565 lines
18 KiB
Fortran
565 lines
18 KiB
Fortran
*> \brief <b> ZHEEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for HE matrices</b>
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*
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* =========== DOCUMENTATION ===========
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*
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* Online html documentation available at
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* http://www.netlib.org/lapack/explore-html/
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*
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*> \htmlonly
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*> Download ZHEEVX + dependencies
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zheevx.f">
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*> [TGZ]</a>
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zheevx.f">
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*> [ZIP]</a>
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zheevx.f">
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*> [TXT]</a>
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*> \endhtmlonly
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*
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* Definition:
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* ===========
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*
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* SUBROUTINE ZHEEVX( JOBZ, RANGE, UPLO, N, A, LDA, VL, VU, IL, IU,
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* ABSTOL, M, W, Z, LDZ, WORK, LWORK, RWORK,
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* IWORK, IFAIL, INFO )
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*
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* .. Scalar Arguments ..
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* CHARACTER JOBZ, RANGE, UPLO
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* INTEGER IL, INFO, IU, LDA, LDZ, LWORK, M, N
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* DOUBLE PRECISION ABSTOL, VL, VU
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* ..
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* .. Array Arguments ..
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* INTEGER IFAIL( * ), IWORK( * )
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* DOUBLE PRECISION RWORK( * ), W( * )
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* COMPLEX*16 A( LDA, * ), WORK( * ), Z( LDZ, * )
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* ..
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*
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*
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*> \par Purpose:
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* =============
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*>
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*> \verbatim
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*>
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*> ZHEEVX computes selected eigenvalues and, optionally, eigenvectors
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*> of a complex Hermitian matrix A. Eigenvalues and eigenvectors can
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*> be selected by specifying either a range of values or a range of
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*> indices for the desired eigenvalues.
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*> \endverbatim
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*
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* Arguments:
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* ==========
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*
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*> \param[in] JOBZ
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*> \verbatim
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*> JOBZ is CHARACTER*1
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*> = 'N': Compute eigenvalues only;
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*> = 'V': Compute eigenvalues and eigenvectors.
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*> \endverbatim
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*>
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*> \param[in] RANGE
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*> \verbatim
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*> RANGE is CHARACTER*1
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*> = 'A': all eigenvalues will be found.
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*> = 'V': all eigenvalues in the half-open interval (VL,VU]
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*> will be found.
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*> = 'I': the IL-th through IU-th eigenvalues will be found.
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*> \endverbatim
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*>
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*> \param[in] UPLO
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*> \verbatim
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*> UPLO is CHARACTER*1
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*> = 'U': Upper triangle of A is stored;
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*> = 'L': Lower triangle of A is stored.
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*> \endverbatim
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*>
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*> \param[in] N
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*> \verbatim
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*> N is INTEGER
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*> The order of the matrix A. N >= 0.
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*> \endverbatim
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*>
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*> \param[in,out] A
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*> \verbatim
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*> A is COMPLEX*16 array, dimension (LDA, N)
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*> On entry, the Hermitian matrix A. If UPLO = 'U', the
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*> leading N-by-N upper triangular part of A contains the
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*> upper triangular part of the matrix A. If UPLO = 'L',
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*> the leading N-by-N lower triangular part of A contains
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*> the lower triangular part of the matrix A.
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*> On exit, the lower triangle (if UPLO='L') or the upper
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*> triangle (if UPLO='U') of A, including the diagonal, is
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*> destroyed.
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*> \endverbatim
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*>
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*> \param[in] LDA
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*> \verbatim
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*> LDA is INTEGER
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*> The leading dimension of the array A. LDA >= max(1,N).
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*> \endverbatim
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*>
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*> \param[in] VL
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*> \verbatim
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*> VL is DOUBLE PRECISION
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*> If RANGE='V', the lower bound of the interval to
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*> be searched for eigenvalues. VL < VU.
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*> Not referenced if RANGE = 'A' or 'I'.
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*> \endverbatim
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*>
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*> \param[in] VU
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*> \verbatim
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*> VU is DOUBLE PRECISION
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*> If RANGE='V', the upper bound of the interval to
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*> be searched for eigenvalues. VL < VU.
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*> Not referenced if RANGE = 'A' or 'I'.
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*> \endverbatim
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*>
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*> \param[in] IL
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*> \verbatim
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*> IL is INTEGER
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*> If RANGE='I', the index of the
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*> smallest eigenvalue to be returned.
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*> 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
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*> Not referenced if RANGE = 'A' or 'V'.
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*> \endverbatim
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*>
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*> \param[in] IU
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*> \verbatim
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*> IU is INTEGER
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*> If RANGE='I', the index of the
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*> largest eigenvalue to be returned.
