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1432 lines
52 KiB
Python
1432 lines
52 KiB
Python
5 years ago
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#
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# Author: Pearu Peterson, March 2002
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#
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# additions by Travis Oliphant, March 2002
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# additions by Eric Jones, June 2002
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# additions by Johannes Loehnert, June 2006
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# additions by Bart Vandereycken, June 2006
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# additions by Andrew D Straw, May 2007
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# additions by Tiziano Zito, November 2008
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#
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# April 2010: Functions for LU, QR, SVD, Schur and Cholesky decompositions were
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# moved to their own files. Still in this file are functions for eigenstuff
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# and for the Hessenberg form.
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from __future__ import division, print_function, absolute_import
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__all__ = ['eig', 'eigvals', 'eigh', 'eigvalsh',
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'eig_banded', 'eigvals_banded',
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'eigh_tridiagonal', 'eigvalsh_tridiagonal', 'hessenberg', 'cdf2rdf']
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import numpy
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from numpy import (array, isfinite, inexact, nonzero, iscomplexobj, cast,
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flatnonzero, conj, asarray, argsort, empty, newaxis,
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argwhere, iscomplex, eye, zeros, einsum)
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# Local imports
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from scipy._lib.six import xrange
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from scipy._lib._util import _asarray_validated
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from scipy._lib.six import string_types
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from .misc import LinAlgError, _datacopied, norm
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from .lapack import get_lapack_funcs, _compute_lwork
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_I = cast['F'](1j)
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def _make_complex_eigvecs(w, vin, dtype):
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"""
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Produce complex-valued eigenvectors from LAPACK DGGEV real-valued output
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"""
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# - see LAPACK man page DGGEV at ALPHAI
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v = numpy.array(vin, dtype=dtype)
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m = (w.imag > 0)
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m[:-1] |= (w.imag[1:] < 0) # workaround for LAPACK bug, cf. ticket #709
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for i in flatnonzero(m):
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v.imag[:, i] = vin[:, i+1]
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conj(v[:, i], v[:, i+1])
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return v
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def _make_eigvals(alpha, beta, homogeneous_eigvals):
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if homogeneous_eigvals:
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if beta is None:
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return numpy.vstack((alpha, numpy.ones_like(alpha)))
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else:
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return numpy.vstack((alpha, beta))
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else:
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if beta is None:
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return alpha
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else:
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w = numpy.empty_like(alpha)
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alpha_zero = (alpha == 0)
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beta_zero = (beta == 0)
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beta_nonzero = ~beta_zero
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w[beta_nonzero] = alpha[beta_nonzero]/beta[beta_nonzero]
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# Use numpy.inf for complex values too since
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# 1/numpy.inf = 0, i.e. it correctly behaves as projective
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# infinity.
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w[~alpha_zero & beta_zero] = numpy.inf
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if numpy.all(alpha.imag == 0):
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w[alpha_zero & beta_zero] = numpy.nan
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else:
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w[alpha_zero & beta_zero] = complex(numpy.nan, numpy.nan)
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return w
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def _geneig(a1, b1, left, right, overwrite_a, overwrite_b,
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homogeneous_eigvals):
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ggev, = get_lapack_funcs(('ggev',), (a1, b1))
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cvl, cvr = left, right
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res = ggev(a1, b1, lwork=-1)
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lwork = res[-2][0].real.astype(numpy.int)
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if ggev.typecode in 'cz':
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alpha, beta, vl, vr, work, info = ggev(a1, b1, cvl, cvr, lwork,
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overwrite_a, overwrite_b)
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w = _make_eigvals(alpha, beta, homogeneous_eigvals)
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else:
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alphar, alphai, beta, vl, vr, work, info = ggev(a1, b1, cvl, cvr,
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lwork, overwrite_a,
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overwrite_b)
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alpha = alphar + _I * alphai
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w = _make_eigvals(alpha, beta, homogeneous_eigvals)
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_check_info(info, 'generalized eig algorithm (ggev)')
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only_real = numpy.all(w.imag == 0.0)
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if not (ggev.typecode in 'cz' or only_real):
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t = w.dtype.char
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if left:
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vl = _make_complex_eigvecs(w, vl, t)
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if right:
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vr = _make_complex_eigvecs(w, vr, t)
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# the eigenvectors returned by the lapack function are NOT normalized
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for i in xrange(vr.shape[0]):
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if right:
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vr[:, i] /= norm(vr[:, i])
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if left:
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vl[:, i] /= norm(vl[:, i])
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if not (left or right):
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return w
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if left:
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if right:
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return w, vl, vr
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return w, vl
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return w, vr
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def eig(a, b=None, left=False, right=True, overwrite_a=False,
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overwrite_b=False, check_finite=True, homogeneous_eigvals=False):
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"""
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Solve an ordinary or generalized eigenvalue problem of a square matrix.
