zheevd.f

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*> \brief <b> ZHEEVD computes the eigenvalues and, optionally, the left and/or right eigenvectors for HE matrices</b>
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
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*
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*
*  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