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*> 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
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*> Not referenced if RANGE = 'A' or 'V'.
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*> \endverbatim
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*>
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*> \param[in] ABSTOL
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*> \verbatim
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*> ABSTOL is DOUBLE PRECISION
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*> The absolute error tolerance for the eigenvalues.
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*> An approximate eigenvalue is accepted as converged
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*> when it is determined to lie in an interval [a,b]
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*> of width less than or equal to
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*>
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*> ABSTOL + EPS * max( |a|,|b| ) ,
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*>
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*> where EPS is the machine precision. If ABSTOL is less than
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*> or equal to zero, then EPS*|T| will be used in its place,
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*> where |T| is the 1-norm of the tridiagonal matrix obtained
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*> by reducing A to tridiagonal form.
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*>
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*> Eigenvalues will be computed most accurately when ABSTOL is
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*> set to twice the underflow threshold 2*DLAMCH('S'), not zero.
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*> If this routine returns with INFO>0, indicating that some
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*> eigenvectors did not converge, try setting ABSTOL to
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*> 2*DLAMCH('S').
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*>
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*> See "Computing Small Singular Values of Bidiagonal Matrices
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*> with Guaranteed High Relative Accuracy," by Demmel and
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*> Kahan, LAPACK Working Note #3.
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*> \endverbatim
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*>
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*> \param[out] M
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*> \verbatim
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*> M is INTEGER
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*> The total number of eigenvalues found. 0 <= M <= N.
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*> If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.
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*> \endverbatim
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*>
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*> \param[out] W
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*> \verbatim
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*> W is DOUBLE PRECISION array, dimension (N)
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*> On normal exit, the first M elements contain the selected
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*> eigenvalues in ascending order.
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*> \endverbatim
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*>
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*> \param[out] Z
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*> \verbatim
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*> Z is COMPLEX*16 array, dimension (LDZ, max(1,M))
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*> If JOBZ = 'V', then if INFO = 0, the first M columns of Z
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*> contain the orthonormal eigenvectors of the matrix A
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*> corresponding to the selected eigenvalues, with the i-th
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*> column of Z holding the eigenvector associated with W(i).
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*> If an eigenvector fails to converge, then that column of Z
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*> contains the latest approximation to the eigenvector, and the
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*> index of the eigenvector is returned in IFAIL.
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*> If JOBZ = 'N', then Z is not referenced.
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*> Note: the user must ensure that at least max(1,M) columns are
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*> supplied in the array Z; if RANGE = 'V', the exact value of M
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*> is not known in advance and an upper bound must be used.
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*> \endverbatim
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*>
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*> \param[in] LDZ
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*> \verbatim
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*> LDZ is INTEGER
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*> The leading dimension of the array Z. LDZ >= 1, and if
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*> JOBZ = 'V', LDZ >= max(1,N).
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*> \endverbatim
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*>
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*> \param[out] WORK
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*> \verbatim
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*> WORK is COMPLEX*16 array, dimension (MAX(1,LWORK))
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*> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
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*> \endverbatim
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*>
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*> \param[in] LWORK
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*> \verbatim
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*> LWORK is INTEGER
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*> The length of the array WORK. LWORK >= 1, when N <= 1;
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*> otherwise 2*N.
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*> For optimal efficiency, LWORK >= (NB+1)*N,
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*> where NB is the max of the blocksize for ZHETRD and for
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*> ZUNMTR as returned by ILAENV.
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*>
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*> If LWORK = -1, then a workspace query is assumed; the routine
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*> only calculates the optimal size of the WORK array, returns
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*> this value as the first entry of the WORK array, and no error
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*> message related to LWORK is issued by XERBLA.
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*> \endverbatim
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*>
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*> \param[out] RWORK
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*> \verbatim
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*> RWORK is DOUBLE PRECISION array, dimension (7*N)
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*> \endverbatim
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*>
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*> \param[out] IWORK
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*> \verbatim
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*> IWORK is INTEGER array, dimension (5*N)
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*> \endverbatim
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*>
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*> \param[out] IFAIL
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*> \verbatim
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*> IFAIL is INTEGER array, dimension (N)
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*> If JOBZ = 'V', then if INFO = 0, the first M elements of
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*> IFAIL are zero. If INFO > 0, then IFAIL contains the
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*> indices of the eigenvectors that failed to converge.
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*> If JOBZ = 'N', then IFAIL is not referenced.
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*> \endverbatim
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*>
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*> \param[out] INFO
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*> \verbatim
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*> INFO is INTEGER
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*> = 0: successful exit
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*> < 0: if INFO = -i, the i-th argument had an illegal value
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*> > 0: if INFO = i, then i eigenvectors failed to converge.
