# -*- coding: utf-8 -*-
#
# Author: Pearu Peterson, March 2002
#
# additions by Travis Oliphant, March 2002
# additions by Eric Jones, June 2002
# additions by Johannes Loehnert, June 2006
# additions by Bart Vandereycken, June 2006
# additions by Andrew D Straw, May 2007
# additions by Tiziano Zito, November 2008
#
# April 2010: Functions for LU, QR, SVD, Schur, and Cholesky decompositions
# were moved to their own files. Still in this file are functions for
# eigenstuff and for the Hessenberg form.
__all__ = ['eig', 'eigvals', 'eigh', 'eigvalsh',
'eig_banded', 'eigvals_banded',
'eigh_tridiagonal', 'eigvalsh_tridiagonal', 'hessenberg', 'cdf2rdf']
import numpy
from numpy import (array, isfinite, inexact, nonzero, iscomplexobj, cast,
flatnonzero, conj, asarray, argsort, empty,
iscomplex, zeros, einsum, eye, inf)
# Local imports
from scipy._lib._util import _asarray_validated
from ._misc import LinAlgError, _datacopied, norm
from .lapack import get_lapack_funcs, _compute_lwork
_I = cast['F'](1j)
def _make_complex_eigvecs(w, vin, dtype):
"""
Produce complex-valued eigenvectors from LAPACK DGGEV real-valued output
"""
# - see LAPACK man page DGGEV at ALPHAI
v = numpy.array(vin, dtype=dtype)
m = (w.imag > 0)
m[:-1] |= (w.imag[1:] < 0) # workaround for LAPACK bug, cf. ticket #709
for i in flatnonzero(m):
v.imag[:, i] = vin[:, i+1]
conj(v[:, i], v[:, i+1])
return v
def _make_eigvals(alpha, beta, homogeneous_eigvals):
if homogeneous_eigvals:
if beta is None:
return numpy.vstack((alpha, numpy.ones_like(alpha)))
else:
return numpy.vstack((alpha, beta))
else:
if beta is None:
return alpha
else:
w = numpy.empty_like(alpha)
alpha_zero = (alpha == 0)
beta_zero = (beta == 0)
beta_nonzero = ~beta_zero
w[beta_nonzero] = alpha[beta_nonzero]/beta[beta_nonzero]
# Use numpy.inf for complex values too since
# 1/numpy.inf = 0, i.e., it correctly behaves as projective
# infinity.
w[~alpha_zero & beta_zero] = numpy.inf
if numpy.all(alpha.imag == 0):
w[alpha_zero & beta_zero] = numpy.nan
else:
w[alpha_zero & beta_zero] = complex(numpy.nan, numpy.nan)
return w
def _geneig(a1, b1, left, right, overwrite_a, overwrite_b,
homogeneous_eigvals):
ggev, = get_lapack_funcs(('ggev',), (a1, b1))
cvl, cvr = left, right
res = ggev(a1, b1, lwork=-1)
lwork = res[-2][0].real.astype(numpy.int_)
if ggev.typecode in 'cz':
alpha, beta, vl, vr, work, info = ggev(a1, b1, cvl, cvr, lwork,
overwrite_a, overwrite_b)
w = _make_eigvals(alpha, beta, homogeneous_eigvals)
else:
alphar, alphai, beta, vl, vr, work, info = ggev(a1, b1, cvl, cvr,
lwork, overwrite_a,
overwrite_b)
alpha = alphar + _I * alphai
w = _make_eigvals(alpha, beta, homogeneous_eigvals)
_check_info(info, 'generalized eig algorithm (ggev)')
only_real = numpy.all(w.imag == 0.0)
if not (ggev.typecode in 'cz' or only_real):
t = w.dtype.char
if left:
vl = _make_complex_eigvecs(w, vl, t)
if right:
vr = _make_complex_eigvecs(w, vr, t)
# the eigenvectors returned by the lapack function are NOT normalized
for i in range(vr.shape[0]):
if right:
vr[:, i] /= norm(vr[:, i])
if left:
vl[:, i] /= norm(vl[:, i])
if not (left or right):
return w
if left:
if right:
return w, vl, vr
return w, vl
return w, vr
[docs]def eig(a, b=None, left=False, right=True, overwrite_a=False,
overwrite_b=False, check_finite=True, homogeneous_eigvals=False):
"""
Solve an ordinary or generalized eigenvalue problem of a square matrix.
Find eigenvalues w and right or left eigenvectors of a general matrix::
a vr[:,i] = w[i] b vr[:,i]
a.H vl[:,i] = w[i].conj() b.H vl[:,i]
where ``.H`` is the Hermitian conjugation.
Parameters
----------
a : (M, M) array_like
A complex or real matrix whose eigenvalues and eigenvectors
will be computed.
b : (M, M) array_like, optional
Right-hand side matrix in a generalized eigenvalue problem.
Default is None, identity matrix is assumed.
left : bool, optional
Whether to calculate and return left eigenvectors. Default is False.
right : bool, optional
Whether to calculate and return right eigenvectors. Default is True.
overwrite_a : bool, optional
Whether to overwrite `a`; may improve performance. Default is False.
overwrite_b : bool, optional
Whether to overwrite `b`; may improve 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.
homogeneous_eigvals : bool, optional
If True, return the eigenvalues in homogeneous coordinates.
In this case ``w`` is a (2, M) array so that::
w[1,i] a vr[:,i] = w[0,i] b vr[:,i]
Default is False.
Returns
-------
w : (M,) or (2, M) double or complex ndarray
The eigenvalues, each repeated according to its
multiplicity. The shape is (M,) unless
``homogeneous_eigvals=True``.
vl : (M, M) double or complex ndarray
The normalized left eigenvector corresponding to the eigenvalue
``w[i]`` is the column vl[:,i]. Only returned if ``left=True``.
vr : (M, M) double or complex ndarray
The normalized right eigenvector corresponding to the eigenvalue
``w[i]`` is the column ``vr[:,i]``. Only returned if ``right=True``.
Raises
------
LinAlgError
If eigenvalue computation does not converge.
See Also
--------
eigvals : eigenvalues of general arrays
eigh : Eigenvalues and right eigenvectors for symmetric/Hermitian arrays.
eig_banded : eigenvalues and right eigenvectors for symmetric/Hermitian
band matrices
eigh_tridiagonal : eigenvalues and right eiegenvectors for
symmetric/Hermitian tridiagonal matrices
Examples
--------
>>> from scipy import linalg
>>> a = np.array([[0., -1.], [1., 0.]])
>>> linalg.eigvals(a)
array([0.+1.j, 0.-1.j])
>>> b = np.array([[0., 1.], [1., 1.]])
>>> linalg.eigvals(a, b)
array([ 1.+0.j, -1.+0.j])
>>> a = np.array([[3., 0., 0.], [0., 8., 0.], [0., 0., 7.]])