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Find eigenvalues w and right or left eigenvectors of a general matrix::
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a vr[:,i] = w[i] b vr[:,i]
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a.H vl[:,i] = w[i].conj() b.H vl[:,i]
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where ``.H`` is the Hermitian conjugation.
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Parameters
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----------
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a : (M, M) array_like
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A complex or real matrix whose eigenvalues and eigenvectors
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will be computed.
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b : (M, M) array_like, optional
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Right-hand side matrix in a generalized eigenvalue problem.
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Default is None, identity matrix is assumed.
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left : bool, optional
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Whether to calculate and return left eigenvectors. Default is False.
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right : bool, optional
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Whether to calculate and return right eigenvectors. Default is True.
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overwrite_a : bool, optional
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Whether to overwrite `a`; may improve performance. Default is False.
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overwrite_b : bool, optional
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Whether to overwrite `b`; may improve performance. Default is False.
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check_finite : bool, optional
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Whether to check that the input matrices contain only finite numbers.
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Disabling may give a performance gain, but may result in problems
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(crashes, non-termination) if the inputs do contain infinities or NaNs.
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homogeneous_eigvals : bool, optional
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If True, return the eigenvalues in homogeneous coordinates.
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In this case ``w`` is a (2, M) array so that::
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w[1,i] a vr[:,i] = w[0,i] b vr[:,i]
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Default is False.
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Returns
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-------
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w : (M,) or (2, M) double or complex ndarray
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The eigenvalues, each repeated according to its
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multiplicity. The shape is (M,) unless
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``homogeneous_eigvals=True``.
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vl : (M, M) double or complex ndarray
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The normalized left eigenvector corresponding to the eigenvalue
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``w[i]`` is the column vl[:,i]. Only returned if ``left=True``.
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vr : (M, M) double or complex ndarray
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The normalized right eigenvector corresponding to the eigenvalue
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``w[i]`` is the column ``vr[:,i]``. Only returned if ``right=True``.
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Raises
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------
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LinAlgError
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If eigenvalue computation does not converge.
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See Also
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--------
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eigvals : eigenvalues of general arrays
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eigh : Eigenvalues and right eigenvectors for symmetric/Hermitian arrays.
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eig_banded : eigenvalues and right eigenvectors for symmetric/Hermitian
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band matrices
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eigh_tridiagonal : eigenvalues and right eiegenvectors for
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symmetric/Hermitian tridiagonal matrices
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Examples
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--------
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>>> from scipy import linalg
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>>> a = np.array([[0., -1.], [1., 0.]])
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>>> linalg.eigvals(a)
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array([0.+1.j, 0.-1.j])
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>>> b = np.array([[0., 1.], [1., 1.]])
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>>> linalg.eigvals(a, b)
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array([ 1.+0.j, -1.+0.j])
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>>> a = np.array([[3., 0., 0.], [0., 8., 0.], [0., 0., 7.]])
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>>> linalg.eigvals(a, homogeneous_eigvals=True)
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array([[3.+0.j, 8.+0.j, 7.+0.j],
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[1.+0.j, 1.+0.j, 1.+0.j]])
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>>> a = np.array([[0., -1.], [1., 0.]])
