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basic.py
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basic.py
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#
# Author: Pearu Peterson, March 2002
#
# w/ additions by Travis Oliphant, March 2002
# and Jake Vanderplas, August 2012
from __future__ import division, print_function, absolute_import
import warnings
import numpy as np
from numpy import atleast_1d, atleast_2d
from .flinalg import get_flinalg_funcs
from .lapack import get_lapack_funcs, _compute_lwork
from .misc import LinAlgError, _datacopied
from .decomp import _asarray_validated
from . import decomp, decomp_svd
from ._solve_toeplitz import levinson
__all__ = ['solve', 'solve_triangular', 'solveh_banded', 'solve_banded',
'solve_toeplitz', 'solve_circulant', 'inv', 'det', 'lstsq',
'pinv', 'pinv2', 'pinvh', 'matrix_balance']
# Linear equations
def solve(a, b, sym_pos=False, lower=False, overwrite_a=False,
overwrite_b=False, debug=None, check_finite=True, assume_a='gen',
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
=================== ========
generic matrix 'gen'
symmetric 'sym'
hermitian 'her'
positive definite 'pos'
=================== ========
If omitted, ``'gen'`` is the default structure.
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.
Parameters
----------
a : (N, N) array_like
Square input data
b : (N, NRHS) array_like
Input data for the right hand side.
sym_pos : bool, optional
Assume `a` is symmetric and positive definite. This key is deprecated
and assume_a = 'pos' keyword is recommended instead. The functionality
is the same. It will be removed in the future.
lower : bool, optional
If True, only the data contained in the lower triangle of `a`. Default
is to use upper triangle. (ignored for ``'gen'``)
overwrite_a : bool, optional
Allow overwriting data in `a` (may enhance performance).
Default is False.
overwrite_b : bool, optional
Allow overwriting 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.
assume_a : str, optional
Valid entries are explained above.
transposed: bool, optional
If True, depending on the data type ``a^T x = b`` or ``a^H x = b`` is
solved (only taken into account for ``'gen'``).
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.
RuntimeWarning
If an ill-conditioned input a is detected.
Examples
--------
Given `a` and `b`, solve for `x`:
>>> 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)
Notes
-----
If the input b matrix is a 1D 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 1D array.
The generic, symmetric, hermitian and positive definite solutions are
obtained via calling ?GESVX, ?SYSVX, ?HESVX, and ?POSVX routines of
LAPACK respectively.
"""
# Flags for 1D or nD right hand side
b_is_1D = False
b_is_ND = False
a1 = atleast_2d(_asarray_validated(a, check_finite=check_finite))
b1 = atleast_1d(_asarray_validated(b, check_finite=check_finite))
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 1D b arrays
if not (n == 1 and b1.size != 0):
raise ValueError('Input b has to have same number of rows as '
'input a')
# accomodate empty arrays
if b1.size == 0:
return np.asfortranarray(b1.copy())
# regularize 1D b arrays to 2D and catch nD RHS arrays
if b1.ndim == 1:
if n == 1:
b1 = b1[None, :]
else:
b1 = b1[:, None]
b_is_1D = True
elif b1.ndim > 2:
b_is_ND = True
r_or_c = complex if np.iscomplexobj(a1) else float
if assume_a in ('gen', 'sym', 'her', 'pos'):
_structure = assume_a
else:
raise ValueError('{} is not a recognized matrix structure'
''.format(assume_a))
# Deprecate keyword "debug"
if debug is not None:
warnings.warn('Use of the "debug" keyword is deprecated '
'and this keyword will be removed in the future '
'versions of SciPy.', DeprecationWarning)