>>> linalg.eigvals(a, homogeneous_eigvals=True)
array([[3.+0.j, 8.+0.j, 7.+0.j],
[1.+0.j, 1.+0.j, 1.+0.j]])
>>> a = np.array([[0., -1.], [1., 0.]])
>>> linalg.eigvals(a) == linalg.eig(a)[0]
array([ True, True])
>>> linalg.eig(a, left=True, right=False)[1] # normalized left eigenvector
array([[-0.70710678+0.j , -0.70710678-0.j ],
[-0. +0.70710678j, -0. -0.70710678j]])
>>> linalg.eig(a, left=False, right=True)[1] # normalized right eigenvector
array([[0.70710678+0.j , 0.70710678-0.j ],
[0. -0.70710678j, 0. +0.70710678j]])
"""
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))
if b is not None:
b1 = _asarray_validated(b, check_finite=check_finite)
overwrite_b = overwrite_b or _datacopied(b1, b)
if len(b1.shape) != 2 or b1.shape[0] != b1.shape[1]:
raise ValueError('expected square matrix')
if b1.shape != a1.shape:
raise ValueError('a and b must have the same shape')
return _geneig(a1, b1, left, right, overwrite_a, overwrite_b,
homogeneous_eigvals)
geev, geev_lwork = get_lapack_funcs(('geev', 'geev_lwork'), (a1,))
compute_vl, compute_vr = left, right
lwork = _compute_lwork(geev_lwork, a1.shape[0],
compute_vl=compute_vl,
compute_vr=compute_vr)
if geev.typecode in 'cz':
w, vl, vr, info = geev(a1, lwork=lwork,
compute_vl=compute_vl,
compute_vr=compute_vr,
overwrite_a=overwrite_a)
w = _make_eigvals(w, None, homogeneous_eigvals)
else:
wr, wi, vl, vr, info = geev(a1, lwork=lwork,
compute_vl=compute_vl,
compute_vr=compute_vr,
overwrite_a=overwrite_a)
t = {'f': 'F', 'd': 'D'}[wr.dtype.char]
w = wr + _I * wi
w = _make_eigvals(w, None, homogeneous_eigvals)
_check_info(info, 'eig algorithm (geev)',
positive='did not converge (only eigenvalues '
'with order >= %d have converged)')
only_real = numpy.all(w.imag == 0.0)
if not (geev.typecode in 'cz' or only_real):
t = w.dtype.char
if left:
vl = _make_complex_eigvecs(w, vl, t)
if right:
vr = _make_complex_eigvecs(w, vr, t)
if not (left or right):
return w
if left:
if right:
return w, vl, vr
return w, vl
return w, vr
def eigh(a, b=None, lower=True, eigvals_only=False, overwrite_a=False,
overwrite_b=False, turbo=True, eigvals=None, type=1,
check_finite=True, subset_by_index=None, subset_by_value=None,
driver=None):
"""
Solve a standard or generalized eigenvalue problem for a complex
Hermitian or real symmetric matrix.
Find eigenvalues array ``w`` and optionally eigenvectors array ``v`` of
array ``a``, where ``b`` is positive definite such that for every
eigenvalue λ (i-th entry of w) and its eigenvector ``vi`` (i-th column of
``v``) satisfies::
a @ vi = λ * b @ vi
vi.conj().T @ a @ vi = λ
vi.conj().T @ b @ vi = 1
In the standard problem, ``b`` is assumed to be the identity matrix.
Parameters
----------
a : (M, M) array_like
A complex Hermitian or real symmetric matrix whose eigenvalues and
eigenvectors will be computed.
b : (M, M) array_like, optional
A complex Hermitian or real symmetric definite positive matrix in.
If omitted, identity matrix is assumed.
lower : bool, optional
Whether the pertinent array data is taken from the lower or upper
triangle of ``a`` and, if applicable, ``b``. (Default: lower)
eigvals_only : bool, optional
Whether to calculate only eigenvalues and no eigenvectors.
(Default: both are calculated)
subset_by_index : iterable, optional
If provided, this two-element iterable defines the start and the end
indices of the desired eigenvalues (ascending order and 0-indexed).
To return only the second smallest to fifth smallest eigenvalues,
``[1, 4]`` is used. ``[n-3, n-1]`` returns the largest three. Only
available with "evr", "evx", and "gvx" drivers. The entries are
directly converted to integers via ``int()``.
subset_by_value : iterable, optional
If provided, this two-element iterable defines the half-open interval
``(a, b]`` that, if any, only the eigenvalues between these values
are returned. Only available with "evr", "evx", and "gvx" drivers. Use
``np.inf`` for the unconstrained ends.
driver : str, optional
Defines which LAPACK driver should be used. Valid options are "ev",
"evd", "evr", "evx" for standard problems and "gv", "gvd", "gvx" for
generalized (where b is not None) problems. See the Notes section.
The default for standard problems is "evr". For generalized problems,
"gvd" is used for full set, and "gvx" for subset requested cases.
type : int, optional
For the generalized problems, this keyword specifies the problem type
to be solved for ``w`` and ``v`` (only takes 1, 2, 3 as possible
inputs)::
1 => a @ v = w @ b @ v
2 => a @ b @ v = w @ v
3 => b @ a @ v = w @ v
This keyword is ignored for standard problems.
overwrite_a : bool, optional
Whether to overwrite data in ``a`` (may improve performance). Default
is False.
overwrite_b : bool, optional
Whether to overwrite data in ``b`` (may improve 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.
turbo : bool, optional
*Deprecated since v1.5.0, use ``driver=gvd`` keyword instead*.
Use divide and conquer algorithm (faster but expensive in memory, only
for generalized eigenvalue problem and if full set of eigenvalues are
requested.). Has no significant effect if eigenvectors are not
requested.
eigvals : tuple (lo, hi), optional
*Deprecated since v1.5.0, use ``subset_by_index`` keyword instead*.
Indexes of the smallest and largest (in ascending order) eigenvalues
and corresponding eigenvectors to be returned: 0 <= lo <= hi <= M-1.
If omitted, all eigenvalues and eigenvectors are returned.
Returns
-------
w : (N,) ndarray
The N (1<=N<=M) selected eigenvalues, in ascending order, each
repeated according to its multiplicity.
v : (M, N) ndarray
(if ``eigvals_only == False``)
Raises
------
LinAlgError
If eigenvalue computation does not converge, an error occurred, or
b matrix is not definite positive. Note that if input matrices are
not symmetric or Hermitian, no error will be reported but results will
be wrong.
See Also
--------
eigvalsh : eigenvalues of symmetric or Hermitian arrays
eig : eigenvalues and right eigenvectors for non-symmetric arrays
eigh_tridiagonal : eigenvalues and right eiegenvectors for
symmetric/Hermitian tridiagonal matrices
Notes
-----
This function does not check the input array for being Hermitian/symmetric
in order to allow for representing arrays with only their upper/lower
triangular parts. Also, note that even though not taken into account,
finiteness check applies to the whole array and unaffected by "lower"
keyword.
