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#
# Author: Pearu Peterson, March 2002
#
# w/ additions by Travis Oliphant, March 2002
# and Jake Vanderplas, August 2012
import warnings
from warnings import warn
from itertools import product
import numpy as np
from numpy import atleast_1d, atleast_2d
from .lapack import get_lapack_funcs, _compute_lwork
from ._misc import LinAlgError, _datacopied, LinAlgWarning
from ._decomp import _asarray_validated
from . import _decomp, _decomp_svd
from ._solve_toeplitz import levinson
from ._cythonized_array_utils import (find_det_from_lu, bandwidth, issymmetric,
ishermitian)
__all__ = ['solve', 'solve_triangular', 'solveh_banded', 'solve_banded',
'solve_toeplitz', 'solve_circulant', 'inv', 'det', 'lstsq',
'pinv', 'pinvh', 'matrix_balance', 'matmul_toeplitz']
# The numpy facilities for type-casting checks are too slow for small sized
# arrays and eat away the time budget for the checkups. Here we set a
# precomputed dict container of the numpy.can_cast() table.
# It can be used to determine quickly what a dtype can be cast to LAPACK
# compatible types, i.e., 'float32, float64, complex64, complex128'.
# Then it can be checked via "casting_dict[arr.dtype.char]"
lapack_cast_dict = {x: ''.join([y for y in 'fdFD' if np.can_cast(x, y)])
for x in np.typecodes['All']}
# Linear equations
def _solve_check(n, info, lamch=None, rcond=None):
""" Check arguments during the different steps of the solution phase """
if info < 0:
raise ValueError(f'LAPACK reported an illegal value in {-info}-th argument.')
elif 0 < info:
raise LinAlgError('Matrix is singular.')
if lamch is None:
return
E = lamch('E')
if rcond < E:
warn(f'Ill-conditioned matrix (rcond={rcond:.6g}): '
'result may not be accurate.',
LinAlgWarning, stacklevel=3)
def _find_matrix_structure(a):
n = a.shape[0]
n_below, n_above = bandwidth(a)
if n_below == n_above == 0:
kind = 'diagonal'
elif n_above == 0:
kind = 'lower triangular'
elif n_below == 0:
kind = 'upper triangular'
elif n_above <= 1 and n_below <= 1 and n > 3:
kind = 'tridiagonal'
elif np.issubdtype(a.dtype, np.complexfloating) and ishermitian(a):
kind = 'hermitian'
elif issymmetric(a):
kind = 'symmetric'
else:
kind = 'general'
return kind, n_below, n_above
def solve(a, b, lower=False, overwrite_a=False,
overwrite_b=False, check_finite=True, assume_a=None,
transposed=False):
"""
Solves the linear equation set ``a @ x == b`` for the unknown ``x``
for square `a` matrix.
If the data matrix is known to be a particular type then supplying the
corresponding string to ``assume_a`` key chooses the dedicated solver.
The available options are
=================== ================================
diagonal 'diagonal'
tridiagonal 'tridiagonal'
banded 'banded'
upper triangular 'upper triangular'
lower triangular 'lower triangular'
symmetric 'symmetric' (or 'sym')
hermitian 'hermitian' (or 'her')
positive definite 'positive definite' (or 'pos')
general 'general' (or 'gen')
=================== ================================
Parameters
----------
a : (N, N) array_like
Square input data
b : (N, NRHS) array_like
Input data for the right hand side.
lower : bool, default: False
Ignored unless ``assume_a`` is one of ``'sym'``, ``'her'``, or ``'pos'``.
If True, the calculation uses only the data in the lower triangle of `a`;
entries above the diagonal are ignored. If False (default), the
calculation uses only the data in the upper triangle of `a`; entries
below the diagonal are ignored.
overwrite_a : bool, default: False
Allow overwriting data in `a` (may enhance performance).
overwrite_b : bool, default: False
Allow overwriting data in `b` (may enhance performance).
check_finite : bool, default: True
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.
assume_a : str, optional
Valid entries are described above.
If omitted or ``None``, checks are performed to identify structure so the
appropriate solver can be called.
transposed : bool, default: False
If True, solve ``a.T @ x == b``. Raises `NotImplementedError`
for complex `a`.
Returns
-------
x : (N, NRHS) ndarray
The solution array.
Raises
------
ValueError
If size mismatches detected or input a is not square.
LinAlgError
If the matrix is singular.
LinAlgWarning
If an ill-conditioned input a is detected.
NotImplementedError
If transposed is True and input a is a complex matrix.
Notes
-----
If the input b matrix is a 1-D array with N elements, when supplied
together with an NxN input a, it is assumed as a valid column vector
despite the apparent size mismatch. This is compatible with the
numpy.dot() behavior and the returned result is still 1-D array.
The general, symmetric, Hermitian and positive definite solutions are
obtained via calling ?GESV, ?SYSV, ?HESV, and ?POSV routines of
LAPACK respectively.
The datatype of the arrays define which solver is called regardless
of the values. In other words, even when the complex array entries have
precisely zero imaginary parts, the complex solver will be called based
on the data type of the array.
Examples
--------
Given `a` and `b`, solve for `x`:
>>> import numpy as np
>>> a = np.array([[3, 2, 0], [1, -1, 0], [0, 5, 1]])
>>> b = np.array([2, 4, -1])
>>> from scipy import linalg
>>> x = linalg.solve(a, b)
>>> x
array([ 2., -2., 9.])
>>> np.dot(a, x) == b
array([ True, True, True], dtype=bool)
"""
# Flags for 1-D or N-D right-hand side
b_is_1D = False
# check finite after determining structure
a1 = atleast_2d(_asarray_validated(a, check_finite=False))
b1 = atleast_1d(_asarray_validated(b, check_finite=False))
a1, b1 = _ensure_dtype_cdsz(a1, b1)
n = a1.shape[0]
overwrite_a = overwrite_a or _datacopied(a1, a)
overwrite_b = overwrite_b or _datacopied(b1, b)
if a1.shape[0] != a1.shape[1]:
raise ValueError('Input a needs to be a square matrix.')
if n != b1.shape[0]:
# Last chance to catch 1x1 scalar a and 1-D b arrays
if not (n == 1 and b1.size != 0):
raise ValueError('Input b has to have same number of rows as '
'input a')
# accommodate empty arrays
if b1.size == 0:
dt = solve(np.eye(2, dtype=a1.dtype), np.ones(2, dtype=b1.dtype)).dtype
return np.empty_like(b1, dtype=dt)
# regularize 1-D b arrays to 2D
if b1.ndim == 1:
if n == 1:
b1 = b1[None, :]
else:
b1 = b1[:, None]
b_is_1D = True
if assume_a not in {None, 'diagonal', 'tridiagonal', 'banded', 'lower triangular',
'upper triangular', 'symmetric', 'hermitian',
'positive definite', 'general', 'sym', 'her', 'pos', 'gen'}:
raise ValueError(f'{assume_a} is not a recognized matrix structure')
# for a real matrix, describe it as "symmetric", not "hermitian"
# (lapack doesn't know what to do with real hermitian matrices)
if assume_a in {'hermitian', 'her'} and not np.iscomplexobj(a1):
assume_a = 'symmetric'
n_below, n_above = None, None
if assume_a is None:
assume_a, n_below, n_above = _find_matrix_structure(a1)
# Get the correct lamch function.
# The LAMCH functions only exists for S and D
# So for complex values we have to convert to real/double.
if a1.dtype.char in 'fF': # single precision
lamch = get_lapack_funcs('lamch', dtype='f')
else:
lamch = get_lapack_funcs('lamch', dtype='d')
# Currently we do not have the other forms of the norm calculators
# lansy, lanpo, lanhe.
# However, in any case they only reduce computations slightly...
if assume_a == 'diagonal':
_matrix_norm = _matrix_norm_diagonal
elif assume_a == 'tridiagonal':
_matrix_norm = _matrix_norm_tridiagonal
elif assume_a in {'lower triangular', 'upper triangular'}:
_matrix_norm = _matrix_norm_triangular(assume_a)
else:
_matrix_norm = _matrix_norm_general
# Since the I-norm and 1-norm are the same for symmetric matrices
# we can collect them all in this one call
# Note however, that when issuing 'gen' and form!='none', then
# the I-norm should be used
if transposed:
trans = 1
norm = 'I'
if np.iscomplexobj(a1):
raise NotImplementedError('scipy.linalg.solve can currently '
'not solve a^T x = b or a^H x = b '
'for complex matrices.')