# Backwards compatibility - old keyword.
if sym_pos:
assume_a = 'pos'
if _structure == 'gen':
gesvx = get_lapack_funcs('gesvx', (a1, b1))
trans_conj = 'N'
if transposed:
trans_conj = 'T' if r_or_c is float else 'H'
(_, _, _, _, _, _, _,
x, rcond, _, _, info) = gesvx(a1, b1,
trans=trans_conj,
overwrite_a=overwrite_a,
overwrite_b=overwrite_b
)
elif _structure == 'sym':
sysvx, sysvx_lw = get_lapack_funcs(('sysvx', 'sysvx_lwork'), (a1, b1))
lwork = _compute_lwork(sysvx_lw, n, lower)
_, _, _, _, x, rcond, _, _, info = sysvx(a1, b1, lwork=lwork,
lower=lower,
overwrite_a=overwrite_a,
overwrite_b=overwrite_b
)
elif _structure == 'her':
hesvx, hesvx_lw = get_lapack_funcs(('hesvx', 'hesvx_lwork'), (a1, b1))
lwork = _compute_lwork(hesvx_lw, n, lower)
_, _, x, rcond, _, _, info = hesvx(a1, b1, lwork=lwork,
lower=lower,
overwrite_a=overwrite_a,
overwrite_b=overwrite_b
)
else:
posvx = get_lapack_funcs('posvx', (a1, b1))
_, _, _, _, _, x, rcond, _, _, info = posvx(a1, b1,
lower=lower,
overwrite_a=overwrite_a,
overwrite_b=overwrite_b
)
# Unlike ?xxSV, ?xxSVX writes the solution x to a separate array, and
# overwrites b with its scaled version which is thrown away. Thus, the
# solution does not admit the same shape with the original b. For
# backwards compatibility, we reshape it manually.
if b_is_1D:
x = x.ravel()
if b_is_ND:
x = x.reshape(*b1.shape, order='F')
if info < 0:
raise ValueError('LAPACK reported an illegal value in {}-th argument'
'.'.format(-info))
elif info == 0:
return x
elif 0 < info <= n:
raise LinAlgError('Matrix is singular.')
elif info > n:
warnings.warn('scipy.linalg.solve\nIll-conditioned matrix detected.'
' Result is not guaranteed to be accurate.\nReciprocal'
' condition number: {}'.format(rcond), RuntimeWarning)
return x
def solve_triangular(a, b, trans=0, lower=False, unit_diagonal=False,
overwrite_b=False, debug=None, 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
"""
# Deprecate keyword "debug"
if debug is not None:
warnings.warn('Use of the "debug" keyword is deprecated '
'and this keyword will be removed in the future '
'versions of SciPy.', DeprecationWarning)
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('incompatible dimensions')
overwrite_b = overwrite_b or _datacopied(b1, b)
if debug:
print('solve:overwrite_b=', overwrite_b)
trans = {'N': 0, 'T': 1, 'C': 2}.get(trans, trans)
trtrs, = get_lapack_funcs(('trtrs',), (a1, b1))
x, info = trtrs(a1, b1, overwrite_b=overwrite_b, lower=lower,
trans=trans, unitdiag=unit_diagonal)
if info == 0:
return x
if info > 0:
raise LinAlgError("singular matrix: resolution failed at diagonal %d" %
(info-1))
raise ValueError('illegal value in %d-th argument of internal trtrs' %
(-info))
def solve_banded(l_and_u, ab, b, overwrite_ab=False, overwrite_b=False,
debug=None, 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`.
"""
# Deprecate keyword "debug"
if debug is not None:
warnings.warn('Use of the "debug" keyword is deprecated '
'and this keyword will be removed in the future '
'versions of SciPy.', DeprecationWarning)
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.")
(l, u) = l_and_u
if l + u + 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)" % (l+u+1, ab.shape[0]))
overwrite_b = overwrite_b or _datacopied(b1, b)
if a1.shape[-1] == 1:
b2 = np.array(b1, copy=(not overwrite_b))
b2 /= a1[1, 0]
return b2
if l == u == 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*l+u+1, a1.shape[1]), dtype=gbsv.dtype)
a2[l:, :] = a1
lu, piv, x, info = gbsv(l, u, 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.
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 * *
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`.
"""
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.")
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("%d-th leading minor not positive definite" % info)
if info < 0:
raise ValueError('illegal value in %d-th argument of internal '
'pbsv' % -info)
return x
def solve_toeplitz(c_or_cr, b, check_finite=True):
"""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.
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, 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`.
Notes
-----
The solution is computed using Levinson-Durbin recursion, which is faster
than generic least-squares methods, but can be less numerically stable.