This function uses LAPACK drivers for computations in all possible keyword
combinations, prefixed with ``sy`` if arrays are real and ``he`` if
complex, e.g., a float array with "evr" driver is solved via
"syevr", complex arrays with "gvx" driver problem is solved via "hegvx"
etc.
As a brief summary, the slowest and the most robust driver is the
classical ``<sy/he>ev`` which uses symmetric QR. ``<sy/he>evr`` is seen as
the optimal choice for the most general cases. However, there are certain
occasions that ``<sy/he>evd`` computes faster at the expense of more
memory usage. ``<sy/he>evx``, while still being faster than ``<sy/he>ev``,
often performs worse than the rest except when very few eigenvalues are
requested for large arrays though there is still no performance guarantee.
For the generalized problem, normalization with respect to the given
type argument::
type 1 and 3 : v.conj().T @ a @ v = w
type 2 : inv(v).conj().T @ a @ inv(v) = w
type 1 or 2 : v.conj().T @ b @ v = I
type 3 : v.conj().T @ inv(b) @ v = I
Examples
--------
>>> from scipy.linalg import eigh
>>> A = np.array([[6, 3, 1, 5], [3, 0, 5, 1], [1, 5, 6, 2], [5, 1, 2, 2]])
>>> w, v = eigh(A)
>>> np.allclose(A @ v - v @ np.diag(w), np.zeros((4, 4)))
True
Request only the eigenvalues
>>> w = eigh(A, eigvals_only=True)
Request eigenvalues that are less than 10.
>>> A = np.array([[34, -4, -10, -7, 2],
... [-4, 7, 2, 12, 0],
... [-10, 2, 44, 2, -19],
... [-7, 12, 2, 79, -34],
... [2, 0, -19, -34, 29]])
>>> eigh(A, eigvals_only=True, subset_by_value=[-np.inf, 10])
array([6.69199443e-07, 9.11938152e+00])
Request the second smallest eigenvalue and its eigenvector
>>> w, v = eigh(A, subset_by_index=[1, 1])
>>> w
array([9.11938152])
>>> v.shape # only a single column is returned
(5, 1)
"""
# set lower
uplo = 'L' if lower else 'U'
# Set job for Fortran routines
_job = 'N' if eigvals_only else 'V'
drv_str = [None, "ev", "evd", "evr", "evx", "gv", "gvd", "gvx"]
if driver not in drv_str:
raise ValueError('"{}" is unknown. Possible values are "None", "{}".'
''.format(driver, '", "'.join(drv_str[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 "a" matrix')
overwrite_a = overwrite_a or (_datacopied(a1, a))
cplx = True if iscomplexobj(a1) else False
n = a1.shape[0]
drv_args = {'overwrite_a': overwrite_a}
if b is not None:
b1 = _asarray_validated(b, check_finite=check_finite)
overwrite_b = overwrite_b or _datacopied(b1, b)
if len(b1.shape) != 2 or b1.shape[0] != b1.shape[1]:
raise ValueError('expected square "b" matrix')
if b1.shape != a1.shape:
raise ValueError("wrong b dimensions {}, should "
"be {}".format(b1.shape, a1.shape))
if type not in [1, 2, 3]:
raise ValueError('"type" keyword only accepts 1, 2, and 3.')
cplx = True if iscomplexobj(b1) else (cplx or False)
drv_args.update({'overwrite_b': overwrite_b, 'itype': type})
# backwards-compatibility handling
subset_by_index = subset_by_index if (eigvals is None) else eigvals
subset = (subset_by_index is not None) or (subset_by_value is not None)
# Both subsets can't be given
if subset_by_index and subset_by_value:
raise ValueError('Either index or value subset can be requested.')
# Take turbo into account if all conditions are met otherwise ignore
if turbo and b is not None:
driver = 'gvx' if subset else 'gvd'
# Check indices if given
if subset_by_index:
lo, hi = [int(x) for x in subset_by_index]
if not (0 <= lo <= hi < n):
raise ValueError('Requested eigenvalue indices are not valid. '
'Valid range is [0, {}] and start <= end, but '
'start={}, end={} is given'.format(n-1, lo, hi))
# fortran is 1-indexed
drv_args.update({'range': 'I', 'il': lo + 1, 'iu': hi + 1})
if subset_by_value:
lo, hi = subset_by_value
if not (-inf <= lo < hi <= inf):
raise ValueError('Requested eigenvalue bounds are not valid. '
'Valid range is (-inf, inf) and low < high, but '
'low={}, high={} is given'.format(lo, hi))
drv_args.update({'range': 'V', 'vl': lo, 'vu': hi})
# fix prefix for lapack routines
pfx = 'he' if cplx else 'sy'
# decide on the driver if not given
# first early exit on incompatible choice
if driver:
if b is None and (driver in ["gv", "gvd", "gvx"]):
raise ValueError('{} requires input b array to be supplied '
'for generalized eigenvalue problems.'
''.format(driver))
if (b is not None) and (driver in ['ev', 'evd', 'evr', 'evx']):
raise ValueError('"{}" does not accept input b array '
'for standard eigenvalue problems.'
''.format(driver))
if subset and (driver in ["ev", "evd", "gv", "gvd"]):
raise ValueError('"{}" cannot compute subsets of eigenvalues'
''.format(driver))
# Default driver is evr and gvd
else:
driver = "evr" if b is None else ("gvx" if subset else "gvd")
lwork_spec = {
'syevd': ['lwork', 'liwork'],
'syevr': ['lwork', 'liwork'],
'heevd': ['lwork', 'liwork', 'lrwork'],
'heevr': ['lwork', 'lrwork', 'liwork'],
}
if b is None: # Standard problem
drv, drvlw = get_lapack_funcs((pfx + driver, pfx+driver+'_lwork'),
[a1])
clw_args = {'n': n, 'lower': lower}
if driver == 'evd':
clw_args.update({'compute_v': 0 if _job == "N" else 1})
lw = _compute_lwork(drvlw, **clw_args)
# Multiple lwork vars
if isinstance(lw, tuple):
lwork_args = dict(zip(lwork_spec[pfx+driver], lw))
else:
lwork_args = {'lwork': lw}
drv_args.update({'lower': lower, 'compute_v': 0 if _job == "N" else 1})
w, v, *other_args, info = drv(a=a1, **drv_args, **lwork_args)
else: # Generalized problem
# 'gvd' doesn't have lwork query
if driver == "gvd":
drv = get_lapack_funcs(pfx + "gvd", [a1, b1])
lwork_args = {}
else:
drv, drvlw = get_lapack_funcs((pfx + driver, pfx+driver+'_lwork'),
[a1, b1])
# generalized drivers use uplo instead of lower
lw = _compute_lwork(drvlw, n, uplo=uplo)
lwork_args = {'lwork': lw}
drv_args.update({'uplo': uplo, 'jobz': _job})
w, v, *other_args, info = drv(a=a1, b=b1, **drv_args, **lwork_args)
# m is always the first extra argument
w = w[:other_args[0]] if subset else w
v = v[:, :other_args[0]] if (subset and not eigvals_only) else v
# Check if we had a successful exit
if info == 0:
if eigvals_only:
return w
else:
return w, v
else:
if info < -1:
raise LinAlgError('Illegal value in argument {} of internal {}'
''.format(-info, drv.typecode + pfx + driver))
elif info > n:
raise LinAlgError('The leading minor of order {} of B is not '
'positive definite. The factorization of B '
'could not be completed and no eigenvalues '
'or eigenvectors were computed.'.format(info-n))
else:
drv_err = {'ev': 'The algorithm failed to converge; {} '
'off-diagonal elements of an intermediate '
'tridiagonal form did not converge to zero.',
'evx': '{} eigenvectors failed to converge.',
'evd': 'The algorithm failed to compute an eigenvalue '
'while working on the submatrix lying in rows '
'and columns {0}/{1} through mod({0},{1}).',
'evr': 'Internal Error.'