else:
trans = 0
norm = '1'
anorm = _matrix_norm(norm, a1, check_finite)
info, rcond = 0, np.inf
# Generalized case 'gesv'
if assume_a in {'general', 'gen'}:
gecon, getrf, getrs = get_lapack_funcs(('gecon', 'getrf', 'getrs'),
(a1, b1))
lu, ipvt, info = getrf(a1, overwrite_a=overwrite_a)
_solve_check(n, info)
x, info = getrs(lu, ipvt, b1,
trans=trans, overwrite_b=overwrite_b)
_solve_check(n, info)
rcond, info = gecon(lu, anorm, norm=norm)
# Hermitian case 'hesv'
elif assume_a in {'hermitian', 'her'}:
hecon, hesv, hesv_lw = get_lapack_funcs(('hecon', 'hesv',
'hesv_lwork'), (a1, b1))
lwork = _compute_lwork(hesv_lw, n, lower)
lu, ipvt, x, info = hesv(a1, b1, lwork=lwork,
lower=lower,
overwrite_a=overwrite_a,
overwrite_b=overwrite_b)
_solve_check(n, info)
rcond, info = hecon(lu, ipvt, anorm)
# Symmetric case 'sysv'
elif assume_a in {'symmetric', 'sym'}:
sycon, sysv, sysv_lw = get_lapack_funcs(('sycon', 'sysv',
'sysv_lwork'), (a1, b1))
lwork = _compute_lwork(sysv_lw, n, lower)
lu, ipvt, x, info = sysv(a1, b1, lwork=lwork,
lower=lower,
overwrite_a=overwrite_a,
overwrite_b=overwrite_b)
_solve_check(n, info)
rcond, info = sycon(lu, ipvt, anorm)
# Diagonal case
elif assume_a == 'diagonal':
diag_a = np.diag(a1)
x = (b1.T / diag_a).T
abs_diag_a = np.abs(diag_a)
rcond = abs_diag_a.min() / abs_diag_a.max()
# Tri-diagonal case
elif assume_a == 'tridiagonal':
a1 = a1.T if transposed else a1
dl, d, du = np.diag(a1, -1), np.diag(a1, 0), np.diag(a1, 1)
_gttrf, _gttrs, _gtcon = get_lapack_funcs(('gttrf', 'gttrs', 'gtcon'), (a1, b1))
dl, d, du, du2, ipiv, info = _gttrf(dl, d, du)
_solve_check(n, info)
x, info = _gttrs(dl, d, du, du2, ipiv, b1, overwrite_b=overwrite_b)
_solve_check(n, info)
rcond, info = _gtcon(dl, d, du, du2, ipiv, anorm)
# Banded case
elif assume_a == 'banded':
a1, n_below, n_above = ((a1.T, n_above, n_below) if transposed
else (a1, n_below, n_above))
n_below, n_above = bandwidth(a1) if n_below is None else (n_below, n_above)
ab = _to_banded(n_below, n_above, a1)
gbsv, = get_lapack_funcs(('gbsv',), (a1, b1))
# Next two lines copied from `solve_banded`
a2 = np.zeros((2*n_below + n_above + 1, ab.shape[1]), dtype=gbsv.dtype)
a2[n_below:, :] = ab
_, _, x, info = gbsv(n_below, n_above, a2, b1,
overwrite_ab=True, overwrite_b=overwrite_b)
_solve_check(n, info)
# TODO: wrap gbcon and use to get rcond
# Triangular case
elif assume_a in {'lower triangular', 'upper triangular'}:
lower = assume_a == 'lower triangular'
x, info = _solve_triangular(a1, b1, lower=lower, overwrite_b=overwrite_b,
trans=transposed)
_solve_check(n, info)
_trcon = get_lapack_funcs(('trcon'), (a1, b1))
rcond, info = _trcon(a1, uplo='L' if lower else 'U')
# Positive definite case 'posv'
else:
pocon, posv = get_lapack_funcs(('pocon', 'posv'),
(a1, b1))
lu, x, info = posv(a1, b1, lower=lower,
overwrite_a=overwrite_a,
overwrite_b=overwrite_b)
_solve_check(n, info)
rcond, info = pocon(lu, anorm)
_solve_check(n, info, lamch, rcond)
if b_is_1D:
x = x.ravel()
return x
def _matrix_norm_diagonal(_, a, check_finite):
# Equivalent of dlange for diagonal matrix, assuming
# norm is either 'I' or '1' (really just not the Frobenius norm)
d = np.diag(a)
d = np.asarray_chkfinite(d) if check_finite else d
return np.abs(d).max()
def _matrix_norm_tridiagonal(norm, a, check_finite):
# Equivalent of dlange for tridiagonal matrix, assuming
# norm is either 'I' or '1'
if norm == 'I':
a = a.T
# Context to avoid warning before error in cases like -inf + inf
with np.errstate(invalid='ignore'):
d = np.abs(np.diag(a))
d[1:] += np.abs(np.diag(a, 1))
d[:-1] += np.abs(np.diag(a, -1))
d = np.asarray_chkfinite(d) if check_finite else d
return d.max()
def _matrix_norm_triangular(structure):
def fun(norm, a, check_finite):
a = np.asarray_chkfinite(a) if check_finite else a
lantr = get_lapack_funcs('lantr', (a,))
return lantr(norm, a, 'L' if structure == 'lower triangular' else 'U' )
return fun
def _matrix_norm_general(norm, a, check_finite):
a = np.asarray_chkfinite(a) if check_finite else a
lange = get_lapack_funcs('lange', (a,))
return lange(norm, a)
def _to_banded(n_below, n_above, a):
n = a.shape[0]
rows = n_above + n_below + 1
ab = np.zeros((rows, n), dtype=a.dtype)
ab[n_above] = np.diag(a)
for i in range(1, n_above + 1):
ab[n_above - i, i:] = np.diag(a, i)
for i in range(1, n_below + 1):
ab[n_above + i, :-i] = np.diag(a, -i)
return ab
def _ensure_dtype_cdsz(*arrays):
# Ensure that the dtype of arrays is one of the standard types
# compatible with LAPACK functions (single or double precision
# real or complex).
dtype = np.result_type(*arrays)
if not np.issubdtype(dtype, np.inexact):
return (array.astype(np.float64) for array in arrays)
complex = np.issubdtype(dtype, np.complexfloating)
if np.finfo(dtype).bits <= 32:
dtype = np.complex64 if complex else np.float32
elif np.finfo(dtype).bits >= 64:
dtype = np.complex128 if complex else np.float64
return (array.astype(dtype, copy=False) for array in arrays)
def solve_triangular(a, b, trans=0, lower=False, unit_diagonal=False,
overwrite_b=False, check_finite=True):
"""
Solve the equation ``a x = b`` for `x`, assuming a is a triangular matrix.
Parameters
----------
a : (M, M) array_like
A triangular matrix
b : (M,) or (M, N) array_like
Right-hand side matrix in ``a x = b``
lower : bool, optional
Use only data contained in the lower triangle of `a`.
Default is to use upper triangle.
trans : {0, 1, 2, 'N', 'T', 'C'}, optional
Type of system to solve:
======== =========
trans system
======== =========
0 or 'N' a x = b
1 or 'T' a^T x = b
2 or 'C' a^H x = b
======== =========
unit_diagonal : bool, optional
If True, diagonal elements of `a` are assumed to be 1 and
will not be referenced.
overwrite_b : bool, optional
Allow overwriting data in `b` (may enhance 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 : (M,) or (M, N) ndarray
Solution to the system ``a x = b``. Shape of return matches `b`.
Raises
------
LinAlgError
If `a` is singular
Notes
-----
.. versionadded:: 0.9.0
Examples
--------
Solve the lower triangular system a x = b, where::
[3 0 0 0] [4]
a = [2 1 0 0] b = [2]
[1 0 1 0] [4]
[1 1 1 1] [2]
>>> import numpy as np
>>> from scipy.linalg import solve_triangular
>>> a = np.array([[3, 0, 0, 0], [2, 1, 0, 0], [1, 0, 1, 0], [1, 1, 1, 1]])
>>> b = np.array([4, 2, 4, 2])
>>> x = solve_triangular(a, b, lower=True)
>>> x
array([ 1.33333333, -0.66666667, 2.66666667, -1.33333333])
>>> a.dot(x) # Check the result
array([ 4., 2., 4., 2.])
"""
a1 = _asarray_validated(a, check_finite=check_finite)
b1 = _asarray_validated(b, check_finite=check_finite)
if len(a1.shape) != 2 or a1.shape[0] != a1.shape[1]:
raise ValueError('expected square matrix')
if a1.shape[0] != b1.shape[0]:
raise ValueError(f'shapes of a {a1.shape} and b {b1.shape} are incompatible')
# accommodate empty arrays
if b1.size == 0:
dt_nonempty = solve_triangular(
np.eye(2, dtype=a1.dtype), np.ones(2, dtype=b1.dtype)
).dtype
return np.empty_like(b1, dtype=dt_nonempty)
overwrite_b = overwrite_b or _datacopied(b1, b)
x, _ = _solve_triangular(a1, b1, trans, lower, unit_diagonal, overwrite_b)
return x
# solve_triangular without the input validation
def _solve_triangular(a1, b1, trans=0, lower=False, unit_diagonal=False,
overwrite_b=False):
trans = {'N': 0, 'T': 1, 'C': 2}.get(trans, trans)
trtrs, = get_lapack_funcs(('trtrs',), (a1, b1))
if a1.flags.f_contiguous or trans == 2:
x, info = trtrs(a1, b1, overwrite_b=overwrite_b, lower=lower,
trans=trans, unitdiag=unit_diagonal)
else:
# transposed system is solved since trtrs expects Fortran ordering
x, info = trtrs(a1.T, b1, overwrite_b=overwrite_b, lower=not lower,
trans=not trans, unitdiag=unit_diagonal)
if info == 0:
return x, info
if info > 0:
raise LinAlgError("singular matrix: resolution failed at diagonal %d" %
(info-1))
raise ValueError('illegal value in %dth argument of internal trtrs' %
(-info))
def solve_banded(l_and_u, ab, b, overwrite_ab=False, overwrite_b=False,
check_finite=True):
"""
Solve the equation a x = b for x, assuming a is banded matrix.