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*> Their indices are stored in array IFAIL.
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*> \endverbatim
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*
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* Authors:
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* ========
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*
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*> \author Univ. of Tennessee
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*> \author Univ. of California Berkeley
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*> \author Univ. of Colorado Denver
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*> \author NAG Ltd.
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*
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*> \date June 2016
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*
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*> \ingroup complex16HEeigen
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*
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* =====================================================================
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SUBROUTINE ZHEEVX( JOBZ, RANGE, UPLO, N, A, LDA, VL, VU, IL, IU,
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$ ABSTOL, M, W, Z, LDZ, WORK, LWORK, RWORK,
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$ IWORK, IFAIL, INFO )
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*
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* -- LAPACK driver routine (version 3.7.0) --
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* -- LAPACK is a software package provided by Univ. of Tennessee, --
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* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
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* June 2016
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*
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* .. Scalar Arguments ..
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CHARACTER JOBZ, RANGE, UPLO
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INTEGER IL, INFO, IU, LDA, LDZ, LWORK, M, N
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DOUBLE PRECISION ABSTOL, VL, VU
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* ..
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* .. Array Arguments ..
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INTEGER IFAIL( * ), IWORK( * )
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DOUBLE PRECISION RWORK( * ), W( * )
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COMPLEX*16 A( LDA, * ), WORK( * ), Z( LDZ, * )
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* ..
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*
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* =====================================================================
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*
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* .. Parameters ..
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DOUBLE PRECISION ZERO, ONE
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PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
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COMPLEX*16 CONE
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PARAMETER ( CONE = ( 1.0D+0, 0.0D+0 ) )
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* ..
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* .. Local Scalars ..
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LOGICAL ALLEIG, INDEIG, LOWER, LQUERY, TEST, VALEIG,
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$ WANTZ
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CHARACTER ORDER
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INTEGER I, IINFO, IMAX, INDD, INDE, INDEE, INDIBL,
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$ INDISP, INDIWK, INDRWK, INDTAU, INDWRK, ISCALE,
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$ ITMP1, J, JJ, LLWORK, LWKMIN, LWKOPT, NB,
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$ NSPLIT
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DOUBLE PRECISION ABSTLL, ANRM, BIGNUM, EPS, RMAX, RMIN, SAFMIN,
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$ SIGMA, SMLNUM, TMP1, VLL, VUU
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* ..
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* .. External Functions ..
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LOGICAL LSAME
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INTEGER ILAENV
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DOUBLE PRECISION DLAMCH, ZLANHE
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EXTERNAL LSAME, ILAENV, DLAMCH, ZLANHE
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* ..
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* .. External Subroutines ..
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EXTERNAL DCOPY, DSCAL, DSTEBZ, DSTERF, XERBLA, ZDSCAL,
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$ ZHETRD, ZLACPY, ZSTEIN, ZSTEQR, ZSWAP, ZUNGTR,
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$ ZUNMTR
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* ..
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* .. Intrinsic Functions ..
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INTRINSIC DBLE, MAX, MIN, SQRT
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* ..
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* .. Executable Statements ..
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*
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* Test the input parameters.