}
if driver in ['ev', 'gv']:
msg = drv_err['ev'].format(info)
elif driver in ['evx', 'gvx']:
msg = drv_err['evx'].format(info)
elif driver in ['evd', 'gvd']:
if eigvals_only:
msg = drv_err['ev'].format(info)
else:
msg = drv_err['evd'].format(info, n+1)
else:
msg = drv_err['evr']
raise LinAlgError(msg)
_conv_dict = {0: 0, 1: 1, 2: 2,
'all': 0, 'value': 1, 'index': 2,
'a': 0, 'v': 1, 'i': 2}
def _check_select(select, select_range, max_ev, max_len):
"""Check that select is valid, convert to Fortran style."""
if isinstance(select, str):
select = select.lower()
try:
select = _conv_dict[select]
except KeyError as e:
raise ValueError('invalid argument for select') from e
vl, vu = 0., 1.
il = iu = 1
if select != 0: # (non-all)
sr = asarray(select_range)
if sr.ndim != 1 or sr.size != 2 or sr[1] < sr[0]:
raise ValueError('select_range must be a 2-element array-like '
'in nondecreasing order')
if select == 1: # (value)
vl, vu = sr
if max_ev == 0:
max_ev = max_len
else: # 2 (index)
if sr.dtype.char.lower() not in 'hilqp':
raise ValueError('when using select="i", select_range must '
'contain integers, got dtype %s (%s)'
% (sr.dtype, sr.dtype.char))
# translate Python (0 ... N-1) into Fortran (1 ... N) with + 1
il, iu = sr + 1
if min(il, iu) < 1 or max(il, iu) > max_len:
raise ValueError('select_range out of bounds')
max_ev = iu - il + 1
return select, vl, vu, il, iu, max_ev
def eig_banded(a_band, lower=False, eigvals_only=False, overwrite_a_band=False,
select='a', select_range=None, max_ev=0, check_finite=True):
"""
Solve real symmetric or complex Hermitian band matrix eigenvalue problem.
Find eigenvalues w and optionally right eigenvectors v of a::
a v[:,i] = w[i] v[:,i]
v.H v = identity
The matrix a is stored in a_band either in lower diagonal or upper
diagonal ordered form:
a_band[u + i - j, j] == a[i,j] (if upper form; i <= j)
a_band[ i - j, j] == a[i,j] (if lower form; i >= j)
where u is the number of bands above the diagonal.
Example of a_band (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
----------
a_band : (u+1, M) array_like
The bands of the M by M matrix a.
lower : bool, optional
Is the matrix in the lower form. (Default is upper form)
eigvals_only : bool, optional
Compute only the eigenvalues and no eigenvectors.
(Default: calculate also eigenvectors)
overwrite_a_band : bool, optional
Discard data in a_band (may enhance performance)
select : {'a', 'v', 'i'}, optional
Which eigenvalues to calculate
====== ========================================
select calculated
====== ========================================
'a' All eigenvalues
'v' Eigenvalues in the interval (min, max]
'i' Eigenvalues with indices min <= i <= max
====== ========================================
select_range : (min, max), optional
Range of selected eigenvalues
max_ev : int, optional
For select=='v', maximum number of eigenvalues expected.
For other values of select, has no meaning.
In doubt, leave this parameter untouched.
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
-------
w : (M,) ndarray
The eigenvalues, in ascending order, each repeated according to its
multiplicity.
v : (M, M) float or complex ndarray
The normalized eigenvector corresponding to the eigenvalue w[i] is
the column v[:,i].
Raises
------
LinAlgError
If eigenvalue computation does not converge.
See Also
--------
eigvals_banded : eigenvalues for symmetric/Hermitian band matrices
eig : eigenvalues and right eigenvectors of general arrays.
eigh : eigenvalues and right eigenvectors for symmetric/Hermitian arrays
eigh_tridiagonal : eigenvalues and right eigenvectors for
symmetric/Hermitian tridiagonal matrices
Examples
--------
>>> from scipy.linalg import eig_banded
>>> A = np.array([[1, 5, 2, 0], [5, 2, 5, 2], [2, 5, 3, 5], [0, 2, 5, 4]])
>>> Ab = np.array([[1, 2, 3, 4], [5, 5, 5, 0], [2, 2, 0, 0]])
>>> w, v = eig_banded(Ab, lower=True)
>>> np.allclose(A @ v - v @ np.diag(w), np.zeros((4, 4)))
True
>>> w = eig_banded(Ab, lower=True, eigvals_only=True)
>>> w
array([-4.26200532, -2.22987175, 3.95222349, 12.53965359])
Request only the eigenvalues between ``[-3, 4]``
>>> w, v = eig_banded(Ab, lower=True, select='v', select_range=[-3, 4])
>>> w
array([-2.22987175, 3.95222349])
"""
if eigvals_only or overwrite_a_band:
a1 = _asarray_validated(a_band, check_finite=check_finite)
overwrite_a_band = overwrite_a_band or (_datacopied(a1, a_band))
else:
a1 = array(a_band)
if issubclass(a1.dtype.type, inexact) and not isfinite(a1).all():
raise ValueError("array must not contain infs or NaNs")
overwrite_a_band = 1
if len(a1.shape) != 2:
raise ValueError('expected a 2-D array')
select, vl, vu, il, iu, max_ev = _check_select(
select, select_range, max_ev, a1.shape[1])
del select_range
if select == 0:
if a1.dtype.char in 'GFD':