"""
# If numerical stability of this algorithm is a problem, a future
# developer might consider implementing other O(N^2) Toeplitz solvers,
# such as GKO (http://www.jstor.org/stable/2153371) or Bareiss.
if isinstance(c_or_cr, tuple):
c, r = c_or_cr
c = _asarray_validated(c, check_finite=check_finite).ravel()
r = _asarray_validated(r, check_finite=check_finite).ravel()
else:
c = _asarray_validated(c_or_cr, check_finite=check_finite).ravel()
r = c.conjugate()
# Form a 1D 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')
b = _asarray_validated(b)
if vals.shape[0] != (2*b.shape[0] - 1):
raise ValueError('incompatible dimensions')
if np.iscomplexobj(vals) or np.iscomplexobj(b):
vals = np.asarray(vals, dtype=np.complex128, order='c')
b = np.asarray(b, dtype=np.complex128)
else:
vals = np.asarray(vals, dtype=np.double, order='c')
b = np.asarray(b, dtype=np.double)
if b.ndim == 1:
x, _ = levinson(vals, np.ascontiguousarray(b))
else:
b_shape = b.shape
b = b.reshape(b.shape[0], -1)
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("'%saxis' entry is out of bounds" % (aname,))
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
Notes
-----
For a one-dimensional 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
--------
>>> 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('Incompatible c and b axis lengths')
fc = np.fft.fft(np.rollaxis(c, caxis, c.ndim), 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.rollaxis(b, baxis, b.ndim), 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.rollaxis(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 2-dimensional.
Examples
--------
>>> 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')
overwrite_a = overwrite_a or _datacopied(a1, a)
#XXX: I found no advantage or disadvantage of using finv.
# finv, = get_flinalg_funcs(('inv',),(a1,))
# if finv is not None:
# a_inv,info = finv(a1,overwrite_a=overwrite_a)
# if info==0:
# return a_inv
# if info>0: raise LinAlgError, "singular matrix"
# if info<0: raise ValueError('illegal value in %d-th argument of '
# 'internal inv.getrf|getri'%(-info))
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 of a square matrix is a value derived arithmetically
from the coefficients of the matrix.
The determinant for a 3x3 matrix, for example, is computed as follows::
a b c
d e f = A
g h i
det(A) = a*e*i + b*f*g + c*d*h - c*e*g - b*d*i - a*f*h
Parameters
----------
a : (M, M) array_like
A square matrix.
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`.
Notes
-----
The determinant is computed via LU factorization, LAPACK routine z/dgetrf.
Examples
--------
>>> from scipy import linalg
>>> a = np.array([[1,2,3], [4,5,6], [7,8,9]])
>>> linalg.det(a)
0.0
>>> a = np.array([[0,2,3], [4,5,6], [7,8,9]])
>>> linalg.det(a)
3.0
"""
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')
overwrite_a = overwrite_a or _datacopied(a1, a)
fdet, = get_flinalg_funcs(('det',), (a1,))
a_det, info = fdet(a1, overwrite_a=overwrite_a)
if info < 0:
raise ValueError('illegal value in %d-th argument of internal '
'det.getrf' % -info)
return a_det
# Linear Least Squares
class LstsqLapackError(LinAlgError):
pass
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 matrix (2-D array).
b : (M,) or (M, K) array_like
Right hand side matrix or vector (1-D or 2-D array).
cond : float, optional
Cutoff for 'small' singular values; used to determine effective
rank of a. Singular values smaller than
``rcond * 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. Return shape matches shape of `b`.
residues : () or (1,) or (K,) ndarray
Sums of residues, squared 2-norm for each column in ``b - a x``.
If rank of matrix a is ``< N`` or ``> M``, or ``'gelsy'`` is used,
this is an empty array. If b was 1-D, this is an (1,) shape array,
otherwise the shape is (K,).
rank : int
Effective rank of matrix `a`.
s : (min(M,N),) ndarray or None
Singular values of `a`. The condition number of a is
``abs(s[0] / s[-1])``. None is returned when ``'gelsy'`` is used.
Raises
------
LinAlgError
If computation does not converge.
ValueError
When parameters are wrong.
See Also
--------
optimize.nnls : linear least squares with non-negativity constraint
"""
a1 = _asarray_validated(a, check_finite=check_finite)
b1 = _asarray_validated(b, check_finite=check_finite)
if len(a1.shape) != 2:
raise ValueError('expected matrix')
m, n = a1.shape
if len(b1.shape) == 2:
nrhs = b1.shape[1]
else:
nrhs = 1
if m != b1.shape[0]:
raise ValueError('incompatible dimensions')
driver = lapack_driver
if driver is None:
driver = lstsq.default_lapack_driver
if driver not in ('gelsd', 'gelsy', 'gelss'):
raise ValueError('LAPACK driver "%s" is not found' % driver)
lapack_func, lapack_lwork = get_lapack_funcs((driver,
'%s_lwork' % driver),
(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)