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>>> linalg.eigvals(a) == linalg.eig(a)[0]
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array([ True, True])
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>>> linalg.eig(a, left=True, right=False)[1] # normalized left eigenvector
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array([[-0.70710678+0.j , -0.70710678-0.j ],
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[-0. +0.70710678j, -0. -0.70710678j]])
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>>> linalg.eig(a, left=False, right=True)[1] # normalized right eigenvector
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array([[0.70710678+0.j , 0.70710678-0.j ],
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[0. -0.70710678j, 0. +0.70710678j]])
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"""
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a1 = _asarray_validated(a, check_finite=check_finite)
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if len(a1.shape) != 2 or a1.shape[0] != a1.shape[1]:
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raise ValueError('expected square matrix')
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overwrite_a = overwrite_a or (_datacopied(a1, a))
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if b is not None:
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b1 = _asarray_validated(b, check_finite=check_finite)
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overwrite_b = overwrite_b or _datacopied(b1, b)
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if len(b1.shape) != 2 or b1.shape[0] != b1.shape[1]:
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raise ValueError('expected square matrix')
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if b1.shape != a1.shape:
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raise ValueError('a and b must have the same shape')
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return _geneig(a1, b1, left, right, overwrite_a, overwrite_b,
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homogeneous_eigvals)
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geev, geev_lwork = get_lapack_funcs(('geev', 'geev_lwork'), (a1,))
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compute_vl, compute_vr = left, right
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lwork = _compute_lwork(geev_lwork, a1.shape[0],
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compute_vl=compute_vl,
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compute_vr=compute_vr)
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if geev.typecode in 'cz':
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w, vl, vr, info = geev(a1, lwork=lwork,
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compute_vl=compute_vl,
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compute_vr=compute_vr,
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overwrite_a=overwrite_a)
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w = _make_eigvals(w, None, homogeneous_eigvals)
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else:
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wr, wi, vl, vr, info = geev(a1, lwork=lwork,
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compute_vl=compute_vl,
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compute_vr=compute_vr,
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overwrite_a=overwrite_a)
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t = {'f': 'F', 'd': 'D'}[wr.dtype.char]
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w = wr + _I * wi
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w = _make_eigvals(w, None, homogeneous_eigvals)
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_check_info(info, 'eig algorithm (geev)',
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positive='did not converge (only eigenvalues '
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'with order >= %d have converged)')
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only_real = numpy.all(w.imag == 0.0)
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if not (geev.typecode in 'cz' or only_real):
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t = w.dtype.char
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if left:
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vl = _make_complex_eigvecs(w, vl, t)
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if right:
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vr = _make_complex_eigvecs(w, vr, t)
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if not (left or right):
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return w
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if left:
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if right:
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return w, vl, vr
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return w, vl
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return w, vr
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def eigh(a, b=None, lower=True, eigvals_only=False, overwrite_a=False,
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overwrite_b=False, turbo=True, eigvals=None, type=1,
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check_finite=True):
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"""
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Solve an ordinary or generalized eigenvalue problem for a complex
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Hermitian or real symmetric matrix.
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Find eigenvalues w and optionally eigenvectors v of matrix `a`, where
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`b` is positive definite::
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a v[:,i] = w[i] b v[:,i]
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v[i,:].conj() a v[:,i] = w[i]
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v[i,:].conj() b v[:,i] = 1
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Parameters
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----------
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a : (M, M) array_like
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A complex Hermitian or real symmetric matrix whose eigenvalues and
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eigenvectors will be computed.
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b : (M, M) array_like, optional
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A complex Hermitian or real symmetric definite positive matrix in.
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If omitted, identity matrix is assumed.
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lower : bool, optional
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Whether the pertinent array data is taken from the lower or upper
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triangle of `a`. (Default: lower)
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eigvals_only : bool, optional
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Whether to calculate only eigenvalues and no eigenvectors.
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(Default: both are calculated)
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turbo : bool, optional
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Use divide and conquer algorithm (faster but expensive in memory,
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only for generalized eigenvalue problem and if eigvals=None)
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eigvals : tuple (lo, hi), optional
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Indexes of the smallest and largest (in ascending order) eigenvalues
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and corresponding eigenvectors to be returned: 0 <= lo <= hi <= M-1.
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If omitted, all eigenvalues and eigenvectors are returned.
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type : int, optional
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Specifies the problem type to be solved:
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type = 1: a v[:,i] = w[i] b v[:,i]
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type = 2: a b v[:,i] = w[i] v[:,i]
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type = 3: b a v[:,i] = w[i] v[:,i]
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overwrite_a : bool, optional
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Whether to overwrite data in `a` (may improve performance)
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overwrite_b : bool, optional
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Whether to overwrite data in `b` (may improve performance)
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check_finite : bool, optional
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||
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Whether to check that the input matrices contain only finite numbers.
|
||
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Disabling may give a performance gain, but may result in problems
|
||
|
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(crashes, non-termination) if the inputs do contain infinities or NaNs.
|
||
|
|
|
||
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Returns
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-------
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w : (N,) float ndarray
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The N (1<=N<=M) selected eigenvalues, in ascending order, each
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repeated according to its multiplicity.
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v : (M, N) complex ndarray
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(if eigvals_only == False)
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The normalized selected eigenvector corresponding to the
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eigenvalue w[i] is the column v[:,i].