# FIXME: implement this somewhen, for now go with builtin values
# FIXME: calc optimal lwork by calling ?hbevd(lwork=-1)
# or by using calc_lwork.f ???
# lwork = calc_lwork.hbevd(bevd.typecode, a1.shape[0], lower)
internal_name = 'hbevd'
else: # a1.dtype.char in 'fd':
# FIXME: implement this somewhen, for now go with builtin values
# see above
# lwork = calc_lwork.sbevd(bevd.typecode, a1.shape[0], lower)
internal_name = 'sbevd'
bevd, = get_lapack_funcs((internal_name,), (a1,))
w, v, info = bevd(a1, compute_v=not eigvals_only,
lower=lower, overwrite_ab=overwrite_a_band)
else: # select in [1, 2]
if eigvals_only:
max_ev = 1
# calculate optimal abstol for dsbevx (see manpage)
if a1.dtype.char in 'fF': # single precision
lamch, = get_lapack_funcs(('lamch',), (array(0, dtype='f'),))
else:
lamch, = get_lapack_funcs(('lamch',), (array(0, dtype='d'),))
abstol = 2 * lamch('s')
if a1.dtype.char in 'GFD':
internal_name = 'hbevx'
else: # a1.dtype.char in 'gfd'
internal_name = 'sbevx'
bevx, = get_lapack_funcs((internal_name,), (a1,))
w, v, m, ifail, info = bevx(
a1, vl, vu, il, iu, compute_v=not eigvals_only, mmax=max_ev,
range=select, lower=lower, overwrite_ab=overwrite_a_band,
abstol=abstol)
# crop off w and v
w = w[:m]
if not eigvals_only:
v = v[:, :m]
_check_info(info, internal_name)
if eigvals_only:
return w
return w, v
[docs]def eigvals(a, b=None, overwrite_a=False, check_finite=True,
homogeneous_eigvals=False):
"""
Compute eigenvalues from an ordinary or generalized eigenvalue problem.
Find eigenvalues of a general matrix::
a vr[:,i] = w[i] b vr[:,i]
Parameters
----------
a : (M, M) array_like
A complex or real matrix whose eigenvalues and eigenvectors
will be computed.
b : (M, M) array_like, optional
Right-hand side matrix in a generalized eigenvalue problem.
If omitted, identity matrix is assumed.
overwrite_a : bool, optional
Whether to overwrite data in a (may improve performance)
check_finite : bool, optional
Whether to check that the input matrices contain only finite numbers.
Disabling may give a performance gain, but may result in problems
(crashes, non-termination) if the inputs do contain infinities
or NaNs.
homogeneous_eigvals : bool, optional
If True, return the eigenvalues in homogeneous coordinates.
In this case ``w`` is a (2, M) array so that::
w[1,i] a vr[:,i] = w[0,i] b vr[:,i]
Default is False.
Returns
-------
w : (M,) or (2, M) double or complex ndarray
The eigenvalues, each repeated according to its multiplicity
but not in any specific order. The shape is (M,) unless
``homogeneous_eigvals=True``.
Raises
------
LinAlgError
If eigenvalue computation does not converge
See Also
--------
eig : eigenvalues and right eigenvectors of general arrays.
eigvalsh : eigenvalues of symmetric or Hermitian arrays
eigvals_banded : eigenvalues for symmetric/Hermitian band matrices
eigvalsh_tridiagonal : eigenvalues of symmetric/Hermitian tridiagonal
matrices
Examples
--------
>>> from scipy import linalg
>>> a = np.array([[0., -1.], [1., 0.]])
>>> linalg.eigvals(a)
array([0.+1.j, 0.-1.j])
>>> b = np.array([[0., 1.], [1., 1.]])
>>> linalg.eigvals(a, b)
array([ 1.+0.j, -1.+0.j])
>>> a = np.array([[3., 0., 0.], [0., 8., 0.], [0., 0., 7.]])
>>> linalg.eigvals(a, homogeneous_eigvals=True)
array([[3.+0.j, 8.+0.j, 7.+0.j],
[1.+0.j, 1.+0.j, 1.+0.j]])
"""
return eig(a, b=b, left=0, right=0, overwrite_a=overwrite_a,
check_finite=check_finite,
homogeneous_eigvals=homogeneous_eigvals)
def eigvalsh(a, b=None, lower=True, overwrite_a=False,
overwrite_b=False, turbo=True, eigvals=None, type=1,
check_finite=True, subset_by_index=None, subset_by_value=None,
driver=None):
"""
Solves a standard or generalized eigenvalue problem for a complex
Hermitian or real symmetric matrix.
Find eigenvalues array ``w`` of array ``a``, where ``b`` is positive
definite such that for every eigenvalue λ (i-th entry of w) and its
eigenvector vi (i-th column of v) satisfies::
a @ vi = λ * b @ vi
vi.conj().T @ a @ vi = λ
vi.conj().T @ b @ vi = 1
In the standard problem, b is assumed to be the identity matrix.
Parameters
----------
a : (M, M) array_like
A complex Hermitian or real symmetric matrix whose eigenvalues will
be computed.
b : (M, M) array_like, optional
A complex Hermitian or real symmetric definite positive matrix in.
If omitted, identity matrix is assumed.
lower : bool, optional
Whether the pertinent array data is taken from the lower or upper
triangle of ``a`` and, if applicable, ``b``. (Default: lower)
overwrite_a : bool, optional
Whether to overwrite data in ``a`` (may improve performance). Default
is False.
overwrite_b : bool, optional
Whether to overwrite data in ``b`` (may improve performance). Default
is False.
type : int, optional
For the generalized problems, this keyword specifies the problem type
to be solved for ``w`` and ``v`` (only takes 1, 2, 3 as possible
inputs)::
1 => a @ v = w @ b @ v
2 => a @ b @ v = w @ v
3 => b @ a @ v = w @ v
This keyword is ignored for standard problems.
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.
subset_by_index : iterable, optional
If provided, this two-element iterable defines the start and the end
indices of the desired eigenvalues (ascending order and 0-indexed).
To return only the second smallest to fifth smallest eigenvalues,
``[1, 4]`` is used. ``[n-3, n-1]`` returns the largest three. Only
available with "evr", "evx", and "gvx" drivers. The entries are
directly converted to integers via ``int()``.
subset_by_value : iterable, optional
If provided, this two-element iterable defines the half-open interval
``(a, b]`` that, if any, only the eigenvalues between these values
are returned. Only available with "evr", "evx", and "gvx" drivers. Use
``np.inf`` for the unconstrained ends.
driver : str, optional
Defines which LAPACK driver should be used. Valid options are "ev",
"evd", "evr", "evx" for standard problems and "gv", "gvd", "gvx" for
generalized (where b is not None) problems. See the Notes section of
`scipy.linalg.eigh`.
turbo : bool, optional
*Deprecated by ``driver=gvd`` option*. Has no significant effect for
eigenvalue computations since no eigenvectors are requested.
.. deprecated:: 1.5.0
eigvals : tuple (lo, hi), optional
*Deprecated by ``subset_by_index`` keyword*. Indexes of the smallest
and largest (in ascending order) eigenvalues and corresponding
eigenvectors to be returned: 0 <= lo <= hi <= M-1. If omitted, all
eigenvalues and eigenvectors are returned.
.. deprecated:: 1.5.0
Returns
-------
w : (N,) ndarray
The ``N`` (``1<=N<=M``) selected eigenvalues, in ascending order, each
repeated according to its multiplicity.
