# -*- 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 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 ``ev`` which uses symmetric QR. ``evr`` is seen as the optimal choice for the most general cases. However, there are certain occasions that ``evd`` computes faster at the expense of more memory usage. ``evx``, while still being faster than ``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 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