357 lines
12 KiB
Python
357 lines
12 KiB
Python
"""Cholesky decomposition functions."""
|
|
|
|
from numpy import asarray_chkfinite, asarray, atleast_2d
|
|
|
|
# Local imports
|
|
from ._misc import LinAlgError, _datacopied
|
|
from .lapack import get_lapack_funcs
|
|
|
|
__all__ = ['cholesky', 'cho_factor', 'cho_solve', 'cholesky_banded',
|
|
'cho_solve_banded']
|
|
|
|
|
|
def _cholesky(a, lower=False, overwrite_a=False, clean=True,
|
|
check_finite=True):
|
|
"""Common code for cholesky() and cho_factor()."""
|
|
|
|
a1 = asarray_chkfinite(a) if check_finite else asarray(a)
|
|
a1 = atleast_2d(a1)
|
|
|
|
# Dimension check
|
|
if a1.ndim != 2:
|
|
raise ValueError(f'Input array needs to be 2D but received a {a1.ndim}d-array.')
|
|
# Squareness check
|
|
if a1.shape[0] != a1.shape[1]:
|
|
raise ValueError('Input array is expected to be square but has '
|
|
f'the shape: {a1.shape}.')
|
|
|
|
# Quick return for square empty array
|
|
if a1.size == 0:
|
|
return a1.copy(), lower
|
|
|
|
overwrite_a = overwrite_a or _datacopied(a1, a)
|
|
potrf, = get_lapack_funcs(('potrf',), (a1,))
|
|
c, info = potrf(a1, lower=lower, overwrite_a=overwrite_a, clean=clean)
|
|
if info > 0:
|
|
raise LinAlgError("%d-th leading minor of the array is not positive "
|
|
"definite" % info)
|
|
if info < 0:
|
|
raise ValueError(f'LAPACK reported an illegal value in {-info}-th argument'
|
|
'on entry to "POTRF".')
|
|
return c, lower
|
|
|
|
|
|
def cholesky(a, lower=False, overwrite_a=False, check_finite=True):
|
|
"""
|
|
Compute the Cholesky decomposition of a matrix.
|
|
|
|
Returns the Cholesky decomposition, :math:`A = L L^*` or
|
|
:math:`A = U^* U` of a Hermitian positive-definite matrix A.
|
|
|
|
Parameters
|
|
----------
|
|
a : (M, M) array_like
|
|
Matrix to be decomposed
|
|
lower : bool, optional
|
|
Whether to compute the upper- or lower-triangular Cholesky
|
|
factorization. Default is upper-triangular.
|
|
overwrite_a : bool, optional
|
|
Whether to overwrite data in `a` (may improve performance).
|
|
check_finite : bool, optional
|
|
Whether to check that the input matrix contains only finite numbers.
|
|
Disabling may give a performance gain, but may result in problems
|
|
(crashes, non-termination) if the inputs do contain infinities or NaNs.
|
|
|
|
Returns
|
|
-------
|
|
c : (M, M) ndarray
|
|
Upper- or lower-triangular Cholesky factor of `a`.
|
|
|
|
Raises
|
|
------
|
|
LinAlgError : if decomposition fails.
|
|
|
|
Examples
|
|
--------
|
|
>>> import numpy as np
|
|
>>> from scipy.linalg import cholesky
|
|
>>> a = np.array([[1,-2j],[2j,5]])
|
|
>>> L = cholesky(a, lower=True)
|
|
>>> L
|
|
array([[ 1.+0.j, 0.+0.j],
|
|
[ 0.+2.j, 1.+0.j]])
|
|
>>> L @ L.T.conj()
|
|
array([[ 1.+0.j, 0.-2.j],
|
|
[ 0.+2.j, 5.+0.j]])
|
|
|
|
"""
|
|
c, lower = _cholesky(a, lower=lower, overwrite_a=overwrite_a, clean=True,
|
|
check_finite=check_finite)
|
|
return c
|
|
|
|
|
|
def cho_factor(a, lower=False, overwrite_a=False, check_finite=True):
|
|
"""
|
|
Compute the Cholesky decomposition of a matrix, to use in cho_solve
|
|
|
|
Returns a matrix containing the Cholesky decomposition,
|
|
``A = L L*`` or ``A = U* U`` of a Hermitian positive-definite matrix `a`.