Raises
------
LinAlgError
If eigenvalue computation does not converge, an error occurred, or
b matrix is not definite positive. Note that if input matrices are
not symmetric or Hermitian, no error will be reported but results will
be wrong.
See Also
--------
eigh : eigenvalues and right eigenvectors for symmetric/Hermitian arrays
eigvals : eigenvalues of general arrays
eigvals_banded : eigenvalues for symmetric/Hermitian band matrices
eigvalsh_tridiagonal : eigenvalues of symmetric/Hermitian tridiagonal
matrices
Notes
-----
This function does not check the input array for being Hermitian/symmetric
in order to allow for representing arrays with only their upper/lower
triangular parts.
This function serves as a one-liner shorthand for `scipy.linalg.eigh` with
the option ``eigvals_only=True`` to get the eigenvalues and not the
eigenvectors. Here it is kept as a legacy convenience. It might be
beneficial to use the main function to have full control and to be a bit
more pythonic.
Examples
--------
For more examples see `scipy.linalg.eigh`.
>>> from scipy.linalg import eigvalsh
>>> A = np.array([[6, 3, 1, 5], [3, 0, 5, 1], [1, 5, 6, 2], [5, 1, 2, 2]])
>>> w = eigvalsh(A)
>>> w
array([-3.74637491, -0.76263923, 6.08502336, 12.42399079])
"""
return eigh(a, b=b, lower=lower, eigvals_only=True,
overwrite_a=overwrite_a, overwrite_b=overwrite_b,
turbo=turbo, eigvals=eigvals, type=type,
check_finite=check_finite, subset_by_index=subset_by_index,
subset_by_value=subset_by_value, driver=driver)
def eigvals_banded(a_band, lower=False, overwrite_a_band=False,
select='a', select_range=None, check_finite=True):
"""
Solve real symmetric or complex Hermitian band matrix eigenvalue problem.
Find eigenvalues w of a::
a v[:,i] = w[i] v[:,i]
v.H v = identity
The matrix a is stored in a_band either in lower diagonal or upper
diagonal ordered form:
a_band[u + i - j, j] == a[i,j] (if upper form; i <= j)
a_band[ i - j, j] == a[i,j] (if lower form; i >= j)
where u is the number of bands above the diagonal.
Example of a_band (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
----------
a_band : (u+1, M) array_like
The bands of the M by M matrix a.
lower : bool, optional
Is the matrix in the lower form. (Default is upper form)
overwrite_a_band : bool, optional
Discard data in a_band (may enhance performance)
select : {'a', 'v', 'i'}, optional
Which eigenvalues to calculate
====== ========================================
select calculated
====== ========================================
'a' All eigenvalues
'v' Eigenvalues in the interval (min, max]
'i' Eigenvalues with indices min <= i <= max
====== ========================================
select_range : (min, max), optional
Range of selected eigenvalues
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
-------
w : (M,) ndarray
The eigenvalues, in ascending order, each repeated according to its
multiplicity.
Raises
------
LinAlgError
If eigenvalue computation does not converge.
See Also
--------
eig_banded : eigenvalues and right eigenvectors for symmetric/Hermitian
band matrices
eigvalsh_tridiagonal : eigenvalues of symmetric/Hermitian tridiagonal
matrices
eigvals : eigenvalues of general arrays
eigh : eigenvalues and right eigenvectors for symmetric/Hermitian arrays
eig : eigenvalues and right eigenvectors for non-symmetric arrays
Examples
--------
>>> from scipy.linalg import eigvals_banded
>>> A = np.array([[1, 5, 2, 0], [5, 2, 5, 2], [2, 5, 3, 5], [0, 2, 5, 4]])
>>> Ab = np.array([[1, 2, 3, 4], [5, 5, 5, 0], [2, 2, 0, 0]])
>>> w = eigvals_banded(Ab, lower=True)
>>> w
array([-4.26200532, -2.22987175, 3.95222349, 12.53965359])
"""
return eig_banded(a_band, lower=lower, eigvals_only=1,
overwrite_a_band=overwrite_a_band, select=select,
select_range=select_range, check_finite=check_finite)
def eigvalsh_tridiagonal(d, e, select='a', select_range=None,
check_finite=True, tol=0., lapack_driver='auto'):
"""
Solve eigenvalue problem for a real symmetric tridiagonal matrix.
Find eigenvalues `w` of ``a``::
a v[:,i] = w[i] v[:,i]
v.H v = identity
For a real symmetric matrix ``a`` with diagonal elements `d` and
off-diagonal elements `e`.
Parameters
----------
d : ndarray, shape (ndim,)
The diagonal elements of the array.
e : ndarray, shape (ndim-1,)
The off-diagonal elements of the array.
select : {'a', 'v', 'i'}, optional
Which eigenvalues to calculate
====== ========================================
select calculated
====== ========================================
'a' All eigenvalues
'v' Eigenvalues in the interval (min, max]
'i' Eigenvalues with indices min <= i <= max
====== ========================================
select_range : (min, max), optional
Range of selected eigenvalues
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.
tol : float
The absolute tolerance to which each eigenvalue is required
(only used when ``lapack_driver='stebz'``).
An eigenvalue (or cluster) is considered to have converged if it
lies in an interval of this width. If <= 0. (default),
the value ``eps*|a|`` is used where eps is the machine precision,
and ``|a|`` is the 1-norm of the matrix ``a``.
lapack_driver : str
LAPACK function to use, can be 'auto', 'stemr', 'stebz', 'sterf',
or 'stev'. When 'auto' (default), it will use 'stemr' if ``select='a'``
and 'stebz' otherwise. 'sterf' and 'stev' can only be used when
``select='a'``.
Returns
-------
w : (M,) ndarray
The eigenvalues, in ascending order, each repeated according to its
multiplicity.
Raises
------
LinAlgError
If eigenvalue computation does not converge.
See Also
--------
eigh_tridiagonal : eigenvalues and right eiegenvectors for
symmetric/Hermitian tridiagonal matrices
Examples
--------
>>> from scipy.linalg import eigvalsh_tridiagonal, eigvalsh
>>> d = 3*np.ones(4)
>>> e = -1*np.ones(3)
>>> w = eigvalsh_tridiagonal(d, e)
>>> A = np.diag(d) + np.diag(e, k=1) + np.diag(e, k=-1)
>>> w2 = eigvalsh(A) # Verify with other eigenvalue routines
>>> np.allclose(w - w2, np.zeros(4))
True
"""
return eigh_tridiagonal(
d, e, eigvals_only=True, select=select, select_range=select_range,
check_finite=check_finite, tol=tol, lapack_driver=lapack_driver)
def eigh_tridiagonal(d, e, eigvals_only=False, select='a', select_range=None,
check_finite=True, tol=0., lapack_driver='auto'):
"""
Solve eigenvalue problem for a real symmetric tridiagonal matrix.
