes 'full' and 'economic' may
be passed using only the first letter for backwards compatibility,
but all others must be spelled out. See the Notes for more
explanation.
Returns
-------
When mode is 'reduced' or 'complete', the result will be a namedtuple with
the attributes `Q` and `R`.
Q : ndarray of float or complex, optional
A matrix with orthonormal columns. When mode = 'complete' the
result is an orthogonal/unitary matrix depending on whether or not
a is real/complex. The determinant may be either +/- 1 in that
case. In case the number of dimensions in the input array is
greater than 2 then a stack of the matrices with above properties
is returned.
R : ndarray of float or complex, optional
The upper-triangular matrix or a stack of upper-triangular
matrices if the number of dimensions in the input array is greater
than 2.
(h, tau) : ndarrays of np.double or np.cdouble, optional
The array h contains the Householder reflectors that generate q
along with r. The tau array contains scaling factors for the
reflectors. In the deprecated 'economic' mode only h is returned.
Raises
------
LinAlgError
If factoring fails.
See Also
--------
scipy.linalg.qr : Similar function in SciPy.
scipy.linalg.rq : Compute RQ decomposition of a matrix.
Notes
-----