|
|
The return value can be directly used as the first parameter to cho_solve.
|
|
|
|
.. warning::
|
|
The returned matrix also contains random data in the entries not
|
|
used by the Cholesky decomposition. If you need to zero these
|
|
entries, use the function `cholesky` instead.
|
|
|
|
Parameters
|
|
----------
|
|
a : (M, M) array_like
|
|
Matrix to be decomposed
|
|
lower : bool, optional
|
|
Whether to compute the upper or lower triangular Cholesky factorization
|
|
(Default: upper-triangular)
|
|
overwrite_a : bool, optional
|
|
Whether to overwrite data in a (may improve performance)
|
|
check_finite : bool, optional
|
|
Whether to check that the input matrix contains only finite numbers.
|
|
Disabling may give a performance gain, but may result in problems
|
|
(crashes, non-termination) if the inputs do contain infinities or NaNs.
|
|
|
|
Returns
|
|
-------
|
|
c : (M, M) ndarray
|
|
Matrix whose upper or lower triangle contains the Cholesky factor
|
|
of `a`. Other parts of the matrix contain random data.
|
|
lower : bool
|
|
Flag indicating whether the factor is in the lower or upper triangle
|
|
|
|
Raises
|
|
------
|
|
LinAlgError
|
|
Raised if decomposition fails.
|
|
|
|
See Also
|
|
--------
|
|
cho_solve : Solve a linear set equations using the Cholesky factorization
|
|
of a matrix.
|
|
|
|
Examples
|
|
--------
|
|
>>> import numpy as np
|
|
>>> from scipy.linalg import cho_factor
|
|
>>> A = np.array([[9, 3, 1, 5], [3, 7, 5, 1], [1, 5, 9, 2], [5, 1, 2, 6]])
|
|
>>> c, low = cho_factor(A)
|
|
>>> c
|
|
array([[3. , 1. , 0.33333333, 1.66666667],
|
|
[3. , 2.44948974, 1.90515869, -0.27216553],
|
|
[1. , 5. , 2.29330749, 0.8559528 ],
|
|
[5. , 1. , 2. , 1.55418563]])
|
|
>>> np.allclose(np.triu(c).T @ np. triu(c) - A, np.zeros((4, 4)))
|
|
True
|
|
|
|
"""
|
|
c, lower = _cholesky(a, lower=lower, overwrite_a=overwrite_a, clean=False,
|
|
check_finite=check_finite)
|
|
return c, lower
|
|
|
|
|
|
def cho_solve(c_and_lower, b, overwrite_b=False, check_finite=True):
|
|
"""Solve the linear equations A x = b, given the Cholesky factorization of A.
|
|
|
|
Parameters
|
|
----------
|
|
(c, lower) : tuple, (array, bool)
|
|
Cholesky factorization of a, as given by cho_factor
|
|
b : array
|
|
Right-hand side
|
|
overwrite_b : bool, optional
|
|
Whether to overwrite data in b (may improve performance)
|
|
check_finite : bool, optional
|
|
Whether to check that the input matrices contain only finite numbers.
|
|
Disabling may give a performance gain, but may result in problems
|
|
(crashes, non-termination) if the inputs do contain infinities or NaNs.