Find eigenvalues `w` and optionally right eigenvectors `v` of ``a``::
a v[:,i] = w[i] v[:,i]
v.H v = identity
For a real symmetric matrix ``a`` with diagonal elements `d` and
off-diagonal elements `e`.
Parameters
----------
d : ndarray, shape (ndim,)
The diagonal elements of the array.
e : ndarray, shape (ndim-1,)
The off-diagonal elements of the array.
select : {'a', 'v', 'i'}, optional
Which eigenvalues to calculate
====== ========================================
select calculated
====== ========================================
'a' All eigenvalues
'v' Eigenvalues in the interval (min, max]
'i' Eigenvalues with indices min <= i <= max
====== ========================================
select_range : (min, max), optional
Range of selected eigenvalues
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.
tol : float
The absolute tolerance to which each eigenvalue is required
(only used when 'stebz' is the `lapack_driver`).
An eigenvalue (or cluster) is considered to have converged if it
lies in an interval of this width. If <= 0. (default),
the value ``eps*|a|`` is used where eps is the machine precision,
and ``|a|`` is the 1-norm of the matrix ``a``.
lapack_driver : str
LAPACK function to use, can be 'auto', 'stemr', 'stebz', 'sterf',
or 'stev'. When 'auto' (default), it will use 'stemr' if ``select='a'``
and 'stebz' otherwise. When 'stebz' is used to find the eigenvalues and
``eigvals_only=False``, then a second LAPACK call (to ``?STEIN``) is
used to find the corresponding eigenvectors. 'sterf' can only be
used when ``eigvals_only=True`` and ``select='a'``. 'stev' can only
be used when ``select='a'``.
Returns
-------
w : (M,) ndarray
The eigenvalues, in ascending order, each repeated according to its
multiplicity.
v : (M, M) ndarray
The normalized eigenvector corresponding to the eigenvalue ``w[i]`` is
the column ``v[:,i]``.
Raises
------
LinAlgError
If eigenvalue computation does not converge.
See Also
--------
eigvalsh_tridiagonal : eigenvalues of symmetric/Hermitian tridiagonal
matrices
eig : eigenvalues and right eigenvectors for non-symmetric arrays
eigh : eigenvalues and right eigenvectors for symmetric/Hermitian arrays
eig_banded : eigenvalues and right eigenvectors for symmetric/Hermitian
band matrices
Notes
-----
This function makes use of LAPACK ``S/DSTEMR`` routines.
Examples
--------
>>> from scipy.linalg import eigh_tridiagonal
>>> d = 3*np.ones(4)
>>> e = -1*np.ones(3)
>>> w, v = eigh_tridiagonal(d, e)
>>> A = np.diag(d) + np.diag(e, k=1) + np.diag(e, k=-1)
>>> np.allclose(A @ v - v @ np.diag(w), np.zeros((4, 4)))
True
"""
d = _asarray_validated(d, check_finite=check_finite)
e = _asarray_validated(e, check_finite=check_finite)
for check in (d, e):
if check.ndim != 1:
raise ValueError('expected a 1-D array')
if check.dtype.char in 'GFD': # complex
raise TypeError('Only real arrays currently supported')
if d.size != e.size + 1:
raise ValueError('d (%s) must have one more element than e (%s)'
% (d.size, e.size))
select, vl, vu, il, iu, _ = _check_select(
select, select_range, 0, d.size)
if not isinstance(lapack_driver, str):
raise TypeError('lapack_driver must be str')
drivers = ('auto', 'stemr', 'sterf', 'stebz', 'stev')
if lapack_driver not in drivers:
raise ValueError('lapack_driver must be one of %s, got %s'
% (drivers, lapack_driver))
if lapack_driver == 'auto':
lapack_driver = 'stemr' if select == 0 else 'stebz'
func, = get_lapack_funcs((lapack_driver,), (d, e))
compute_v = not eigvals_only
if lapack_driver == 'sterf':
if select != 0:
raise ValueError('sterf can only be used when select == "a"')
if not eigvals_only:
raise ValueError('sterf can only be used when eigvals_only is '
'True')
w, info = func(d, e)
m = len(w)
elif lapack_driver == 'stev':
if select != 0:
raise ValueError('stev can only be used when select == "a"')
w, v, info = func(d, e, compute_v=compute_v)
m = len(w)
elif lapack_driver == 'stebz':
tol = float(tol)
internal_name = 'stebz'
stebz, = get_lapack_funcs((internal_name,), (d, e))
# If getting eigenvectors, needs to be block-ordered (B) instead of
# matrix-ordered (E), and we will reorder later
order = 'E' if eigvals_only else 'B'
m, w, iblock, isplit, info = stebz(d, e, select, vl, vu, il, iu, tol,
order)
else: # 'stemr'
# ?STEMR annoyingly requires size N instead of N-1
e_ = empty(e.size+1, e.dtype)
e_[:-1] = e
stemr_lwork, = get_lapack_funcs(('stemr_lwork',), (d, e))
lwork, liwork, info = stemr_lwork(d, e_, select, vl, vu, il, iu,
compute_v=compute_v)
_check_info(info, 'stemr_lwork')
m, w, v, info = func(d, e_, select, vl, vu, il, iu,
compute_v=compute_v, lwork=lwork, liwork=liwork)
_check_info(info, lapack_driver + ' (eigh_tridiagonal)')
w = w[:m]
if eigvals_only:
return w
else:
# Do we still need to compute the eigenvalues?
if lapack_driver == 'stebz':
func, = get_lapack_funcs(('stein',), (d, e))
v, info = func(d, e, w, iblock, isplit)
_check_info(info, 'stein (eigh_tridiagonal)',
positive='%d eigenvectors failed to converge')
# Convert block-order to matrix-order
order = argsort(w)
w, v = w[order], v[:, order]
else:
v = v[:, :m]
return w, v
def _check_info(info, driver, positive='did not converge (LAPACK info=%d)'):
"""Check info return value."""
if info < 0:
raise ValueError('illegal value in argument %d of internal %s'
% (-info, driver))
if info > 0 and positive:
raise LinAlgError(("%s " + positive) % (driver, info,))
def hessenberg(a, calc_q=False, overwrite_a=False, check_finite=True):
"""
Compute Hessenberg form of a matrix.
The Hessenberg decomposition is::
A = Q H Q^H
where `Q` is unitary/orthogonal and `H` has only zero elements below
the first sub-diagonal.
Parameters
----------
a : (M, M) array_like
Matrix to bring into Hessenberg form.
calc_q : bool, optional
Whether to compute the transformation matrix. Default is False.
overwrite_a : bool, optional
Whether to overwrite `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
-------
H : (M, M) ndarray
Hessenberg form of `a`.
Q : (M, M) ndarray
Unitary/orthogonal similarity transformation matrix ``A = Q H Q^H``.
Only returned if ``calc_q=True``.
