zhpevd.f

*> \brief <b> ZHPEVD computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices</b>
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
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*
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*
*  Definition:
*  ===========
*
*       SUBROUTINE ZHPEVD( JOBZ, UPLO, N, AP, W, Z, LDZ, WORK, LWORK,
*                          RWORK, LRWORK, IWORK, LIWORK, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          JOBZ, UPLO
*       INTEGER            INFO, LDZ, LIWORK, LRWORK, LWORK, N
*       ..
*       .. Array Arguments ..
*       INTEGER            IWORK( * )
*       DOUBLE PRECISION   RWORK( * ), W( * )
*       COMPLEX*16         AP( * ), WORK( * ), Z( LDZ, * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> ZHPEVD computes all the eigenvalues and, optionally, eigenvectors of
*> a complex Hermitian matrix A in packed storage.  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] AP
*> \verbatim
*>          AP is COMPLEX*16 array, dimension (N*(N+1)/2)
*>          On entry, the upper or lower triangle of the Hermitian matrix
*>          A, packed columnwise in a linear array.  The j-th column of A
*>          is stored in the array AP as follows:
*>          if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
*>          if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.
*>
*>          On exit, AP is overwritten by values generated during the
*>          reduction to tridiagonal form.  If UPLO = 'U', the diagonal
*>          and first superdiagonal of the tridiagonal matrix T overwrite
*>          the corresponding elements of A, and if UPLO = 'L', the
*>          diagonal and first subdiagonal of T overwrite the
*>          corresponding elements of A.
*> \endverbatim
*>
*> \param[out] W
*> \verbatim
*>          W is DOUBLE PRECISION array, dimension (N)
*>          If INFO = 0, the eigenvalues in ascending order.
*> \endverbatim
*>
*> \param[out] Z
*> \verbatim
*>          Z is COMPLEX*16 array, dimension (LDZ, N)
*>          If JOBZ = 'V', then if INFO = 0, Z contains the orthonormal
*>          eigenvectors of the matrix A, with the i-th column of Z
*>          holding the eigenvector associated with W(i).
*>          If JOBZ = 'N', then Z is not referenced.
*> \endverbatim
*>
*> \param[in] LDZ
*> \verbatim
*>          LDZ is INTEGER
*>          The leading dimension of the array Z.  LDZ >= 1, and if
*>          JOBZ = 'V', LDZ >= max(1,N).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is COMPLEX*16 array, dimension (MAX(1,LWORK))
*>          On exit, if INFO = 0, WORK(1) returns the required LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*>          LWORK is INTEGER
*>          The dimension of array WORK.
*>          If N <= 1,               LWORK must be at least 1.
*>          If JOBZ = 'N' and N > 1, LWORK must be at least N.
*>          If JOBZ = 'V' and N > 1, LWORK must be at least 2*N.
*>
*>          If LWORK = -1, then a workspace query is assumed; the routine
*>          only calculates the required 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 (MAX(1,LRWORK))
*>          On exit, if INFO = 0, RWORK(1) returns the required LRWORK.
*> \endverbatim
*>
*> \param[in] LRWORK
*> \verbatim
*>          LRWORK is INTEGER
*>          The dimension of 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 required 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 required LIWORK.
*> \endverbatim
*>
*> \param[in] LIWORK
*> \verbatim
*>          LIWORK is INTEGER
*>          The dimension of array IWORK.
*>          If JOBZ  = 'N' or 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 required 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, the algorithm failed to converge; i
*>                off-diagonal elements of an intermediate tridiagonal
*>                form did not converge to zero.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup hpevd
*
*  =====================================================================
      SUBROUTINE zhpevd( JOBZ, UPLO, N, AP, W, Z, LDZ, WORK, LWORK,
     $                   RWORK, LRWORK, IWORK, LIWORK, INFO )
*
*  -- LAPACK driver routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          JOBZ, UPLO
      INTEGER            INFO, LDZ, LIWORK, LRWORK, LWORK, N
*     ..
*     .. Array Arguments ..