|
|
|
|
Returns
|
|
-------
|
|
x : array
|
|
The solution to the system A x = b
|
|
|
|
See Also
|
|
--------
|
|
cho_factor : Cholesky factorization of a matrix
|
|
|
|
Examples
|
|
--------
|
|
>>> import numpy as np
|
|
>>> from scipy.linalg import cho_factor, cho_solve
|
|
>>> A = np.array([[9, 3, 1, 5], [3, 7, 5, 1], [1, 5, 9, 2], [5, 1, 2, 6]])
|
|
>>> c, low = cho_factor(A)
|
|
>>> x = cho_solve((c, low), [1, 1, 1, 1])
|
|
>>> np.allclose(A @ x - [1, 1, 1, 1], np.zeros(4))
|
|
True
|
|
|
|
"""
|
|
(c, lower) = c_and_lower
|
|
if check_finite:
|
|
b1 = asarray_chkfinite(b)
|
|
c = asarray_chkfinite(c)
|
|
else:
|
|
b1 = asarray(b)
|
|
c = asarray(c)
|
|
if c.ndim != 2 or c.shape[0] != c.shape[1]:
|
|
raise ValueError("The factored matrix c is not square.")
|
|
if c.shape[1] != b1.shape[0]:
|
|
raise ValueError(f"incompatible dimensions ({c.shape} and {b1.shape})")
|
|
|
|
overwrite_b = overwrite_b or _datacopied(b1, b)
|
|
|
|
potrs, = get_lapack_funcs(('potrs',), (c, b1))
|
|
x, info = potrs(c, b1, lower=lower, overwrite_b=overwrite_b)
|
|
if info != 0:
|
|
raise ValueError('illegal value in %dth argument of internal potrs'
|
|
% -info)
|
|
return x
|
|
|
|
|
|
def cholesky_banded(ab, overwrite_ab=False, lower=False, check_finite=True):
|
|
"""
|
|
Cholesky decompose a banded Hermitian positive-definite matrix
|
|
|
|
The matrix a is stored in ab either in lower-diagonal or upper-
|
|
diagonal ordered form::
|
|
|
|
ab[u + i - j, j] == a[i,j] (if upper form; i <= j)
|
|
ab[ i - j, j] == a[i,j] (if lower form; i >= j)
|
|
|
|
Example of ab (shape of a is (6,6), u=2)::
|
|
|
|
upper form:
|
|
* * a02 a13 a24 a35
|
|
* a01 a12 a23 a34 a45
|
|
a00 a11 a22 a33 a44 a55
|
|
|
|
lower form:
|
|
a00 a11 a22 a33 a44 a55
|
|
a10 a21 a32 a43 a54 *
|
|
a20 a31 a42 a53 * *
|
|
|
|
Parameters
|
|
----------
|
|
ab : (u + 1, M) array_like
|
|
Banded matrix
|
|
overwrite_ab : bool, optional
|
|
Discard data in ab (may enhance performance)
|
|
lower : bool, optional
|
|
Is the matrix in the lower form. (Default is upper form)
|
|
check_finite : bool, optional
|
|
Whether to check that the input matrix contains only finite numbers.
|
|
Disabling may give a performance gain, but may result in problems
|
|
(crashes, non-termination) if the inputs do contain infinities or NaNs.
|
|
|
|
Returns
|
|
-------
|
|
c : (u + 1, M) ndarray
|
|
Cholesky factorization of a, in the same banded format as ab
|
|
|
|
See Also
|
|
--------
|
|
cho_solve_banded :
|
|
Solve a linear set equations, given the Cholesky factorization
|
|
of a banded Hermitian.