The matrix a is stored in `ab` using the matrix diagonal ordered form::
ab[u + i - j, j] == a[i,j]
Example of `ab` (shape of a is (6,6), `u` =1, `l` =2)::
* a01 a12 a23 a34 a45
a00 a11 a22 a33 a44 a55
a10 a21 a32 a43 a54 *
a20 a31 a42 a53 * *
Parameters
----------
(l, u) : (integer, integer)
Number of non-zero lower and upper diagonals
ab : (`l` + `u` + 1, M) array_like
Banded matrix
b : (M,) or (M, K) array_like
Right-hand side
overwrite_ab : bool, optional
Discard data in `ab` (may enhance performance)
overwrite_b : bool, optional
Discard data in `b` (may enhance 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 : (M,) or (M, K) ndarray
The solution to the system a x = b. Returned shape depends on the
shape of `b`.
Examples
--------
Solve the banded system a x = b, where::
[5 2 -1 0 0] [0]
[1 4 2 -1 0] [1]
a = [0 1 3 2 -1] b = [2]
[0 0 1 2 2] [2]
[0 0 0 1 1] [3]
There is one nonzero diagonal below the main diagonal (l = 1), and
two above (u = 2). The diagonal banded form of the matrix is::
[* * -1 -1 -1]
ab = [* 2 2 2 2]
[5 4 3 2 1]
[1 1 1 1 *]
>>> import numpy as np
>>> from scipy.linalg import solve_banded
>>> ab = np.array([[0, 0, -1, -1, -1],
... [0, 2, 2, 2, 2],
... [5, 4, 3, 2, 1],
... [1, 1, 1, 1, 0]])
>>> b = np.array([0, 1, 2, 2, 3])
>>> x = solve_banded((1, 2), ab, b)
>>> x
array([-2.37288136, 3.93220339, -4. , 4.3559322 , -1.3559322 ])
"""
a1 = _asarray_validated(ab, check_finite=check_finite, as_inexact=True)
b1 = _asarray_validated(b, check_finite=check_finite, as_inexact=True)
# Validate shapes.
if a1.shape[-1] != b1.shape[0]:
raise ValueError("shapes of ab and b are not compatible.")
(nlower, nupper) = l_and_u
if nlower + nupper + 1 != a1.shape[0]:
raise ValueError("invalid values for the number of lower and upper "
"diagonals: l+u+1 (%d) does not equal ab.shape[0] "
"(%d)" % (nlower + nupper + 1, ab.shape[0]))
# accommodate empty arrays
if b1.size == 0:
dt = solve(np.eye(1, dtype=a1.dtype), np.ones(1, dtype=b1.dtype)).dtype
return np.empty_like(b1, dtype=dt)
overwrite_b = overwrite_b or _datacopied(b1, b)
if a1.shape[-1] == 1:
b2 = np.array(b1, copy=(not overwrite_b))
# a1.shape[-1] == 1 -> original matrix is 1x1. Typically, the user
# will pass u = l = 0 and `a1` will be 1x1. However, the rest of the
# function works with unnecessary rows in `a1` as long as
# `a1[u + i - j, j] == a[i,j]`. In the 1x1 case, we want i = j = 0,
# so the diagonal is in row `u` of `a1`. See gh-8906.
b2 /= a1[nupper, 0]
return b2
if nlower == nupper == 1:
overwrite_ab = overwrite_ab or _datacopied(a1, ab)
gtsv, = get_lapack_funcs(('gtsv',), (a1, b1))
du = a1[0, 1:]
d = a1[1, :]
dl = a1[2, :-1]
du2, d, du, x, info = gtsv(dl, d, du, b1, overwrite_ab, overwrite_ab,
overwrite_ab, overwrite_b)
else:
gbsv, = get_lapack_funcs(('gbsv',), (a1, b1))
a2 = np.zeros((2*nlower + nupper + 1, a1.shape[1]), dtype=gbsv.dtype)
a2[nlower:, :] = a1
lu, piv, x, info = gbsv(nlower, nupper, a2, b1, overwrite_ab=True,
overwrite_b=overwrite_b)
if info == 0:
return x
if info > 0:
raise LinAlgError("singular matrix")
raise ValueError('illegal value in %d-th argument of internal '
'gbsv/gtsv' % -info)
def solveh_banded(ab, b, overwrite_ab=False, overwrite_b=False, lower=False,
check_finite=True):
"""
Solve equation a x = b. a is Hermitian positive-definite banded matrix.
Uses Thomas' Algorithm, which is more efficient than standard LU
factorization, but should only be used for Hermitian positive-definite
matrices.
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), number of upper diagonals,
``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 * *
Cells marked with * are not used.
Parameters
----------
ab : (``u`` + 1, M) array_like
Banded matrix
b : (M,) or (M, K) array_like
Right-hand side
overwrite_ab : bool, optional
Discard data in `ab` (may enhance performance)
overwrite_b : bool, optional
Discard data in `b` (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 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 : (M,) or (M, K) ndarray
The solution to the system ``a x = b``. Shape of return matches shape
of `b`.
Notes
-----
In the case of a non-positive definite matrix ``a``, the solver
`solve_banded` may be used.
Examples
--------
Solve the banded system ``A x = b``, where::
[ 4 2 -1 0 0 0] [1]
[ 2 5 2 -1 0 0] [2]
A = [-1 2 6 2 -1 0] b = [2]
[ 0 -1 2 7 2 -1] [3]
[ 0 0 -1 2 8 2] [3]
[ 0 0 0 -1 2 9] [3]
>>> import numpy as np
>>> from scipy.linalg import solveh_banded
``ab`` contains the main diagonal and the nonzero diagonals below the
main diagonal. That is, we use the lower form:
>>> ab = np.array([[ 4, 5, 6, 7, 8, 9],
... [ 2, 2, 2, 2, 2, 0],
... [-1, -1, -1, -1, 0, 0]])
>>> b = np.array([1, 2, 2, 3, 3, 3])
>>> x = solveh_banded(ab, b, lower=True)
>>> x
array([ 0.03431373, 0.45938375, 0.05602241, 0.47759104, 0.17577031,
0.34733894])
Solve the Hermitian banded system ``H x = b``, where::
[ 8 2-1j 0 0 ] [ 1 ]
H = [2+1j 5 1j 0 ] b = [1+1j]
[ 0 -1j 9 -2-1j] [1-2j]
[ 0 0 -2+1j 6 ] [ 0 ]
In this example, we put the upper diagonals in the array ``hb``:
>>> hb = np.array([[0, 2-1j, 1j, -2-1j],
... [8, 5, 9, 6 ]])
>>> b = np.array([1, 1+1j, 1-2j, 0])
>>> x = solveh_banded(hb, b)
>>> x
array([ 0.07318536-0.02939412j, 0.11877624+0.17696461j,
0.10077984-0.23035393j, -0.00479904-0.09358128j])
"""
a1 = _asarray_validated(ab, check_finite=check_finite)
b1 = _asarray_validated(b, check_finite=check_finite)
# Validate shapes.
if a1.shape[-1] != b1.shape[0]:
raise ValueError("shapes of ab and b are not compatible.")
# accommodate empty arrays
if b1.size == 0:
dt = solve(np.eye(1, dtype=a1.dtype), np.ones(1, dtype=b1.dtype)).dtype
return np.empty_like(b1, dtype=dt)
overwrite_b = overwrite_b or _datacopied(b1, b)
overwrite_ab = overwrite_ab or _datacopied(a1, ab)
if a1.shape[0] == 2:
ptsv, = get_lapack_funcs(('ptsv',), (a1, b1))
if lower:
d = a1[0, :].real
e = a1[1, :-1]
else:
d = a1[1, :].real
e = a1[0, 1:].conj()
d, du, x, info = ptsv(d, e, b1, overwrite_ab, overwrite_ab,
overwrite_b)
else:
pbsv, = get_lapack_funcs(('pbsv',), (a1, b1))
c, x, info = pbsv(a1, b1, lower=lower, overwrite_ab=overwrite_ab,
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 '
'pbsv' % -info)
return x
def solve_toeplitz(c_or_cr, b, check_finite=True):
r"""Solve a Toeplitz system using Levinson Recursion
The Toeplitz matrix has constant diagonals, with c as its first column
and r as its first row. If r is not given, ``r == conjugate(c)`` is
assumed.