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*
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LOWER = LSAME( UPLO, 'L' )
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WANTZ = LSAME( JOBZ, 'V' )
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ALLEIG = LSAME( RANGE, 'A' )
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VALEIG = LSAME( RANGE, 'V' )
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INDEIG = LSAME( RANGE, 'I' )
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LQUERY = ( LWORK.EQ.-1 )
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*
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INFO = 0
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IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN
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INFO = -1
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ELSE IF( .NOT.( ALLEIG .OR. VALEIG .OR. INDEIG ) ) THEN
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INFO = -2
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ELSE IF( .NOT.( LOWER .OR. LSAME( UPLO, 'U' ) ) ) THEN
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INFO = -3
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ELSE IF( N.LT.0 ) THEN
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INFO = -4
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ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
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INFO = -6
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ELSE
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IF( VALEIG ) THEN
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IF( N.GT.0 .AND. VU.LE.VL )
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$ INFO = -8
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ELSE IF( INDEIG ) THEN
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IF( IL.LT.1 .OR. IL.GT.MAX( 1, N ) ) THEN
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INFO = -9
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ELSE IF( IU.LT.MIN( N, IL ) .OR. IU.GT.N ) THEN
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INFO = -10
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END IF
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END IF
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END IF
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IF( INFO.EQ.0 ) THEN
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IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.N ) ) THEN
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INFO = -15
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END IF
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END IF
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*
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IF( INFO.EQ.0 ) THEN
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IF( N.LE.1 ) THEN
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LWKMIN = 1
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WORK( 1 ) = LWKMIN
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ELSE
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LWKMIN = 2*N
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NB = ILAENV( 1, 'ZHETRD', UPLO, N, -1, -1, -1 )
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NB = MAX( NB, ILAENV( 1, 'ZUNMTR', UPLO, N, -1, -1, -1 ) )
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LWKOPT = MAX( 1, ( NB + 1 )*N )
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WORK( 1 ) = LWKOPT
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END IF
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*
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IF( LWORK.LT.LWKMIN .AND. .NOT.LQUERY )
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$ INFO = -17
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END IF
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*
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IF( INFO.NE.0 ) THEN
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CALL XERBLA( 'ZHEEVX', -INFO )
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RETURN
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ELSE IF( LQUERY ) THEN
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RETURN
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END IF
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*
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* Quick return if possible
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*
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M = 0
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IF( N.EQ.0 ) THEN
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RETURN
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END IF
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*
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IF( N.EQ.1 ) THEN
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IF( ALLEIG .OR. INDEIG ) THEN
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M = 1
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W( 1 ) = A( 1, 1 )
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ELSE IF( VALEIG ) THEN
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IF( VL.LT.DBLE( A( 1, 1 ) ) .AND. VU.GE.DBLE( A( 1, 1 ) ) )
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$ THEN
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M = 1
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W( 1 ) = A( 1, 1 )
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END IF
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END IF
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IF( WANTZ )
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$ Z( 1, 1 ) = CONE
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RETURN
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END IF
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*
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* Get machine constants.
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*
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SAFMIN = DLAMCH( 'Safe minimum' )
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EPS = DLAMCH( 'Precision' )
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SMLNUM = SAFMIN / EPS
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BIGNUM = ONE / SMLNUM
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RMIN = SQRT( SMLNUM )
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RMAX = MIN( SQRT( BIGNUM ), ONE / SQRT( SQRT( SAFMIN ) ) )
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*
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* Scale matrix to allowable range, if necessary.
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*
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ISCALE = 0
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ABSTLL = ABSTOL
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IF( VALEIG ) THEN
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VLL = VL
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VUU = VU
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END IF
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ANRM = ZLANHE( 'M', UPLO, N, A, LDA, RWORK )
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IF( ANRM.GT.ZERO .AND. ANRM.LT.RMIN ) THEN
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ISCALE = 1
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SIGMA = RMIN / ANRM
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ELSE IF( ANRM.GT.RMAX ) THEN
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ISCALE = 1
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SIGMA = RMAX / ANRM
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END IF
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IF( ISCALE.EQ.1 ) THEN
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IF( LOWER ) THEN
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DO 10 J = 1, N
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CALL ZDSCAL( N-J+1, SIGMA, A( J, J ), 1 )
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10 CONTINUE
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ELSE
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DO 20 J = 1, N
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CALL ZDSCAL( J, SIGMA, A( 1, J ), 1 )
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20 CONTINUE
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END IF
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IF( ABSTOL.GT.0 )
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$ ABSTLL = ABSTOL*SIGMA
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IF( VALEIG ) THEN
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VLL = VL*SIGMA
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VUU = VU*SIGMA
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END IF
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END IF
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*
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* Call ZHETRD to reduce Hermitian matrix to tridiagonal form.
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*
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INDD = 1
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INDE = INDD + N
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INDRWK = INDE + N
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INDTAU = 1
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INDWRK = INDTAU + N
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LLWORK = LWORK - INDWRK + 1
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CALL ZHETRD( UPLO, N, A, LDA, RWORK( INDD ), RWORK( INDE ),
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$ WORK( INDTAU ), WORK( INDWRK ), LLWORK, IINFO )
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*
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* If all eigenvalues are desired and ABSTOL is less than or equal to
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* zero, then call DSTERF or ZUNGTR and ZSTEQR. If this fails for
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* some eigenvalue, then try DSTEBZ.
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*
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TEST = .FALSE.
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IF( INDEIG ) THEN
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IF( IL.EQ.1 .AND. IU.EQ.N ) THEN
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TEST = .TRUE.
