Skip to content

HTTPS clone URL

Subversion checkout URL

You can clone with
or
.
Download ZIP
Fetching contributors…

Cannot retrieve contributors at this time

1058 lines (889 sloc) 35.785 kB
#
# 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
__all__ = ['solve', 'solve_triangular', 'solveh_banded', 'solve_banded',
'solve_toeplitz', 'solve_circulant', 'inv', 'det', 'lstsq',
'pinv', 'pinv2', 'pinvh']
import numpy as np
from .flinalg import get_flinalg_funcs
from .lapack import get_lapack_funcs
from .misc import LinAlgError, _datacopied
from .decomp import _asarray_validated
from . import decomp, decomp_svd
from ._solve_toeplitz import levinson
# Linear equations
def solve(a, b, sym_pos=False, lower=False, overwrite_a=False,
overwrite_b=False, debug=False, check_finite=True):
"""
Solve the equation ``a x = b`` for ``x``.
Parameters
----------
a : (M, M) array_like
A square matrix.
b : (M,) or (M, N) array_like
Right-hand side matrix in ``a x = b``.
sym_pos : bool, optional
Assume `a` is symmetric and positive definite.
lower : bool, optional
Use only data contained in the lower triangle of `a`, if `sym_pos` is
true. Default is to use upper triangle.
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.
Returns
-------
x : (M,) or (M, N) ndarray
Solution to the system ``a x = b``. Shape of the return matches the
shape of `b`.
Raises
------
LinAlgError
If `a` is singular.
ValueError
If `a` is not square
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)
"""
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_a = overwrite_a or _datacopied(a1, a)
overwrite_b = overwrite_b or _datacopied(b1, b)
if debug:
print('solve:overwrite_a=', overwrite_a)
print('solve:overwrite_b=', overwrite_b)
if sym_pos:
posv, = get_lapack_funcs(('posv',), (a1, b1))
c, x, info = posv(a1, b1, lower=lower,
overwrite_a=overwrite_a,
overwrite_b=overwrite_b)
else:
gesv, = get_lapack_funcs(('gesv',), (a1, b1))
lu, piv, x, info = gesv(a1, b1, overwrite_a=overwrite_a,
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 gesv|posv' %
-info)
def solve_triangular(a, b, trans=0, lower=False, unit_diagonal=False,
overwrite_b=False, debug=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
"""
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 %s" %
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=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`.
"""
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=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 algorithim 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')
if vals.shape[0] != (2*b.shape[0] - 1):
raise ValueError('incompatible dimensions')
b = _asarray_validated(b)
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, info = getri_lwork(a1.shape[0])
if info != 0:
raise ValueError('internal getri work space query failed: %d' %
(info,))
lwork = int(lwork.real)
# 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
def lstsq(a, b, cond=None, overwrite_a=False, overwrite_b=False,
check_finite=True):
"""
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.
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 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
Singular values of `a`. The condition number of a is
``abs(s[0]/s[-1])``.
Raises
------
LinAlgError :
If computation does not converge.
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')
gelss, = get_lapack_funcs(('gelss',), (a1, b1))
if n > m:
# 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=gelss.dtype)
b2[:m, :] = b1
else:
b2 = np.zeros(n, dtype=gelss.dtype)
b2[:m] = b1
b1 = b2
overwrite_a = overwrite_a or _datacopied(a1, a)
overwrite_b = overwrite_b or _datacopied(b1, b)
# get optimal work array
work = gelss(a1, b1, lwork=-1)[4]
lwork = work[0].real.astype(int)
v, x, s, rank, work, info = gelss(