.. warning::
Beginning in SciPy 1.17, multidimensional input will be treated as a batch,
not ``ravel``\ ed. To preserve the existing behavior, ``ravel`` arguments
before passing them to `solve_toeplitz`.
Parameters
----------
c_or_cr : array_like or tuple of (array_like, array_like)
The vector ``c``, or a tuple of arrays (``c``, ``r``). If not
supplied, ``r = conjugate(c)`` is assumed; in this case, if c[0] is
real, the Toeplitz matrix is Hermitian. r[0] is ignored; the first row
of the Toeplitz matrix is ``[c[0], r[1:]]``.
b : (M,) or (M, K) array_like
Right-hand side in ``T x = 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
(result entirely NaNs) if the inputs do contain infinities or NaNs.
Returns
-------
x : (M,) or (M, K) ndarray
The solution to the system ``T x = b``. Shape of return matches shape
of `b`.
See Also
--------
toeplitz : Toeplitz matrix
Notes
-----
The solution is computed using Levinson-Durbin recursion, which is faster
than generic least-squares methods, but can be less numerically stable.
Examples
--------
Solve the Toeplitz system T x = b, where::
[ 1 -1 -2 -3] [1]
T = [ 3 1 -1 -2] b = [2]
[ 6 3 1 -1] [2]
[10 6 3 1] [5]
To specify the Toeplitz matrix, only the first column and the first
row are needed.
>>> import numpy as np
>>> c = np.array([1, 3, 6, 10]) # First column of T
>>> r = np.array([1, -1, -2, -3]) # First row of T
>>> b = np.array([1, 2, 2, 5])
>>> from scipy.linalg import solve_toeplitz, toeplitz
>>> x = solve_toeplitz((c, r), b)
>>> x
array([ 1.66666667, -1. , -2.66666667, 2.33333333])
Check the result by creating the full Toeplitz matrix and
multiplying it by `x`. We should get `b`.
>>> T = toeplitz(c, r)
>>> T.dot(x)
array([ 1., 2., 2., 5.])
"""
# If numerical stability of this algorithm is a problem, a future
# developer might consider implementing other O(N^2) Toeplitz solvers,
# such as GKO (https://www.jstor.org/stable/2153371) or Bareiss.
r, c, b, dtype, b_shape = _validate_args_for_toeplitz_ops(
c_or_cr, b, check_finite, keep_b_shape=True)
# accommodate empty arrays
if b.size == 0:
return np.empty_like(b)
# Form a 1-D array of values to be used in the matrix, containing a
# reversed copy of r[1:], followed by c.
vals = np.concatenate((r[-1:0:-1], c))
if b is None:
raise ValueError('illegal value, `b` is a required argument')
if b.ndim == 1:
x, _ = levinson(vals, np.ascontiguousarray(b))
else:
x = np.column_stack([levinson(vals, np.ascontiguousarray(b[:, i]))[0]
for i in range(b.shape[1])])
x = x.reshape(*b_shape)
return x
def _get_axis_len(aname, a, axis):
ax = axis
if ax < 0:
ax += a.ndim
if 0 <= ax < a.ndim:
return a.shape[ax]
raise ValueError(f"'{aname}axis' entry is out of bounds")
def solve_circulant(c, b, singular='raise', tol=None,
caxis=-1, baxis=0, outaxis=0):
"""Solve C x = b for x, where C is a circulant matrix.
`C` is the circulant matrix associated with the vector `c`.
The system is solved by doing division in Fourier space. The
calculation is::
x = ifft(fft(b) / fft(c))
where `fft` and `ifft` are the fast Fourier transform and its inverse,
respectively. For a large vector `c`, this is *much* faster than
solving the system with the full circulant matrix.
Parameters
----------
c : array_like
The coefficients of the circulant matrix.
b : array_like
Right-hand side matrix in ``a x = b``.
singular : str, optional
This argument controls how a near singular circulant matrix is
handled. If `singular` is "raise" and the circulant matrix is
near singular, a `LinAlgError` is raised. If `singular` is
"lstsq", the least squares solution is returned. Default is "raise".
tol : float, optional
If any eigenvalue of the circulant matrix has an absolute value
that is less than or equal to `tol`, the matrix is considered to be
near singular. If not given, `tol` is set to::
tol = abs_eigs.max() * abs_eigs.size * np.finfo(np.float64).eps
where `abs_eigs` is the array of absolute values of the eigenvalues
of the circulant matrix.
caxis : int
When `c` has dimension greater than 1, it is viewed as a collection
of circulant vectors. In this case, `caxis` is the axis of `c` that
holds the vectors of circulant coefficients.
baxis : int
When `b` has dimension greater than 1, it is viewed as a collection
of vectors. In this case, `baxis` is the axis of `b` that holds the
right-hand side vectors.
outaxis : int
When `c` or `b` are multidimensional, the value returned by
`solve_circulant` is multidimensional. In this case, `outaxis` is
the axis of the result that holds the solution vectors.
Returns
-------
x : ndarray
Solution to the system ``C x = b``.
Raises
------
LinAlgError
If the circulant matrix associated with `c` is near singular.
See Also
--------
circulant : circulant matrix
Notes
-----
For a 1-D vector `c` with length `m`, and an array `b`
with shape ``(m, ...)``,
solve_circulant(c, b)
returns the same result as
solve(circulant(c), b)
where `solve` and `circulant` are from `scipy.linalg`.
.. versionadded:: 0.16.0
Examples
--------
>>> import numpy as np
>>> from scipy.linalg import solve_circulant, solve, circulant, lstsq
>>> c = np.array([2, 2, 4])
>>> b = np.array([1, 2, 3])
>>> solve_circulant(c, b)
array([ 0.75, -0.25, 0.25])
Compare that result to solving the system with `scipy.linalg.solve`:
>>> solve(circulant(c), b)
array([ 0.75, -0.25, 0.25])
A singular example:
>>> c = np.array([1, 1, 0, 0])
>>> b = np.array([1, 2, 3, 4])
Calling ``solve_circulant(c, b)`` will raise a `LinAlgError`. For the
least square solution, use the option ``singular='lstsq'``:
>>> solve_circulant(c, b, singular='lstsq')
array([ 0.25, 1.25, 2.25, 1.25])
Compare to `scipy.linalg.lstsq`:
>>> x, resid, rnk, s = lstsq(circulant(c), b)
>>> x
array([ 0.25, 1.25, 2.25, 1.25])
A broadcasting example:
Suppose we have the vectors of two circulant matrices stored in an array
with shape (2, 5), and three `b` vectors stored in an array with shape
(3, 5). For example,
>>> c = np.array([[1.5, 2, 3, 0, 0], [1, 1, 4, 3, 2]])
>>> b = np.arange(15).reshape(-1, 5)
We want to solve all combinations of circulant matrices and `b` vectors,
with the result stored in an array with shape (2, 3, 5). When we
disregard the axes of `c` and `b` that hold the vectors of coefficients,
the shapes of the collections are (2,) and (3,), respectively, which are
not compatible for broadcasting. To have a broadcast result with shape
(2, 3), we add a trivial dimension to `c`: ``c[:, np.newaxis, :]`` has
shape (2, 1, 5). The last dimension holds the coefficients of the
circulant matrices, so when we call `solve_circulant`, we can use the
default ``caxis=-1``. The coefficients of the `b` vectors are in the last
dimension of the array `b`, so we use ``baxis=-1``. If we use the
default `outaxis`, the result will have shape (5, 2, 3), so we'll use
``outaxis=-1`` to put the solution vectors in the last dimension.
>>> x = solve_circulant(c[:, np.newaxis, :], b, baxis=-1, outaxis=-1)
>>> x.shape
(2, 3, 5)
>>> np.set_printoptions(precision=3) # For compact output of numbers.