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END IF
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END IF
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IF( ( ALLEIG .OR. TEST ) .AND. ( ABSTOL.LE.ZERO ) ) THEN
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CALL DCOPY( N, RWORK( INDD ), 1, W, 1 )
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INDEE = INDRWK + 2*N
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IF( .NOT.WANTZ ) THEN
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CALL DCOPY( N-1, RWORK( INDE ), 1, RWORK( INDEE ), 1 )
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CALL DSTERF( N, W, RWORK( INDEE ), INFO )
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ELSE
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CALL ZLACPY( 'A', N, N, A, LDA, Z, LDZ )
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CALL ZUNGTR( UPLO, N, Z, LDZ, WORK( INDTAU ),
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$ WORK( INDWRK ), LLWORK, IINFO )
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CALL DCOPY( N-1, RWORK( INDE ), 1, RWORK( INDEE ), 1 )
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CALL ZSTEQR( JOBZ, N, W, RWORK( INDEE ), Z, LDZ,
|
|
$ RWORK( INDRWK ), INFO )
|
|
IF( INFO.EQ.0 ) THEN
|
|
DO 30 I = 1, N
|
|
IFAIL( I ) = 0
|
|
30 CONTINUE
|
|
END IF
|
|
END IF
|
|
IF( INFO.EQ.0 ) THEN
|
|
M = N
|
|
GO TO 40
|
|
END IF
|
|
INFO = 0
|
|
END IF
|
|
*
|
|
* Otherwise, call DSTEBZ and, if eigenvectors are desired, ZSTEIN.
|
|
*
|
|
IF( WANTZ ) THEN
|
|
ORDER = 'B'
|
|
ELSE
|
|
ORDER = 'E'
|
|
END IF
|
|
INDIBL = 1
|
|
INDISP = INDIBL + N
|
|
INDIWK = INDISP + N
|
|
CALL DSTEBZ( RANGE, ORDER, N, VLL, VUU, IL, IU, ABSTLL,
|
|
$ RWORK( INDD ), RWORK( INDE ), M, NSPLIT, W,
|
|
$ IWORK( INDIBL ), IWORK( INDISP ), RWORK( INDRWK ),
|
|
$ IWORK( INDIWK ), INFO )
|
|
*
|
|
IF( WANTZ ) THEN
|
|
CALL ZSTEIN( N, RWORK( INDD ), RWORK( INDE ), M, W,
|
|
$ IWORK( INDIBL ), IWORK( INDISP ), Z, LDZ,
|
|
$ RWORK( INDRWK ), IWORK( INDIWK ), IFAIL, INFO )
|
|
*
|
|
* Apply unitary matrix used in reduction to tridiagonal
|
|
* form to eigenvectors returned by ZSTEIN.
|
|
*
|
|
CALL ZUNMTR( 'L', UPLO, 'N', N, M, A, LDA, WORK( INDTAU ), Z,
|
|
$ LDZ, WORK( INDWRK ), LLWORK, IINFO )
|
|
END IF
|
|
*
|
|
* If matrix was scaled, then rescale eigenvalues appropriately.
|
|
*
|
|
40 CONTINUE
|
|
IF( ISCALE.EQ.1 ) THEN
|
|
IF( INFO.EQ.0 ) THEN
|
|
IMAX = M
|
|
ELSE
|
|
IMAX = INFO - 1
|
|
END IF
|
|
CALL DSCAL( IMAX, ONE / SIGMA, W, 1 )
|
|
END IF
|
|
*
|
|
* If eigenvalues are not in order, then sort them, along with
|
|
* eigenvectors.
|
|
*
|
|
IF( WANTZ ) THEN
|
|
DO 60 J = 1, M - 1
|
|
I = 0
|
|
TMP1 = W( J )
|
|
DO 50 JJ = J + 1, M
|
|
IF( W( JJ ).LT.TMP1 ) THEN
|
|
I = JJ
|
|
TMP1 = W( JJ )
|
|
END IF
|
|
50 CONTINUE
|
|
*
|
|
IF( I.NE.0 ) THEN
|
|
ITMP1 = IWORK( INDIBL+I-1 )
|
|
W( I ) = W( J )
|
|
IWORK( INDIBL+I-1 ) = IWORK( INDIBL+J-1 )
|
|
W( J ) = TMP1
|
|
IWORK( INDIBL+J-1 ) = ITMP1
|
|
CALL ZSWAP( N, Z( 1, I ), 1, Z( 1, J ), 1 )
|
|
IF( INFO.NE.0 ) THEN
|
|
ITMP1 = IFAIL( I )
|
|
IFAIL( I ) = IFAIL( J )
|
|
IFAIL( J ) = ITMP1
|
|
END IF
|
|
END IF
|
|
60 CONTINUE
|
|
END IF
|
|
*
|
|
* Set WORK(1) to optimal complex workspace size.
|
|
*
|
|
WORK( 1 ) = LWKOPT
|
|
*
|
|
RETURN
|
|
*
|
|
* End of ZHEEVX
|
|
*
|
|
END
|