Examples
--------
>>> from scipy.linalg import hessenberg
>>> A = np.array([[2, 5, 8, 7], [5, 2, 2, 8], [7, 5, 6, 6], [5, 4, 4, 8]])
>>> H, Q = hessenberg(A, calc_q=True)
>>> H
array([[ 2. , -11.65843866, 1.42005301, 0.25349066],
[ -9.94987437, 14.53535354, -5.31022304, 2.43081618],
[ 0. , -1.83299243, 0.38969961, -0.51527034],
[ 0. , 0. , -3.83189513, 1.07494686]])
>>> np.allclose(Q @ H @ Q.conj().T - A, np.zeros((4, 4)))
True
"""
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))
# if 2x2 or smaller: already in Hessenberg
if a1.shape[0] <= 2:
if calc_q:
return a1, eye(a1.shape[0])
return a1
gehrd, gebal, gehrd_lwork = get_lapack_funcs(('gehrd', 'gebal',
'gehrd_lwork'), (a1,))
ba, lo, hi, pivscale, info = gebal(a1, permute=0, overwrite_a=overwrite_a)
_check_info(info, 'gebal (hessenberg)', positive=False)
n = len(a1)
lwork = _compute_lwork(gehrd_lwork, ba.shape[0], lo=lo, hi=hi)
hq, tau, info = gehrd(ba, lo=lo, hi=hi, lwork=lwork, overwrite_a=1)
_check_info(info, 'gehrd (hessenberg)', positive=False)
h = numpy.triu(hq, -1)
if not calc_q:
return h
# use orghr/unghr to compute q
orghr, orghr_lwork = get_lapack_funcs(('orghr', 'orghr_lwork'), (a1,))
lwork = _compute_lwork(orghr_lwork, n, lo=lo, hi=hi)
q, info = orghr(a=hq, tau=tau, lo=lo, hi=hi, lwork=lwork, overwrite_a=1)
_check_info(info, 'orghr (hessenberg)', positive=False)
return h, q
def cdf2rdf(w, v):
"""
Converts complex eigenvalues ``w`` and eigenvectors ``v`` to real
eigenvalues in a block diagonal form ``wr`` and the associated real
eigenvectors ``vr``, such that::
vr @ wr = X @ vr
continues to hold, where ``X`` is the original array for which ``w`` and
``v`` are the eigenvalues and eigenvectors.
.. versionadded:: 1.1.0
Parameters
----------
w : (..., M) array_like
Complex or real eigenvalues, an array or stack of arrays
Conjugate pairs must not be interleaved, else the wrong result
will be produced. So ``[1+1j, 1, 1-1j]`` will give a correct result,
but ``[1+1j, 2+1j, 1-1j, 2-1j]`` will not.
v : (..., M, M) array_like
Complex or real eigenvectors, a square array or stack of square arrays.
Returns
-------
wr : (..., M, M) ndarray
Real diagonal block form of eigenvalues
vr : (..., M, M) ndarray
Real eigenvectors associated with ``wr``
See Also
--------
eig : Eigenvalues and right eigenvectors for non-symmetric arrays
rsf2csf : Convert real Schur form to complex Schur form
Notes
-----
``w``, ``v`` must be the eigenstructure for some *real* matrix ``X``.
For example, obtained by ``w, v = scipy.linalg.eig(X)`` or
``w, v = numpy.linalg.eig(X)`` in which case ``X`` can also represent
stacked arrays.
.. versionadded:: 1.1.0
Examples
--------
>>> X = np.array([[1, 2, 3], [0, 4, 5], [0, -5, 4]])
>>> X
array([[ 1, 2, 3],
[ 0, 4, 5],
[ 0, -5, 4]])
>>> from scipy import linalg
>>> w, v = linalg.eig(X)
>>> w
array([ 1.+0.j, 4.+5.j, 4.-5.j])
>>> v
array([[ 1.00000+0.j , -0.01906-0.40016j, -0.01906+0.40016j],
[ 0.00000+0.j , 0.00000-0.64788j, 0.00000+0.64788j],
[ 0.00000+0.j , 0.64788+0.j , 0.64788-0.j ]])
>>> wr, vr = linalg.cdf2rdf(w, v)
>>> wr
array([[ 1., 0., 0.],
[ 0., 4., 5.],
[ 0., -5., 4.]])
>>> vr
array([[ 1. , 0.40016, -0.01906],
[ 0. , 0.64788, 0. ],
[ 0. , 0. , 0.64788]])
>>> vr @ wr
array([[ 1. , 1.69593, 1.9246 ],
[ 0. , 2.59153, 3.23942],
[ 0. , -3.23942, 2.59153]])
>>> X @ vr
array([[ 1. , 1.69593, 1.9246 ],
[ 0. , 2.59153, 3.23942],
[ 0. , -3.23942, 2.59153]])
"""
w, v = _asarray_validated(w), _asarray_validated(v)
# check dimensions
if w.ndim < 1:
raise ValueError('expected w to be at least 1D')
if v.ndim < 2:
raise ValueError('expected v to be at least 2D')
if v.ndim != w.ndim + 1:
raise ValueError('expected eigenvectors array to have exactly one '
'dimension more than eigenvalues array')
# check shapes
n = w.shape[-1]
M = w.shape[:-1]
if v.shape[-2] != v.shape[-1]:
raise ValueError('expected v to be a square matrix or stacked square '
'matrices: v.shape[-2] = v.shape[-1]')
if v.shape[-1] != n:
raise ValueError('expected the same number of eigenvalues as '
'eigenvectors')
# get indices for each first pair of complex eigenvalues
complex_mask = iscomplex(w)
n_complex = complex_mask.sum(axis=-1)
# check if all complex eigenvalues have conjugate pairs
if not (n_complex % 2 == 0).all():
raise ValueError('expected complex-conjugate pairs of eigenvalues')
# find complex indices
idx = nonzero(complex_mask)
idx_stack = idx[:-1]
idx_elem = idx[-1]
# filter them to conjugate indices, assuming pairs are not interleaved
j = idx_elem[0::2]
k = idx_elem[1::2]
stack_ind = ()
for i in idx_stack:
# should never happen, assuming nonzero orders by the last axis
assert (i[0::2] == i[1::2]).all(),\
"Conjugate pair spanned different arrays!"
stack_ind += (i[0::2],)
# all eigenvalues to diagonal form
wr = zeros(M + (n, n), dtype=w.real.dtype)
di = range(n)
wr[..., di, di] = w.real
# complex eigenvalues to real block diagonal form
wr[stack_ind + (j, k)] = w[stack_ind + (j,)].imag
wr[stack_ind + (k, j)] = w[stack_ind + (k,)].imag
# compute real eigenvectors associated with real block diagonal eigenvalues
u = zeros(M + (n, n), dtype=numpy.cdouble)
u[..., di, di] = 1.0
u[stack_ind + (j, j)] = 0.5j
u[stack_ind + (j, k)] = 0.5
u[stack_ind + (k, j)] = -0.5j
u[stack_ind + (k, k)] = 0.5
# multipy matrices v and u (equivalent to v @ u)
vr = einsum('...ij,...jk->...ik', v, u).real
return wr, vr