>>> x
array([[[-0.118, 0.22 , 1.277, -0.142, 0.302],
[ 0.651, 0.989, 2.046, 0.627, 1.072],
[ 1.42 , 1.758, 2.816, 1.396, 1.841]],
[[ 0.401, 0.304, 0.694, -0.867, 0.377],
[ 0.856, 0.758, 1.149, -0.412, 0.831],
[ 1.31 , 1.213, 1.603, 0.042, 1.286]]])
Check by solving one pair of `c` and `b` vectors (cf. ``x[1, 1, :]``):
>>> solve_circulant(c[1], b[1, :])
array([ 0.856, 0.758, 1.149, -0.412, 0.831])
"""
c = np.atleast_1d(c)
nc = _get_axis_len("c", c, caxis)
b = np.atleast_1d(b)
nb = _get_axis_len("b", b, baxis)
if nc != nb:
raise ValueError(f'Shapes of c {c.shape} and b {b.shape} are incompatible')
# accommodate empty arrays
if b.size == 0:
dt = solve_circulant(np.arange(3, dtype=c.dtype),
np.ones(3, dtype=b.dtype)).dtype
return np.empty_like(b, dtype=dt)
fc = np.fft.fft(np.moveaxis(c, caxis, -1), axis=-1)
abs_fc = np.abs(fc)
if tol is None:
# This is the same tolerance as used in np.linalg.matrix_rank.
tol = abs_fc.max(axis=-1) * nc * np.finfo(np.float64).eps
if tol.shape != ():
tol.shape = tol.shape + (1,)
else:
tol = np.atleast_1d(tol)
near_zeros = abs_fc <= tol
is_near_singular = np.any(near_zeros)
if is_near_singular:
if singular == 'raise':
raise LinAlgError("near singular circulant matrix.")
else:
# Replace the small values with 1 to avoid errors in the
# division fb/fc below.
fc[near_zeros] = 1
fb = np.fft.fft(np.moveaxis(b, baxis, -1), axis=-1)
q = fb / fc
if is_near_singular:
# `near_zeros` is a boolean array, same shape as `c`, that is
# True where `fc` is (near) zero. `q` is the broadcasted result
# of fb / fc, so to set the values of `q` to 0 where `fc` is near
# zero, we use a mask that is the broadcast result of an array
# of True values shaped like `b` with `near_zeros`.
mask = np.ones_like(b, dtype=bool) & near_zeros
q[mask] = 0
x = np.fft.ifft(q, axis=-1)
if not (np.iscomplexobj(c) or np.iscomplexobj(b)):
x = x.real
if outaxis != -1:
x = np.moveaxis(x, -1, outaxis)
return x
# matrix inversion
def inv(a, overwrite_a=False, check_finite=True):
"""
Compute the inverse of a matrix.
Parameters
----------
a : array_like
Square matrix to be inverted.
overwrite_a : bool, optional
Discard data in `a` (may improve performance). Default is False.
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
-------
ainv : ndarray
Inverse of the matrix `a`.
Raises
------
LinAlgError
If `a` is singular.
ValueError
If `a` is not square, or not 2D.
Examples
--------
>>> import numpy as np
>>> from scipy import linalg
>>> a = np.array([[1., 2.], [3., 4.]])
>>> linalg.inv(a)
array([[-2. , 1. ],
[ 1.5, -0.5]])
>>> np.dot(a, linalg.inv(a))
array([[ 1., 0.],
[ 0., 1.]])
"""
a1 = _asarray_validated(a, check_finite=check_finite)
if len(a1.shape) != 2 or a1.shape[0] != a1.shape[1]:
raise ValueError('expected square matrix')
# accommodate empty square matrices
if a1.size == 0:
dt = inv(np.eye(2, dtype=a1.dtype)).dtype
return np.empty_like(a1, dtype=dt)
overwrite_a = overwrite_a or _datacopied(a1, a)
getrf, getri, getri_lwork = get_lapack_funcs(('getrf', 'getri',
'getri_lwork'),
(a1,))
lu, piv, info = getrf(a1, overwrite_a=overwrite_a)
if info == 0:
lwork = _compute_lwork(getri_lwork, a1.shape[0])
# XXX: the following line fixes curious SEGFAULT when
# benchmarking 500x500 matrix inverse. This seems to
# be a bug in LAPACK ?getri routine because if lwork is
# minimal (when using lwork[0] instead of lwork[1]) then
# all tests pass. Further investigation is required if
# more such SEGFAULTs occur.
lwork = int(1.01 * lwork)
inv_a, info = getri(lu, piv, lwork=lwork, overwrite_lu=1)
if info > 0:
raise LinAlgError("singular matrix")
if info < 0:
raise ValueError('illegal value in %d-th argument of internal '
'getrf|getri' % -info)
return inv_a
# Determinant
def det(a, overwrite_a=False, check_finite=True):
"""
Compute the determinant of a matrix
The determinant is a scalar that is a function of the associated square
matrix coefficients. The determinant value is zero for singular matrices.
Parameters
----------
a : (..., M, M) array_like
Input array to compute determinants for.
overwrite_a : bool, optional
Allow overwriting data in a (may enhance 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
-------
det : (...) float or complex
Determinant of `a`. For stacked arrays, a scalar is returned for each
(m, m) slice in the last two dimensions of the input. For example, an
input of shape (p, q, m, m) will produce a result of shape (p, q). If
all dimensions are 1 a scalar is returned regardless of ndim.
Notes
-----
The determinant is computed by performing an LU factorization of the
input with LAPACK routine 'getrf', and then calculating the product of
diagonal entries of the U factor.
Even if the input array is single precision (float32 or complex64), the
result will be returned in double precision (float64 or complex128) to
prevent overflows.
Examples
--------
>>> import numpy as np
>>> from scipy import linalg
>>> a = np.array([[1,2,3], [4,5,6], [7,8,9]]) # A singular matrix
>>> linalg.det(a)
0.0
>>> b = np.array([[0,2,3], [4,5,6], [7,8,9]])
>>> linalg.det(b)
3.0
>>> # An array with the shape (3, 2, 2, 2)
>>> c = np.array([[[[1., 2.], [3., 4.]],
... [[5., 6.], [7., 8.]]],
... [[[9., 10.], [11., 12.]],
... [[13., 14.], [15., 16.]]],
... [[[17., 18.], [19., 20.]],
... [[21., 22.], [23., 24.]]]])
>>> linalg.det(c) # The resulting shape is (3, 2)
array([[-2., -2.],
[-2., -2.],
[-2., -2.]])
>>> linalg.det(c[0, 0]) # Confirm the (0, 0) slice, [[1, 2], [3, 4]]
-2.0
"""
# The goal is to end up with a writable contiguous array to pass to Cython
# First we check and make arrays.
a1 = np.asarray_chkfinite(a) if check_finite else np.asarray(a)
if a1.ndim < 2:
raise ValueError('The input array must be at least two-dimensional.')
if a1.shape[-1] != a1.shape[-2]:
raise ValueError('Last 2 dimensions of the array must be square'
f' but received shape {a1.shape}.')
# Also check if dtype is LAPACK compatible
if a1.dtype.char not in 'fdFD':
dtype_char = lapack_cast_dict[a1.dtype.char]
if not dtype_char: # No casting possible
raise TypeError(f'The dtype "{a1.dtype.name}" cannot be cast '
'to float(32, 64) or complex(64, 128).')
a1 = a1.astype(dtype_char[0]) # makes a copy, free to scratch
overwrite_a = True
# Empty array has determinant 1 because math.
if min(*a1.shape) == 0:
dtyp = np.float64 if a1.dtype.char not in 'FD' else np.complex128
if a1.ndim == 2:
return dtyp(1.0)
else:
return np.ones(shape=a1.shape[:-2], dtype=dtyp)
# Scalar case
if a1.shape[-2:] == (1, 1):
a1 = a1[..., 0, 0]
if a1.ndim == 0:
a1 = a1[()]
# Convert float32 to float64, and complex64 to complex128.
if a1.dtype.char in 'dD':
return a1
return a1.astype('d') if a1.dtype.char == 'f' else a1.astype('D')
# Then check overwrite permission
if not _datacopied(a1, a): # "a" still alive through "a1"
if not overwrite_a:
# Data belongs to "a" so make a copy
a1 = a1.copy(order='C')
# else: Do nothing we'll use "a" if possible
# else: a1 has its own data thus free to scratch
# Then layout checks, might happen that overwrite is allowed but original
# array was read-only or non-C-contiguous.
if not (a1.flags['C_CONTIGUOUS'] and a1.flags['WRITEABLE']):
a1 = a1.copy(order='C')
if a1.ndim == 2:
det = find_det_from_lu(a1)
# Convert float, complex to NumPy scalars
return (np.float64(det) if np.isrealobj(det) else np.complex128(det))
# loop over the stacked array, and avoid overflows for single precision
# Cf. np.linalg.det(np.diag([1e+38, 1e+38]).astype(np.float32))
dtype_char = a1.dtype.char
if dtype_char in 'fF':
dtype_char = 'd' if dtype_char.islower() else 'D'
det = np.empty(a1.shape[:-2], dtype=dtype_char)
for ind in product(*[range(x) for x in a1.shape[:-2]]):
det[ind] = find_det_from_lu(a1[ind])
return det
# Linear Least Squares
def lstsq(a, b, cond=None, overwrite_a=False, overwrite_b=False,
check_finite=True, lapack_driver=None):
"""
Compute least-squares solution to equation Ax = b.
Compute a vector x such that the 2-norm ``|b - A x|`` is minimized.
Parameters
----------
a : (M, N) array_like
Left-hand side array
b : (M,) or (M, K) array_like
Right hand side array
cond : float, optional
Cutoff for 'small' singular values; used to determine effective
rank of a. Singular values smaller than
``cond * largest_singular_value`` are considered zero.
overwrite_a : bool, optional
Discard data in `a` (may enhance performance). Default is False.
overwrite_b : bool, optional
Discard data in `b` (may enhance performance). Default is False.
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.
lapack_driver : str, optional
Which LAPACK driver is used to solve the least-squares problem.