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Normalization:
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type 1 and 3: v.conj() a v = w
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type 2: inv(v).conj() a inv(v) = w
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type = 1 or 2: v.conj() b v = I
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type = 3: v.conj() inv(b) v = I
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Raises
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||
|
|
------
|
||
|
|
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 is 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 : eigenvalues and right eigenvectors for symmetric/Hermitian 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.
|
||
|
|
|
||
|
|
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
|
||
|
|
|
||
|
|
"""
|
||
|
|
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 iscomplexobj(a1):
|
||
|
|
cplx = True
|
||
|
|
else:
|
||
|
|
cplx = False
|
||
|
|
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("wrong b dimensions %s, should "
|
||
|
|
"be %s" % (str(b1.shape), str(a1.shape)))
|
||
|
|
if iscomplexobj(b1):
|
||
|
|
cplx = True
|
||
|
|
else:
|
||
|
|
cplx = cplx or False
|
||
|
|
else:
|
||
|
|
b1 = None
|
||
|
|
|
||
|
|
# Set job for fortran routines
|
||
|
|
_job = (eigvals_only and 'N') or 'V'
|
||
|
|
|
||
|
|
# port eigenvalue range from python to fortran convention
|
||
|
|
if eigvals is not None:
|
||
|
|
lo, hi = eigvals
|
||
|
|
if lo < 0 or hi >= a1.shape[0]:
|
||
|
|
raise ValueError('The eigenvalue range specified is not valid.\n'
|
||
|
|
'Valid range is [%s,%s]' % (0, a1.shape[0]-1))
|
||
|
|
lo += 1
|
||
|
|
hi += 1
|
||
|
|
eigvals = (lo, hi)
|
||
|
|
|
||
|
|
# set lower
|
||
|
|
if lower:
|
||
|
|
uplo = 'L'
|
||
|
|
else:
|
||
|
|
uplo = 'U'
|
||
|
|
|
||
|
|
# fix prefix for lapack routines
|
||
|
|
if cplx:
|
||
|
|
pfx = 'he'
|
||
|
|
else:
|
||
|
|
pfx = 'sy'
|
||
|
|
|
||
|
|
# Standard Eigenvalue Problem
|
||
|
|
# Use '*evr' routines
|
||
|
|
# FIXME: implement calculation of optimal lwork
|
||
|
|
# for all lapack routines
|
||
|
|
if b1 is None:
|
||
|
|
driver = pfx+'evr'
|
||
|
|
(evr,) = get_lapack_funcs((driver,), (a1,))
|
||
|
|
if eigvals is None:
|
||
|
|
w, v, info = evr(a1, uplo=uplo, jobz=_job, range="A", il=1,
|
||
|
|
iu=a1.shape[0], overwrite_a=overwrite_a)
|
||
|
|
else:
|
||
|
|
(lo, hi) = eigvals
|
||
|
|
w_tot, v, info = evr(a1, uplo=uplo, jobz=_job, range="I",
|
||
|
|
il=lo, iu=hi, overwrite_a=overwrite_a)
|
||
|
|
w = w_tot[0:hi-lo+1]
|
||
|
|
|
||
|
|
# Generalized Eigenvalue Problem
|
||
|
|
else:
|
||
|
|
# Use '*gvx' routines if range is specified
|
||
|
|
if eigvals is not None:
|
||
|
|
driver = pfx+'gvx'
|
||
|
|
(gvx,) = get_lapack_funcs((driver,), (a1, b1))
|
||
|
|
(lo, hi) = eigvals
|
||
|
|
w_tot, v, ifail, info = gvx(a1, b1, uplo=uplo, iu=hi,
|
||
|
|
itype=type, jobz=_job, il=lo,
|
||
|
|
overwrite_a=overwrite_a,
|
||
|
|
overwrite_b=overwrite_b)
|
||
|
|
w = w_tot[0:hi-lo+1]
|
||
|
|
# Use '*gvd' routine if turbo is on and no eigvals are specified
|
||
|
|
elif turbo:
|
||
|
|
driver = pfx+'gvd'
|
||
|
|
(gvd,) = get_lapack_funcs((driver,), (a1, b1))
|
||
|
|
v, w, info = gvd(a1, b1, uplo=uplo, itype=type, jobz=_job,
|
||
|
|
overwrite_a=overwrite_a,
|
||
|
|
overwrite_b=overwrite_b)
|
||
|
|
# Use '*gv' routine if turbo is off and no eigvals are specified
|
||
|
|
else:
|
||
|
|
driver = pfx+'gv'
|
||
|
|
(gv,) = get_lapack_funcs((driver,), (a1, b1))
|
||
|
|
v, w, info = gv(a1, b1, uplo=uplo, itype=type, jobz=_job,
|
||
|
|
overwrite_a=overwrite_a,
|
||
|
|
overwrite_b=overwrite_b)
|
||
|
|
|
||
|
|
# Check if we had a successful exit
|
||
|
|
if info == 0:
|
||
|
|
if eigvals_only:
|
||
|
|
return w
|
||
|
|
else:
|
||
|
|
return w, v
|
||
|
|
_check_info(info, driver, positive=False) # triage more specifically
|
||
|
|
if info > 0 and b1 is None:
|
||
|
|
raise LinAlgError("unrecoverable internal error.")