a1, b1, cond=cond, lwork=lwork, overwrite_a=overwrite_a,
overwrite_b=overwrite_b)
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 gelss' %
-info)
resids = np.asarray([], dtype=x.dtype)
if n < m:
x1 = x[:n]
if rank == n:
resids = np.sum(np.abs(x[n:])**2, axis=0)
x = x1
return x, resids, rank, s
def pinv(a, cond=None, rcond=None, return_rank=False, check_finite=True):
"""
Compute the (Moore-Penrose) pseudo-inverse of a matrix.
Calculate a generalized inverse of a matrix using a least-squares
solver.
Parameters
----------
a : (M, N) array_like
Matrix to be pseudo-inverted.
cond, rcond : float, optional
Cutoff for 'small' singular values in the least-squares solver.
Singular values smaller than ``rcond * largest_singular_value``
are considered zero.
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 == True
Raises
------
LinAlgError
If computation does not converge.
Examples
--------
>>> from scipy import linalg
>>> a = np.random.randn(9, 6)
>>> B = linalg.pinv(a)
>>> np.allclose(a, np.dot(a, np.dot(B, a)))
True
>>> np.allclose(B, np.dot(B, np.dot(a, B)))
True
"""
a = _asarray_validated(a, check_finite=check_finite)
b = np.identity(a.shape[0], dtype=a.dtype)
if rcond is not None:
cond = rcond
x, resids, rank, s = lstsq(a, b, cond=cond, check_finite=False)
if return_rank:
return x, rank
else:
return x
def pinv2(a, cond=None, rcond=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 and including all 'large' singular
values.
Parameters
----------
a : (M, N) array_like
Matrix to be pseudo-inverted.
cond, rcond : float or None
Cutoff for 'small' singular values.
Singular values smaller than ``rcond*largest_singular_value``
are considered zero.
If None or -1, suitable machine precision is used.
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 == True
Raises
------
LinAlgError
If SVD computation does not converge.
Examples
--------
>>> from scipy import linalg
>>> a = np.random.randn(9, 6)
>>> B = linalg.pinv2(a)
>>> np.allclose(a, np.dot(a, np.dot(B, a)))
True
>>> np.allclose(B, np.dot(B, np.dot(a, B)))
True
"""
a = _asarray_validated(a, check_finite=check_finite)
u, s, vh = decomp_svd.svd(a, full_matrices=False, check_finite=False)
if rcond is not None:
cond = rcond
if cond in [None, -1]:
t = u.dtype.char.lower()
factor = {'f': 1E3, 'd': 1E6}
cond = factor[t] * np.finfo(t).eps
rank = np.sum(s > cond * np.max(s))
u = u[:, :rank]
u /= s[:rank]
B = np.transpose(np.conjugate(np.dot(u, vh[:rank])))
if return_rank:
return B, rank
else:
return B
def pinvh(a, cond=None, rcond=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 Hermitian or 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
cond, rcond : float or None
Cutoff for 'small' eigenvalues.
Singular values smaller than rcond * largest_eigenvalue are considered
zero.
If None or -1, suitable machine precision is used.
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 == True
Raises
------
LinAlgError
If eigenvalue does not converge
Examples
--------
>>> from scipy.linalg import pinvh
>>> a = np.random.randn(9, 6)
>>> a = np.dot(a, a.T)
>>> B = pinvh(a)
>>> np.allclose(a, np.dot(a, np.dot(B, a)))
True
>>> np.allclose(B, np.dot(B, np.dot(a, B)))
True
"""
a = _asarray_validated(a, check_finite=check_finite)
s, u = decomp.eigh(a, lower=lower, check_finite=False)
if rcond is not None:
cond = rcond
if cond in [None, -1]:
t = u.dtype.char.lower()
factor = {'f': 1E3, 'd': 1E6}
cond = factor[t] * np.finfo(t).eps
# For Hermitian matrices, singular values equal abs(eigenvalues)
above_cutoff = (abs(s) > cond * np.max(abs(s)))
psigma_diag = 1.0 / s[above_cutoff]
u = u[:, above_cutoff]
B = np.dot(u * psigma_diag, np.conjugate(u).T)
if return_rank:
return B, len(psigma_diag)
else:
return B
Jump to Line
Something went wrong with that request. Please try again.