Options are ``'gelsd'``, ``'gelsy'``, ``'gelss'``. Default
(``'gelsd'``) is a good choice. However, ``'gelsy'`` can be slightly
faster on many problems. ``'gelss'`` was used historically. It is
generally slow but uses less memory.
.. versionadded:: 0.17.0
Returns
-------
x : (N,) or (N, K) ndarray
Least-squares solution.
residues : (K,) ndarray or float
Square of the 2-norm for each column in ``b - a x``, if ``M > N`` and
``rank(A) == n`` (returns a scalar if ``b`` is 1-D). Otherwise a
(0,)-shaped array is returned.
rank : int
Effective rank of `a`.
s : (min(M, N),) ndarray or None
Singular values of `a`. The condition number of ``a`` is
``s[0] / s[-1]``.
Raises
------
LinAlgError
If computation does not converge.
ValueError
When parameters are not compatible.
See Also
--------
scipy.optimize.nnls : linear least squares with non-negativity constraint
Notes
-----
When ``'gelsy'`` is used as a driver, `residues` is set to a (0,)-shaped
array and `s` is always ``None``.
Examples
--------
>>> import numpy as np
>>> from scipy.linalg import lstsq
>>> import matplotlib.pyplot as plt
Suppose we have the following data:
>>> x = np.array([1, 2.5, 3.5, 4, 5, 7, 8.5])
>>> y = np.array([0.3, 1.1, 1.5, 2.0, 3.2, 6.6, 8.6])
We want to fit a quadratic polynomial of the form ``y = a + b*x**2``
to this data. We first form the "design matrix" M, with a constant
column of 1s and a column containing ``x**2``:
>>> M = x[:, np.newaxis]**[0, 2]
>>> M
array([[ 1. , 1. ],
[ 1. , 6.25],
[ 1. , 12.25],
[ 1. , 16. ],
[ 1. , 25. ],
[ 1. , 49. ],
[ 1. , 72.25]])
We want to find the least-squares solution to ``M.dot(p) = y``,
where ``p`` is a vector with length 2 that holds the parameters
``a`` and ``b``.
>>> p, res, rnk, s = lstsq(M, y)
>>> p
array([ 0.20925829, 0.12013861])
Plot the data and the fitted curve.
>>> plt.plot(x, y, 'o', label='data')
>>> xx = np.linspace(0, 9, 101)
>>> yy = p[0] + p[1]*xx**2
>>> plt.plot(xx, yy, label='least squares fit, $y = a + bx^2$')
>>> plt.xlabel('x')
>>> plt.ylabel('y')
>>> plt.legend(framealpha=1, shadow=True)
>>> plt.grid(alpha=0.25)
>>> plt.show()
"""
a1 = _asarray_validated(a, check_finite=check_finite)
b1 = _asarray_validated(b, check_finite=check_finite)
if len(a1.shape) != 2:
raise ValueError('Input array a should be 2D')
m, n = a1.shape
if len(b1.shape) == 2:
nrhs = b1.shape[1]
else:
nrhs = 1
if m != b1.shape[0]:
raise ValueError('Shape mismatch: a and b should have the same number'
f' of rows ({m} != {b1.shape[0]}).')
if m == 0 or n == 0: # Zero-sized problem, confuses LAPACK
x = np.zeros((n,) + b1.shape[1:], dtype=np.common_type(a1, b1))
if n == 0:
residues = np.linalg.norm(b1, axis=0)**2
else:
residues = np.empty((0,))
return x, residues, 0, np.empty((0,))
driver = lapack_driver
if driver is None:
driver = lstsq.default_lapack_driver
if driver not in ('gelsd', 'gelsy', 'gelss'):
raise ValueError(f'LAPACK driver "{driver}" is not found')
lapack_func, lapack_lwork = get_lapack_funcs((driver,
f'{driver}_lwork'),
(a1, b1))
real_data = True if (lapack_func.dtype.kind == 'f') else False
if m < n:
# need to extend b matrix as it will be filled with
# a larger solution matrix
if len(b1.shape) == 2:
b2 = np.zeros((n, nrhs), dtype=lapack_func.dtype)
b2[:m, :] = b1
else:
b2 = np.zeros(n, dtype=lapack_func.dtype)
b2[:m] = b1
b1 = b2
overwrite_a = overwrite_a or _datacopied(a1, a)
overwrite_b = overwrite_b or _datacopied(b1, b)
if cond is None:
cond = np.finfo(lapack_func.dtype).eps
if driver in ('gelss', 'gelsd'):
if driver == 'gelss':
lwork = _compute_lwork(lapack_lwork, m, n, nrhs, cond)
v, x, s, rank, work, info = lapack_func(a1, b1, cond, lwork,
overwrite_a=overwrite_a,
overwrite_b=overwrite_b)
elif driver == 'gelsd':
if real_data:
lwork, iwork = _compute_lwork(lapack_lwork, m, n, nrhs, cond)
x, s, rank, info = lapack_func(a1, b1, lwork,
iwork, cond, False, False)
else: # complex data
lwork, rwork, iwork = _compute_lwork(lapack_lwork, m, n,
nrhs, cond)
x, s, rank, info = lapack_func(a1, b1, lwork, rwork, iwork,
cond, False, False)
if info > 0:
raise LinAlgError("SVD did not converge in Linear Least Squares")
if info < 0:
raise ValueError('illegal value in %d-th argument of internal %s'
% (-info, lapack_driver))
resids = np.asarray([], dtype=x.dtype)
if m > n:
x1 = x[:n]
if rank == n:
resids = np.sum(np.abs(x[n:])**2, axis=0)
x = x1
return x, resids, rank, s
elif driver == 'gelsy':
lwork = _compute_lwork(lapack_lwork, m, n, nrhs, cond)
jptv = np.zeros((a1.shape[1], 1), dtype=np.int32)
v, x, j, rank, info = lapack_func(a1, b1, jptv, cond,
lwork, False, False)
if info < 0:
raise ValueError("illegal value in %d-th argument of internal "
"gelsy" % -info)
if m > n:
x1 = x[:n]
x = x1
return x, np.array([], x.dtype), rank, None
lstsq.default_lapack_driver = 'gelsd'
def pinv(a, *, atol=None, rtol=None, return_rank=False, check_finite=True):
"""
Compute the (Moore-Penrose) pseudo-inverse of a matrix.
Calculate a generalized inverse of a matrix using its
singular-value decomposition ``U @ S @ V`` in the economy mode and picking
up only the columns/rows that are associated with significant singular
values.
If ``s`` is the maximum singular value of ``a``, then the
significance cut-off value is determined by ``atol + rtol * s``. Any
singular value below this value is assumed insignificant.
Parameters
----------
a : (M, N) array_like
Matrix to be pseudo-inverted.
atol : float, optional
Absolute threshold term, default value is 0.
.. versionadded:: 1.7.0
rtol : float, optional
Relative threshold term, default value is ``max(M, N) * eps`` where
``eps`` is the machine precision value of the datatype of ``a``.
.. versionadded:: 1.7.0
return_rank : bool, optional
If True, return the effective rank of the matrix.
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
-------
B : (N, M) ndarray
The pseudo-inverse of matrix `a`.
rank : int
The effective rank of the matrix. Returned if `return_rank` is True.
Raises
------
LinAlgError
If SVD computation does not converge.
See Also
--------
pinvh : Moore-Penrose pseudoinverse of a hermitian matrix.
Notes
-----
If ``A`` is invertible then the Moore-Penrose pseudoinverse is exactly
the inverse of ``A`` [1]_. If ``A`` is not invertible then the
Moore-Penrose pseudoinverse computes the ``x`` solution to ``Ax = b`` such
that ``||Ax - b||`` is minimized [1]_.
References
----------
.. [1] Penrose, R. (1956). On best approximate solutions of linear matrix
equations. Mathematical Proceedings of the Cambridge Philosophical
Society, 52(1), 17-19. doi:10.1017/S0305004100030929
Examples
--------
Given an ``m x n`` matrix ``A`` and an ``n x m`` matrix ``B`` the four
Moore-Penrose conditions are:
1. ``ABA = A`` (``B`` is a generalized inverse of ``A``),
2. ``BAB = B`` (``A`` is a generalized inverse of ``B``),
3. ``(AB)* = AB`` (``AB`` is hermitian),
4. ``(BA)* = BA`` (``BA`` is hermitian) [1]_.
Here, ``A*`` denotes the conjugate transpose. The Moore-Penrose
pseudoinverse is a unique ``B`` that satisfies all four of these
conditions and exists for any ``A``. Note that, unlike the standard
matrix inverse, ``A`` does not have to be a square matrix or have
linearly independent columns/rows.
As an example, we can calculate the Moore-Penrose pseudoinverse of a
random non-square matrix and verify it satisfies the four conditions.