      INTEGER            IWORK( * )
      DOUBLE PRECISION   RWORK( * ), W( * )
      COMPLEX*16         AP( * ), WORK( * ), Z( ldz, * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO, ONE
      parameter( zero = 0.0d+0, one = 1.0d+0 )
      COMPLEX*16         CONE
      parameter( cone = ( 1.0d+0, 0.0d+0 ) )
*     ..
*     .. Local Scalars ..
      LOGICAL            LQUERY, WANTZ
      INTEGER            IINFO, IMAX, INDE, INDRWK, INDTAU, INDWRK,
     $                   iscale, liwmin, llrwk, llwrk, lrwmin, lwmin
      DOUBLE PRECISION   ANRM, BIGNUM, EPS, RMAX, RMIN, SAFMIN, SIGMA,
     $                   smlnum
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      DOUBLE PRECISION   DLAMCH, ZLANHP
      EXTERNAL           lsame, dlamch, zlanhp
*     ..
*     .. External Subroutines ..
      EXTERNAL           dscal, dsterf, xerbla, zdscal, zhptrd, zstedc,
     $                   zupmtr
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          sqrt
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
      wantz = lsame( jobz, 'V' )
      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.( lsame( uplo, 'L' ) .OR. lsame( uplo, 'U' ) ) )
     $          THEN
         info = -2
      ELSE IF( n.LT.0 ) THEN
         info = -3
      ELSE IF( ldz.LT.1 .OR. ( wantz .AND. ldz.LT.n ) ) THEN
         info = -7
      END IF
*
      IF( info.EQ.0 ) THEN
         IF( n.LE.1 ) THEN
            lwmin = 1
            liwmin = 1
            lrwmin = 1
         ELSE
            IF( wantz ) THEN
               lwmin = 2*n
               lrwmin = 1 + 5*n + 2*n**2
               liwmin = 3 + 5*n
            ELSE
               lwmin = n
               lrwmin = n
               liwmin = 1
            END IF
         END IF
         work( 1 ) = lwmin
         rwork( 1 ) = lrwmin
         iwork( 1 ) = liwmin
*
         IF( lwork.LT.lwmin .AND. .NOT.lquery ) THEN
            info = -9
         ELSE IF( lrwork.LT.lrwmin .AND. .NOT.lquery ) THEN
            info = -11
         ELSE IF( liwork.LT.liwmin .AND. .NOT.lquery ) THEN
            info = -13
         END IF
      END IF
*
      IF( info.NE.0 ) THEN
         CALL xerbla( 'ZHPEVD', -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 ) = dble( ap( 1 ) )
         IF( wantz )
     $      z( 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 = zlanhp( 'M', uplo, n, ap, 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 ) THEN
         CALL zdscal( ( n*( n+1 ) ) / 2, sigma, ap, 1 )
      END IF
*
*     Call ZHPTRD to reduce Hermitian packed matrix to tridiagonal form.
*
      inde = 1
      indtau = 1
      indrwk = inde + n
      indwrk = indtau + n
      llwrk = lwork - indwrk + 1
      llrwk = lrwork - indrwk + 1
      CALL zhptrd( uplo, n, ap, w, rwork( inde ), work( indtau ),
     $             iinfo )
*
*     For eigenvalues only, call DSTERF.  For eigenvectors, first call
*     ZUPGTR to generate the orthogonal matrix, then call ZSTEDC.
*
      IF( .NOT.wantz ) THEN
         CALL dsterf( n, w, rwork( inde ), info )
      ELSE
         CALL zstedc( 'I', n, w, rwork( inde ), z, ldz, work( indwrk ),
     $                llwrk, rwork( indrwk ), llrwk, iwork, liwork,
     $                info )
         CALL zupmtr( 'L', uplo, 'N', n, n, ap, work( indtau ), z, ldz,
     $                work( indwrk ), iinfo )
      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 ) = lwmin
      rwork( 1 ) = lrwmin
      iwork( 1 ) = liwmin
      RETURN
*
*     End of ZHPEVD
*
      END