|
||
|
|
|
||
|
|
# The algorithm failed to converge.
|
||
|
|
elif 0 < info <= b1.shape[0]:
|
||
|
|
if eigvals is not None:
|
||
|
|
raise LinAlgError("the eigenvectors %s failed to"
|
||
|
|
" converge." % nonzero(ifail)-1)
|
||
|
|
else:
|
||
|
|
raise LinAlgError("internal fortran routine failed to converge: "
|
||
|
|
"%i off-diagonal elements of an "
|
||
|
|
"intermediate tridiagonal form did not converge"
|
||
|
|
" to zero." % info)
|
||
|
|
|
||
|
|
# This occurs when b is not positive definite
|
||
|
|
else:
|
||
|
|
raise LinAlgError("the leading minor of order %i"
|
||
|
|
" of 'b' is not positive definite. The"
|
||
|
|
" factorization of 'b' could not be completed"
|
||
|
|
" and no eigenvalues or eigenvectors were"
|
||
|
|
" computed." % (info-b1.shape[0]))
|
||
|
|
|
||
|
|
|
||
|
|
_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, string_types):
|
||
|
|
select = select.lower()
|
||
|
|
try:
|
||
|
|
select = _conv_dict[select]
|
||
|
|
except KeyError:
|
||
|
|
raise ValueError('invalid argument for select')
|
||
|
|
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 eiegenvectors 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 two-dimensional 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
|
||
|
|
|
||
|
|
|
||
|
|
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):
|
||
|
|
"""
|
||
|
|
Solve an ordinary or generalized eigenvalue problem for a complex
|
||
|
|
Hermitian or real symmetric matrix.
|
||
|
|
|
||
|
|
Find eigenvalues w of matrix a, where b is positive definite::
|
||
|
|
|
||
|
|
a v[:,i] = w[i] b v[:,i]
|
||
|
|
v[i,:].conj() a v[:,i] = w[i]
|
||
|
|
v[i,:].conj() b v[:,i] = 1
|
||
|
|
|
||
|
|
|
||
|
|
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`. (Default: lower)
|
||
|
|
turbo : bool, optional
|
||
|
|
Use divide and conquer algorithm (faster but expensive in memory,
|
||
|
|
only for generalized eigenvalue problem and if eigvals=None)
|
||
|
|
eigvals : tuple (lo, hi), optional
|
||
|
|
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.
|
||
|
|
type : int, optional
|
||
|
|
Specifies the problem type to be solved:
|
||
|
|
|
||
|
|
type = 1: a v[:,i] = w[i] b v[:,i]
|
||
|
|
|
||
|
|
type = 2: a b v[:,i] = w[i] v[:,i]
|
||
|
|
|
||
|
|
type = 3: b a v[:,i] = w[i] v[:,i]
|
||
|
|
overwrite_a : bool, optional
|
||
|
|
Whether to overwrite data in `a` (may improve performance)
|
||
|
|
overwrite_b : bool, optional
|
||
|
|
Whether to overwrite data in `b` (may improve performance)
|
||
|
|
check_finite : bool, optional
|
||
|
|
Whether to check that the input matrices contain only finite numbers.
|
||
|
|
Disabling may give a performance gain, but may result in problems
|
||
|
|
(crashes, non-termination) if the inputs do contain infinities or NaNs.
|
||
|
|
|
||
|
|
Returns
|
||
|
|
-------
|
||
|
|
w : (N,) float 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 is 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.
|
||
|
|
|
||
|
|
Examples
|
||
|
|
--------
|
||
|
|
>>> 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)
|
||
|
|
|
||
|
|
|
||
|
|
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 one-dimensional 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, string_types):
|
||
|
|
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
|
||
|
|
# matirx-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, numpy.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 one-dimensional')
|
||
|
|
if v.ndim < 2:
|
||
|
|
raise ValueError('expected v to be at least two-dimensional')
|
||
|
|
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
|