>>> import numpy as np
>>> from scipy import linalg
>>> rng = np.random.default_rng()
>>> A = rng.standard_normal((9, 6))
>>> B = linalg.pinv(A)
>>> np.allclose(A @ B @ A, A) # Condition 1
True
>>> np.allclose(B @ A @ B, B) # Condition 2
True
>>> np.allclose((A @ B).conj().T, A @ B) # Condition 3
True
>>> np.allclose((B @ A).conj().T, B @ A) # Condition 4
True
"""
a = _asarray_validated(a, check_finite=check_finite)
u, s, vh = _decomp_svd.svd(a, full_matrices=False, check_finite=False)
t = u.dtype.char.lower()
maxS = np.max(s, initial=0.)
atol = 0. if atol is None else atol
rtol = max(a.shape) * np.finfo(t).eps if (rtol is None) else rtol
if (atol < 0.) or (rtol < 0.):
raise ValueError("atol and rtol values must be positive.")
val = atol + maxS * rtol
rank = np.sum(s > val)
u = u[:, :rank]
u /= s[:rank]
B = (u @ vh[:rank]).conj().T
if return_rank:
return B, rank
else:
return B
def pinvh(a, atol=None, rtol=None, lower=True, return_rank=False,
check_finite=True):
"""
Compute the (Moore-Penrose) pseudo-inverse of a Hermitian matrix.
Calculate a generalized inverse of a complex Hermitian/real symmetric
matrix using its eigenvalue decomposition and including all eigenvalues
with 'large' absolute value.
Parameters
----------
a : (N, N) array_like
Real symmetric or complex hermetian matrix to be pseudo-inverted
atol : float, optional
Absolute threshold term, default value is 0.
.. versionadded:: 1.7.0
rtol : float, optional
Relative threshold term, default value is ``N * eps`` where
``eps`` is the machine precision value of the datatype of ``a``.
.. versionadded:: 1.7.0
lower : bool, optional
Whether the pertinent array data is taken from the lower or upper
triangle of `a`. (Default: lower)
return_rank : bool, optional
If True, return the effective rank of the matrix.
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
-------
B : (N, N) ndarray
The pseudo-inverse of matrix `a`.
rank : int
The effective rank of the matrix. Returned if `return_rank` is True.
Raises
------
LinAlgError
If eigenvalue algorithm does not converge.
See Also
--------
pinv : Moore-Penrose pseudoinverse of a matrix.
Examples
--------
For a more detailed example see `pinv`.
>>> import numpy as np
>>> from scipy.linalg import pinvh
>>> rng = np.random.default_rng()
>>> a = rng.standard_normal((9, 6))
>>> a = np.dot(a, a.T)
>>> B = pinvh(a)
>>> np.allclose(a, a @ B @ a)
True
>>> np.allclose(B, B @ a @ B)
True
"""
a = _asarray_validated(a, check_finite=check_finite)
s, u = _decomp.eigh(a, lower=lower, check_finite=False, driver='ev')
t = u.dtype.char.lower()
maxS = np.max(np.abs(s), initial=0.)
atol = 0. if atol is None else atol
rtol = max(a.shape) * np.finfo(t).eps if (rtol is None) else rtol
if (atol < 0.) or (rtol < 0.):
raise ValueError("atol and rtol values must be positive.")
val = atol + maxS * rtol
above_cutoff = (abs(s) > val)
psigma_diag = 1.0 / s[above_cutoff]
u = u[:, above_cutoff]
B = (u * psigma_diag) @ u.conj().T
if return_rank:
return B, len(psigma_diag)
else:
return B
def matrix_balance(A, permute=True, scale=True, separate=False,
overwrite_a=False):
"""
Compute a diagonal similarity transformation for row/column balancing.
The balancing tries to equalize the row and column 1-norms by applying
a similarity transformation such that the magnitude variation of the
matrix entries is reflected to the scaling matrices.
Moreover, if enabled, the matrix is first permuted to isolate the upper
triangular parts of the matrix and, again if scaling is also enabled,
only the remaining subblocks are subjected to scaling.
The balanced matrix satisfies the following equality
.. math::
B = T^{-1} A T
The scaling coefficients are approximated to the nearest power of 2
to avoid round-off errors.
Parameters
----------
A : (n, n) array_like
Square data matrix for the balancing.
permute : bool, optional
The selector to define whether permutation of A is also performed
prior to scaling.
scale : bool, optional
The selector to turn on and off the scaling. If False, the matrix
will not be scaled.
separate : bool, optional
This switches from returning a full matrix of the transformation
to a tuple of two separate 1-D permutation and scaling arrays.
overwrite_a : bool, optional
This is passed to xGEBAL directly. Essentially, overwrites the result
to the data. It might increase the space efficiency. See LAPACK manual
for details. This is False by default.
Returns
-------
B : (n, n) ndarray
Balanced matrix
T : (n, n) ndarray
A possibly permuted diagonal matrix whose nonzero entries are
integer powers of 2 to avoid numerical truncation errors.
scale, perm : (n,) ndarray
If ``separate`` keyword is set to True then instead of the array
``T`` above, the scaling and the permutation vectors are given
separately as a tuple without allocating the full array ``T``.
Notes
-----
This algorithm is particularly useful for eigenvalue and matrix
decompositions and in many cases it is already called by various
LAPACK routines.
The algorithm is based on the well-known technique of [1]_ and has
been modified to account for special cases. See [2]_ for details
which have been implemented since LAPACK v3.5.0. Before this version
there are corner cases where balancing can actually worsen the
conditioning. See [3]_ for such examples.
The code is a wrapper around LAPACK's xGEBAL routine family for matrix
balancing.
.. versionadded:: 0.19.0
References
----------
.. [1] B.N. Parlett and C. Reinsch, "Balancing a Matrix for
Calculation of Eigenvalues and Eigenvectors", Numerische Mathematik,
Vol.13(4), 1969, :doi:`10.1007/BF02165404`
.. [2] R. James, J. Langou, B.R. Lowery, "On matrix balancing and
eigenvector computation", 2014, :arxiv:`1401.5766`
.. [3] D.S. Watkins. A case where balancing is harmful.
Electron. Trans. Numer. Anal, Vol.23, 2006.
Examples
--------
>>> import numpy as np
>>> from scipy import linalg
>>> x = np.array([[1,2,0], [9,1,0.01], [1,2,10*np.pi]])
>>> y, permscale = linalg.matrix_balance(x)
>>> np.abs(x).sum(axis=0) / np.abs(x).sum(axis=1)
array([ 3.66666667, 0.4995005 , 0.91312162])
>>> np.abs(y).sum(axis=0) / np.abs(y).sum(axis=1)
array([ 1.2 , 1.27041742, 0.92658316]) # may vary
>>> permscale # only powers of 2 (0.5 == 2^(-1))
array([[ 0.5, 0. , 0. ], # may vary
[ 0. , 1. , 0. ],
[ 0. , 0. , 1. ]])
"""
A = np.atleast_2d(_asarray_validated(A, check_finite=True))
if not np.equal(*A.shape):
raise ValueError('The data matrix for balancing should be square.')
# accommodate empty arrays
if A.size == 0:
b_n, t_n = matrix_balance(np.eye(2, dtype=A.dtype))
B = np.empty_like(A, dtype=b_n.dtype)
if separate:
scaling = np.ones_like(A, shape=len(A))
perm = np.arange(len(A))
return B, (scaling, perm)
return B, np.empty_like(A, dtype=t_n.dtype)
gebal = get_lapack_funcs(('gebal'), (A,))
B, lo, hi, ps, info = gebal(A, scale=scale, permute=permute,
overwrite_a=overwrite_a)
if info < 0:
raise ValueError('xGEBAL exited with the internal error '
f'"illegal value in argument number {-info}.". See '
'LAPACK documentation for the xGEBAL error codes.')
# Separate the permutations from the scalings and then convert to int
scaling = np.ones_like(ps, dtype=float)
scaling[lo:hi+1] = ps[lo:hi+1]
# gebal uses 1-indexing
ps = ps.astype(int, copy=False) - 1
n = A.shape[0]
perm = np.arange(n)
# LAPACK permutes with the ordering n --> hi, then 0--> lo
if hi < n:
for ind, x in enumerate(ps[hi+1:][::-1], 1):
if n-ind == x:
continue
perm[[x, n-ind]] = perm[[n-ind, x]]
if lo > 0:
for ind, x in enumerate(ps[:lo]):
if ind == x:
continue
perm[[x, ind]] = perm[[ind, x]]
if separate:
return B, (scaling, perm)
# get the inverse permutation
iperm = np.empty_like(perm)
iperm[perm] = np.arange(n)
return B, np.diag(scaling)[iperm, :]
def _validate_args_for_toeplitz_ops(c_or_cr, b, check_finite, keep_b_shape,
enforce_square=True):
"""Validate arguments and format inputs for toeplitz functions
Parameters
----------
c_or_cr : array_like or tuple of (array_like, array_like)
The vector ``c``, or a tuple of arrays (``c``, ``r``). Whatever the
actual shape of ``c``, it will be converted to a 1-D array. If not
supplied, ``r = conjugate(c)`` is assumed; in this case, if c[0] is
real, the Toeplitz matrix is Hermitian. r[0] is ignored; the first row
of the Toeplitz matrix is ``[c[0], r[1:]]``. Whatever the actual shape
of ``r``, it will be converted to a 1-D array.
b : (M,) or (M, K) array_like
Right-hand side in ``T x = b``.
check_finite : bool
Whether to check that the input matrices contain only finite numbers.