|
|
|
|
Examples
|
|
--------
|
|
>>> import numpy as np
|
|
>>> from scipy.linalg import cholesky_banded
|
|
>>> from numpy import allclose, zeros, diag
|
|
>>> Ab = np.array([[0, 0, 1j, 2, 3j], [0, -1, -2, 3, 4], [9, 8, 7, 6, 9]])
|
|
>>> A = np.diag(Ab[0,2:], k=2) + np.diag(Ab[1,1:], k=1)
|
|
>>> A = A + A.conj().T + np.diag(Ab[2, :])
|
|
>>> c = cholesky_banded(Ab)
|
|
>>> C = np.diag(c[0, 2:], k=2) + np.diag(c[1, 1:], k=1) + np.diag(c[2, :])
|
|
>>> np.allclose(C.conj().T @ C - A, np.zeros((5, 5)))
|
|
True
|
|
|
|
"""
|
|
if check_finite:
|
|
ab = asarray_chkfinite(ab)
|
|
else:
|
|
ab = asarray(ab)
|
|
|
|
pbtrf, = get_lapack_funcs(('pbtrf',), (ab,))
|
|
c, info = pbtrf(ab, lower=lower, overwrite_ab=overwrite_ab)
|
|
if info > 0:
|
|
raise LinAlgError("%d-th leading minor not positive definite" % info)
|
|
if info < 0:
|
|
raise ValueError('illegal value in %d-th argument of internal pbtrf'
|
|
% -info)
|
|
return c
|
|
|
|
|
|
def cho_solve_banded(cb_and_lower, b, overwrite_b=False, check_finite=True):
|
|
"""
|
|
Solve the linear equations ``A x = b``, given the Cholesky factorization of
|
|
the banded Hermitian ``A``.
|
|
|
|
Parameters
|
|
----------
|
|
(cb, lower) : tuple, (ndarray, bool)
|
|
`cb` is the Cholesky factorization of A, as given by cholesky_banded.
|
|
`lower` must be the same value that was given to cholesky_banded.
|
|
b : array_like
|
|
Right-hand side
|
|
overwrite_b : bool, optional
|
|
If True, the function will overwrite the values in `b`.
|
|
check_finite : bool, optional
|
|
Whether to check that the input matrices contain only finite numbers.
|
|
Disabling may give a performance gain, but may result in problems
|
|
(crashes, non-termination) if the inputs do contain infinities or NaNs.
|
|
|
|
Returns
|
|
-------
|
|
x : array
|
|
The solution to the system A x = b
|
|
|
|
See Also
|
|
--------
|
|
cholesky_banded : Cholesky factorization of a banded matrix
|
|
|
|
Notes
|
|
-----
|
|
|
|
.. versionadded:: 0.8.0
|
|
|
|
Examples
|
|
--------
|
|
>>> import numpy as np
|
|
>>> from scipy.linalg import cholesky_banded, cho_solve_banded
|
|
>>> Ab = np.array([[0, 0, 1j, 2, 3j], [0, -1, -2, 3, 4], [9, 8, 7, 6, 9]])
|
|
>>> A = np.diag(Ab[0,2:], k=2) + np.diag(Ab[1,1:], k=1)
|
|
>>> A = A + A.conj().T + np.diag(Ab[2, :])
|
|
>>> c = cholesky_banded(Ab)
|
|
>>> x = cho_solve_banded((c, False), np.ones(5))
|
|
>>> np.allclose(A @ x - np.ones(5), np.zeros(5))
|
|
True
|
|
|
|
"""
|
|
(cb, lower) = cb_and_lower
|
|
if check_finite:
|
|
cb = asarray_chkfinite(cb)
|
|
b = asarray_chkfinite(b)
|
|
else:
|
|
cb = asarray(cb)
|
|
b = asarray(b)
|
|
|
|
# Validate shapes.
|
|
if cb.shape[-1] != b.shape[0]:
|
|
raise ValueError("shapes of cb and b are not compatible.")
|
|
|
|
pbtrs, = get_lapack_funcs(('pbtrs',), (cb, b))
|
|
x, info = pbtrs(cb, b, lower=lower, overwrite_b=overwrite_b)
|
|
if info > 0:
|
|
raise LinAlgError("%dth leading minor not positive definite" % info)
|
|
if info < 0:
|
|
raise ValueError('illegal value in %dth argument of internal pbtrs'
|
|
% -info)
|
|
return x
|