*> \brief ZHEEVD computes the eigenvalues and, optionally, the left and/or right eigenvectors for HE matrices * * =========== DOCUMENTATION =========== * * Online html documentation available at * http://www.netlib.org/lapack/explore-html/ * *> \htmlonly *> Download ZHEEVD + dependencies *> *> [TGZ] *> *> [ZIP] *> *> [TXT] *> \endhtmlonly * * Definition: * =========== * * SUBROUTINE ZHEEVD( JOBZ, UPLO, N, A, LDA, W, WORK, LWORK, RWORK, * LRWORK, IWORK, LIWORK, INFO ) * * .. Scalar Arguments .. * CHARACTER JOBZ, UPLO * INTEGER INFO, LDA, LIWORK, LRWORK, LWORK, N * .. * .. Array Arguments .. * INTEGER IWORK( * ) * DOUBLE PRECISION RWORK( * ), W( * ) * COMPLEX*16 A( LDA, * ), WORK( * ) * .. * * *> \par Purpose: * ============= *> *> \verbatim *> *> ZHEEVD computes all eigenvalues and, optionally, eigenvectors of a *> complex Hermitian matrix A. If eigenvectors are desired, it uses a *> divide and conquer algorithm. *> *> The divide and conquer algorithm makes very mild assumptions about *> floating point arithmetic. It will work on machines with a guard *> digit in add/subtract, or on those binary machines without guard *> digits which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or *> Cray-2. It could conceivably fail on hexadecimal or decimal machines *> without guard digits, but we know of none. *> \endverbatim * * Arguments: * ========== * *> \param[in] JOBZ *> \verbatim *> JOBZ is CHARACTER*1 *> = 'N': Compute eigenvalues only; *> = 'V': Compute eigenvalues and eigenvectors. *> \endverbatim *> *> \param[in] UPLO *> \verbatim *> UPLO is CHARACTER*1 *> = 'U': Upper triangle of A is stored; *> = 'L': Lower triangle of A is stored. *> \endverbatim *> *> \param[in] N *> \verbatim *> N is INTEGER *> The order of the matrix A. N >= 0. *> \endverbatim *> *> \param[in,out] A *> \verbatim *> A is COMPLEX*16 array, dimension (LDA, N) *> On entry, the Hermitian matrix A. If UPLO = 'U', the *> leading N-by-N upper triangular part of A contains the *> upper triangular part of the matrix A. If UPLO = 'L', *> the leading N-by-N lower triangular part of A contains *> the lower triangular part of the matrix A. *> On exit, if JOBZ = 'V', then if INFO = 0, A contains the *> orthonormal eigenvectors of the matrix A. *> If JOBZ = 'N', then on exit the lower triangle (if UPLO='L') *> or the upper triangle (if UPLO='U') of A, including the *> diagonal, is destroyed. *> \endverbatim *> *> \param[in] LDA *> \verbatim *> LDA is INTEGER *> The leading dimension of the array A. LDA >= max(1,N). *> \endverbatim *> *> \param[out] W *> \verbatim *> W is DOUBLE PRECISION array, dimension (N) *> If INFO = 0, the eigenvalues in ascending order. *> \endverbatim *> *> \param[out] WORK *> \verbatim *> WORK is COMPLEX*16 array, dimension (MAX(1,LWORK)) *> On exit, if INFO = 0, WORK(1) returns the optimal LWORK. *> \endverbatim *> *> \param[in] LWORK *> \verbatim *> LWORK is INTEGER *> The length of the array WORK. *> If N <= 1, LWORK must be at least 1. *> If JOBZ = 'N' and N > 1, LWORK must be at least N + 1. *> If JOBZ = 'V' and N > 1, LWORK must be at least 2*N + N**2. *> *> If LWORK = -1, then a workspace query is assumed; the routine *> only calculates the optimal sizes of the WORK, RWORK and *> IWORK arrays, returns these values as the first entries of *> the WORK, RWORK and IWORK arrays, and no error message *> related to LWORK or LRWORK or LIWORK is issued by XERBLA. *> \endverbatim *> *> \param[out] RWORK *> \verbatim *> RWORK is DOUBLE PRECISION array, *> dimension (LRWORK) *> On exit, if INFO = 0, RWORK(1) returns the optimal LRWORK. *> \endverbatim *> *> \param[in] LRWORK *> \verbatim *> LRWORK is INTEGER *> The dimension of the array RWORK. *> If N <= 1, LRWORK must be at least 1. *> If JOBZ = 'N' and N > 1, LRWORK must be at least N. *> If JOBZ = 'V' and N > 1, LRWORK must be at least *> 1 + 5*N + 2*N**2. *> *> If LRWORK = -1, then a workspace query is assumed; the *> routine only calculates the optimal sizes of the WORK, RWORK *> and IWORK arrays, returns these values as the first entries *> of the WORK, RWORK and IWORK arrays, and no error message *> related to LWORK or LRWORK or LIWORK is issued by XERBLA. *> \endverbatim *> *> \param[out] IWORK *> \verbatim *> IWORK is INTEGER array, dimension (MAX(1,LIWORK)) *> On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK. *> \endverbatim *> *> \param[in] LIWORK *> \verbatim *> LIWORK is INTEGER *> The dimension of the array IWORK. *> If N <= 1, LIWORK must be at least 1. *> If JOBZ = 'N' and N > 1, LIWORK must be at least 1. *> If JOBZ = 'V' and N > 1, LIWORK must be at least 3 + 5*N. *> *> If LIWORK = -1, then a workspace query is assumed; the *> routine only calculates the optimal sizes of the WORK, RWORK *> and IWORK arrays, returns these values as the first entries *> of the WORK, RWORK and IWORK arrays, and no error message *> related to LWORK or LRWORK or LIWORK is issued by XERBLA. *> \endverbatim *> *> \param[out] INFO *> \verbatim *> INFO is INTEGER *> = 0: successful exit *> < 0: if INFO = -i, the i-th argument had an illegal value *> > 0: if INFO = i and JOBZ = 'N', then the algorithm failed *> to converge; i off-diagonal elements of an intermediate *> tridiagonal form did not converge to zero; *> if INFO = i and JOBZ = 'V', then the algorithm failed *> to compute an eigenvalue while working on the submatrix *> lying in rows and columns INFO/(N+1) through *> mod(INFO,N+1). *> \endverbatim * * Authors: * ======== * *> \author Univ. of Tennessee *> \author Univ. of California Berkeley *> \author Univ. of Colorado Denver *> \author NAG Ltd. * *> \date December 2016 * *> \ingroup complex16HEeigen * *> \par Further Details: * ===================== *> *> Modified description of INFO. Sven, 16 Feb 05. * *> \par Contributors: * ================== *> *> Jeff Rutter, Computer Science Division, University of California *> at Berkeley, USA *> * ===================================================================== SUBROUTINE ZHEEVD( JOBZ, UPLO, N, A, LDA, W, WORK, LWORK, RWORK, \$ LRWORK, IWORK, LIWORK, INFO ) * * -- LAPACK driver routine (version 3.7.0) -- * -- LAPACK is a software package provided by Univ. of Tennessee, -- * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- * December 2016 * * .. Scalar Arguments .. CHARACTER JOBZ, UPLO INTEGER INFO, LDA, LIWORK, LRWORK, LWORK, N * .. * .. Array Arguments .. INTEGER IWORK( * ) DOUBLE PRECISION RWORK( * ), W( * ) COMPLEX*16 A( LDA, * ), WORK( * ) * .. * * ===================================================================== * * .. Parameters .. DOUBLE PRECISION ZERO, ONE PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 ) COMPLEX*16 CONE PARAMETER ( CONE = ( 1.0D0, 0.0D0 ) ) * .. * .. Local Scalars .. LOGICAL LOWER, LQUERY, WANTZ INTEGER IINFO, IMAX, INDE, INDRWK, INDTAU, INDWK2, \$ INDWRK, ISCALE, LIOPT, LIWMIN, LLRWK, LLWORK, \$ LLWRK2, LOPT, LROPT, LRWMIN, LWMIN DOUBLE PRECISION ANRM, BIGNUM, EPS, RMAX, RMIN, SAFMIN, SIGMA, \$ SMLNUM * .. * .. External Functions .. LOGICAL LSAME INTEGER ILAENV DOUBLE PRECISION DLAMCH, ZLANHE EXTERNAL LSAME, ILAENV, DLAMCH, ZLANHE * .. * .. External Subroutines .. EXTERNAL DSCAL, DSTERF, XERBLA, ZHETRD, ZLACPY, ZLASCL, \$ ZSTEDC, ZUNMTR * .. * .. Intrinsic Functions .. INTRINSIC MAX, SQRT * .. * .. Executable Statements .. * * Test the input parameters. * WANTZ = LSAME( JOBZ, 'V' ) LOWER = LSAME( UPLO, 'L' ) LQUERY = ( LWORK.EQ.-1 .OR. LRWORK.EQ.-1 .OR. LIWORK.EQ.-1 ) * INFO = 0 IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN INFO = -1 ELSE IF( .NOT.( LOWER .OR. LSAME( UPLO, 'U' ) ) ) THEN INFO = -2 ELSE IF( N.LT.0 ) THEN INFO = -3 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN INFO = -5 END IF * IF( INFO.EQ.0 ) THEN IF( N.LE.1 ) THEN LWMIN = 1 LRWMIN = 1 LIWMIN = 1 LOPT = LWMIN LROPT = LRWMIN LIOPT = LIWMIN ELSE IF( WANTZ ) THEN LWMIN = 2*N + N*N LRWMIN = 1 + 5*N + 2*N**2 LIWMIN = 3 + 5*N ELSE LWMIN = N + 1 LRWMIN = N LIWMIN = 1 END IF LOPT = MAX( LWMIN, N + \$ ILAENV( 1, 'ZHETRD', UPLO, N, -1, -1, -1 ) ) LROPT = LRWMIN LIOPT = LIWMIN END IF WORK( 1 ) = LOPT RWORK( 1 ) = LROPT IWORK( 1 ) = LIOPT * IF( LWORK.LT.LWMIN .AND. .NOT.LQUERY ) THEN INFO = -8 ELSE IF( LRWORK.LT.LRWMIN .AND. .NOT.LQUERY ) THEN INFO = -10 ELSE IF( LIWORK.LT.LIWMIN .AND. .NOT.LQUERY ) THEN INFO = -12 END IF END IF * IF( INFO.NE.0 ) THEN CALL XERBLA( 'ZHEEVD', -INFO ) RETURN ELSE IF( LQUERY ) THEN RETURN END IF * * Quick return if possible * IF( N.EQ.0 ) \$ RETURN * IF( N.EQ.1 ) THEN W( 1 ) = A( 1, 1 ) IF( WANTZ ) \$ A( 1, 1 ) = CONE RETURN END IF * * Get machine constants. * SAFMIN = DLAMCH( 'Safe minimum' ) EPS = DLAMCH( 'Precision' ) SMLNUM = SAFMIN / EPS BIGNUM = ONE / SMLNUM RMIN = SQRT( SMLNUM ) RMAX = SQRT( BIGNUM ) * * Scale matrix to allowable range, if necessary. * ANRM = ZLANHE( 'M', UPLO, N, A, LDA, RWORK ) ISCALE = 0 IF( ANRM.GT.ZERO .AND. ANRM.LT.RMIN ) THEN ISCALE = 1 SIGMA = RMIN / ANRM ELSE IF( ANRM.GT.RMAX ) THEN ISCALE = 1 SIGMA = RMAX / ANRM END IF IF( ISCALE.EQ.1 ) \$ CALL ZLASCL( UPLO, 0, 0, ONE, SIGMA, N, N, A, LDA, INFO ) * * Call ZHETRD to reduce Hermitian matrix to tridiagonal form. * INDE = 1 INDTAU = 1 INDWRK = INDTAU + N INDRWK = INDE + N INDWK2 = INDWRK + N*N LLWORK = LWORK - INDWRK + 1 LLWRK2 = LWORK - INDWK2 + 1 LLRWK = LRWORK - INDRWK + 1 CALL ZHETRD( UPLO, N, A, LDA, W, RWORK( INDE ), WORK( INDTAU ), \$ WORK( INDWRK ), LLWORK, IINFO ) * * For eigenvalues only, call DSTERF. For eigenvectors, first call * ZSTEDC to generate the eigenvector matrix, WORK(INDWRK), of the * tridiagonal matrix, then call ZUNMTR to multiply it to the * Householder transformations represented as Householder vectors in * A. * IF( .NOT.WANTZ ) THEN CALL DSTERF( N, W, RWORK( INDE ), INFO ) ELSE CALL ZSTEDC( 'I', N, W, RWORK( INDE ), WORK( INDWRK ), N, \$ WORK( INDWK2 ), LLWRK2, RWORK( INDRWK ), LLRWK, \$ IWORK, LIWORK, INFO ) CALL ZUNMTR( 'L', UPLO, 'N', N, N, A, LDA, WORK( INDTAU ), \$ WORK( INDWRK ), N, WORK( INDWK2 ), LLWRK2, IINFO ) CALL ZLACPY( 'A', N, N, WORK( INDWRK ), N, A, LDA ) END IF * * If matrix was scaled, then rescale eigenvalues appropriately. * IF( ISCALE.EQ.1 ) THEN IF( INFO.EQ.0 ) THEN IMAX = N ELSE IMAX = INFO - 1 END IF CALL DSCAL( IMAX, ONE / SIGMA, W, 1 ) END IF * WORK( 1 ) = LOPT RWORK( 1 ) = LROPT IWORK( 1 ) = LIOPT * RETURN * * End of ZHEEVD * END