Disabling may give a performance gain, but may result in problems
(result entirely NaNs) if the inputs do contain infinities or NaNs.
keep_b_shape : bool
Whether to convert a (M,) dimensional b into a (M, 1) dimensional
matrix.
enforce_square : bool, optional
If True (default), this verifies that the Toeplitz matrix is square.
Returns
-------
r : array
1d array corresponding to the first row of the Toeplitz matrix.
c: array
1d array corresponding to the first column of the Toeplitz matrix.
b: array
(M,), (M, 1) or (M, K) dimensional array, post validation,
corresponding to ``b``.
dtype: numpy datatype
``dtype`` stores the datatype of ``r``, ``c`` and ``b``. If any of
``r``, ``c`` or ``b`` are complex, ``dtype`` is ``np.complex128``,
otherwise, it is ``np.float``.
b_shape: tuple
Shape of ``b`` after passing it through ``_asarray_validated``.
"""
if isinstance(c_or_cr, tuple):
c, r = c_or_cr
c = _asarray_validated(c, check_finite=check_finite)
r = _asarray_validated(r, check_finite=check_finite)
else:
c = _asarray_validated(c_or_cr, check_finite=check_finite)
r = c.conjugate()
if c.ndim > 1 or r.ndim > 1:
msg = ("Beginning in SciPy 1.17, multidimensional input will be treated as a "
"batch, not `ravel`ed. To preserve the existing behavior and silence "
"this warning, `ravel` arguments before passing them to "
"`toeplitz`, `matmul_toeplitz`, and `solve_toeplitz`.")
warnings.warn(msg, FutureWarning, stacklevel=2)
c = c.ravel()
r = r.ravel()
if b is None:
raise ValueError('`b` must be an array, not None.')
b = _asarray_validated(b, check_finite=check_finite)
b_shape = b.shape
is_not_square = r.shape[0] != c.shape[0]
if (enforce_square and is_not_square) or b.shape[0] != r.shape[0]:
raise ValueError('Incompatible dimensions.')
is_cmplx = np.iscomplexobj(r) or np.iscomplexobj(c) or np.iscomplexobj(b)
dtype = np.complex128 if is_cmplx else np.float64
r, c, b = (np.asarray(i, dtype=dtype) for i in (r, c, b))
if b.ndim == 1 and not keep_b_shape:
b = b.reshape(-1, 1)
elif b.ndim != 1:
b = b.reshape(b.shape[0], -1 if b.size > 0 else 0)
return r, c, b, dtype, b_shape
def matmul_toeplitz(c_or_cr, x, check_finite=False, workers=None):
r"""Efficient Toeplitz Matrix-Matrix Multiplication using FFT
This function returns the matrix multiplication between a Toeplitz
matrix and a dense matrix.
The Toeplitz matrix has constant diagonals, with c as its first column
and r as its first row. If r is not given, ``r == conjugate(c)`` is
assumed.
.. warning::
Beginning in SciPy 1.17, multidimensional input will be treated as a batch,
not ``ravel``\ ed. To preserve the existing behavior, ``ravel`` arguments
before passing them to `matmul_toeplitz`.
Parameters
----------
c_or_cr : array_like or tuple of (array_like, array_like)
The vector ``c``, or a tuple of arrays (``c``, ``r``). If not
supplied, ``r = conjugate(c)`` is assumed; in this case, if c[0] is
real, the Toeplitz matrix is Hermitian. r[0] is ignored; the first row
of the Toeplitz matrix is ``[c[0], r[1:]]``.
x : (M,) or (M, K) array_like
Matrix with which to multiply.
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
(result entirely NaNs) if the inputs do contain infinities or NaNs.
workers : int, optional
To pass to scipy.fft.fft and ifft. Maximum number of workers to use
for parallel computation. If negative, the value wraps around from
``os.cpu_count()``. See scipy.fft.fft for more details.
Returns
-------
T @ x : (M,) or (M, K) ndarray
The result of the matrix multiplication ``T @ x``. Shape of return
matches shape of `x`.
See Also
--------
toeplitz : Toeplitz matrix
solve_toeplitz : Solve a Toeplitz system using Levinson Recursion
Notes
-----
The Toeplitz matrix is embedded in a circulant matrix and the FFT is used
to efficiently calculate the matrix-matrix product.
Because the computation is based on the FFT, integer inputs will
result in floating point outputs. This is unlike NumPy's `matmul`,
which preserves the data type of the input.
This is partly based on the implementation that can be found in [1]_,
licensed under the MIT license. More information about the method can be
found in reference [2]_. References [3]_ and [4]_ have more reference
implementations in Python.
.. versionadded:: 1.6.0
References
----------
.. [1] Jacob R Gardner, Geoff Pleiss, David Bindel, Kilian
Q Weinberger, Andrew Gordon Wilson, "GPyTorch: Blackbox Matrix-Matrix
Gaussian Process Inference with GPU Acceleration" with contributions
from Max Balandat and Ruihan Wu. Available online:
https://github.com/cornellius-gp/gpytorch
.. [2] J. Demmel, P. Koev, and X. Li, "A Brief Survey of Direct Linear
Solvers". In Z. Bai, J. Demmel, J. Dongarra, A. Ruhe, and H. van der
Vorst, editors. Templates for the Solution of Algebraic Eigenvalue
Problems: A Practical Guide. SIAM, Philadelphia, 2000. Available at:
http://www.netlib.org/utk/people/JackDongarra/etemplates/node384.html
.. [3] R. Scheibler, E. Bezzam, I. Dokmanic, Pyroomacoustics: A Python
package for audio room simulations and array processing algorithms,
Proc. IEEE ICASSP, Calgary, CA, 2018.
https://github.com/LCAV/pyroomacoustics/blob/pypi-release/
pyroomacoustics/adaptive/util.py
.. [4] Marano S, Edwards B, Ferrari G and Fah D (2017), "Fitting
Earthquake Spectra: Colored Noise and Incomplete Data", Bulletin of
the Seismological Society of America., January, 2017. Vol. 107(1),
pp. 276-291.
Examples
--------
Multiply the Toeplitz matrix T with matrix x::
[ 1 -1 -2 -3] [1 10]
T = [ 3 1 -1 -2] x = [2 11]
[ 6 3 1 -1] [2 11]
[10 6 3 1] [5 19]
To specify the Toeplitz matrix, only the first column and the first
row are needed.
>>> import numpy as np
>>> c = np.array([1, 3, 6, 10]) # First column of T
>>> r = np.array([1, -1, -2, -3]) # First row of T
>>> x = np.array([[1, 10], [2, 11], [2, 11], [5, 19]])
>>> from scipy.linalg import toeplitz, matmul_toeplitz
>>> matmul_toeplitz((c, r), x)
array([[-20., -80.],
[ -7., -8.],
[ 9., 85.],
[ 33., 218.]])
Check the result by creating the full Toeplitz matrix and
multiplying it by ``x``.
>>> toeplitz(c, r) @ x
array([[-20, -80],
[ -7, -8],
[ 9, 85],
[ 33, 218]])
The full matrix is never formed explicitly, so this routine
is suitable for very large Toeplitz matrices.
>>> n = 1000000
>>> matmul_toeplitz([1] + [0]*(n-1), np.ones(n))
array([1., 1., 1., ..., 1., 1., 1.], shape=(1000000,))
"""
from ..fft import fft, ifft, rfft, irfft
r, c, x, dtype, x_shape = _validate_args_for_toeplitz_ops(
c_or_cr, x, check_finite, keep_b_shape=False, enforce_square=False)
n, m = x.shape
T_nrows = len(c)
T_ncols = len(r)
p = T_nrows + T_ncols - 1 # equivalent to len(embedded_col)
return_shape = (T_nrows,) if len(x_shape) == 1 else (T_nrows, m)
# accommodate empty arrays
if x.size == 0:
return np.empty_like(x, shape=return_shape)
embedded_col = np.concatenate((c, r[-1:0:-1]))
if np.iscomplexobj(embedded_col) or np.iscomplexobj(x):
fft_mat = fft(embedded_col, axis=0, workers=workers).reshape(-1, 1)
fft_x = fft(x, n=p, axis=0, workers=workers)
mat_times_x = ifft(fft_mat*fft_x, axis=0,
workers=workers)[:T_nrows, :]
else:
# Real inputs; using rfft is faster
fft_mat = rfft(embedded_col, axis=0, workers=workers).reshape(-1, 1)
fft_x = rfft(x, n=p, axis=0, workers=workers)
mat_times_x = irfft(fft_mat*fft_x, axis=0,
workers=workers, n=p)[:T_nrows, :]
return mat_times_x.reshape(*return_shape)
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