zhbevd.f

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*> \brief <b> ZHBEVD computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices</b>
*
*  =========== DOCUMENTATION ===========
*
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*            http://www.netlib.org/lapack/explore-html/
*
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*
*  Definition:
*  ===========
*
*       SUBROUTINE ZHBEVD( JOBZ, UPLO, N, KD, AB, LDAB, W, Z, LDZ, WORK,
*                          LWORK, RWORK, LRWORK, IWORK, LIWORK, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          JOBZ, UPLO
*       INTEGER            INFO, KD, LDAB, LDZ, LIWORK, LRWORK, LWORK, N
*       ..
*       .. Array Arguments ..
*       INTEGER            IWORK( * )
*       DOUBLE PRECISION   RWORK( * ), W( * )
*       COMPLEX*16         AB( LDAB, * ), WORK( * ), Z( LDZ, * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> ZHBEVD computes all the eigenvalues and, optionally, eigenvectors of
*> a complex Hermitian band matrix A.  If eigenvectors are desired, it
*> uses a divide and conquer algorithm.
*>
*> \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] KD
*> \verbatim
*>          KD is INTEGER
*>          The number of superdiagonals of the matrix A if UPLO = 'U',
*>          or the number of subdiagonals if UPLO = 'L'.  KD >= 0.
*> \endverbatim
*>
*> \param[in,out] AB
*> \verbatim
*>          AB is COMPLEX*16 array, dimension (LDAB, N)
*>          On entry, the upper or lower triangle of the Hermitian band
*>          matrix A, stored in the first KD+1 rows of the array.  The
*>          j-th column of A is stored in the j-th column of the array AB
*>          as follows:
*>          if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
*>          if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+kd).
*>
*>          On exit, AB is overwritten by values generated during the
*>          reduction to tridiagonal form.  If UPLO = 'U', the first
*>          superdiagonal and the diagonal of the tridiagonal matrix T
*>          are returned in rows KD and KD+1 of AB, and if UPLO = 'L',
*>          the diagonal and first subdiagonal of T are returned in the
*>          first two rows of AB.
*> \endverbatim
*>
*> \param[in] LDAB
*> \verbatim
*>          LDAB is INTEGER
*>          The leading dimension of the array AB.  LDAB >= KD + 1.
*> \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 optimal LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*>          LWORK is INTEGER
*>          The dimension 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.
*>          If JOBZ = 'V' and N > 1, LWORK must be at least 2*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 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 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 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, 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 hbevd
*
*  =====================================================================
      SUBROUTINE zhbevd( JOBZ, UPLO, N, KD, AB, LDAB, 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, KD, LDAB, LDZ, LIWORK, LRWORK, LWORK, N
*     ..
*     .. Array Arguments ..
      INTEGER            IWORK( * )
      DOUBLE PRECISION   RWORK( * ), W( * )
      COMPLEX*16         AB( LDAB, * ), WORK( * ), Z( LDZ, * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO, ONE
      PARAMETER          ( ZERO = 0.0d0, one = 1.0d0 )
      COMPLEX*16         CZERO, CONE
      parameter( czero = ( 0.0d0, 0.0d0 ),
     $                   cone = ( 1.0d0, 0.0d0 ) )
*     ..
*     .. Local Scalars ..
      LOGICAL            LOWER, LQUERY, WANTZ
      INTEGER            IINFO, IMAX, INDE, INDWK2, INDWRK, ISCALE,
     $                   liwmin, llrwk, llwk2, lrwmin, lwmin
      DOUBLE PRECISION   ANRM, BIGNUM, EPS, RMAX, RMIN, SAFMIN, SIGMA,
     $                   SMLNUM
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      DOUBLE PRECISION   DLAMCH, ZLANHB
      EXTERNAL           lsame, dlamch, zlanhb
*     ..
*     .. External Subroutines ..
      EXTERNAL           dscal, dsterf, xerbla, zgemm, zhbtrd,
     $                   zlacpy,
     $                   zlascl, zstedc
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          sqrt
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
      wantz = lsame( jobz, 'V' )
      lower = lsame( uplo, 'L' )
      lquery = ( lwork.EQ.-1 .OR. liwork.EQ.-1 .OR. lrwork.EQ.-1 )
*
      info = 0
      IF( n.LE.1 ) THEN
         lwmin = 1
         lrwmin = 1
         liwmin = 1
      ELSE
         IF( wantz ) THEN
            lwmin = 2*n**2
            lrwmin = 1 + 5*n + 2*n**2
            liwmin = 3 + 5*n
         ELSE
            lwmin = n
            lrwmin = n
            liwmin = 1
         END IF
      END IF
      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( kd.LT.0 ) THEN
         info = -4
      ELSE IF( ldab.LT.kd+1 ) THEN
         info = -6
      ELSE IF( ldz.LT.1 .OR. ( wantz .AND. ldz.LT.n ) ) THEN
         info = -9
      END IF
*
      IF( info.EQ.0 ) THEN
         work( 1 ) = lwmin
         rwork( 1 ) = real( lrwmin )
         iwork( 1 ) = liwmin
*
         IF( lwork.LT.lwmin .AND. .NOT.lquery ) THEN
            info = -11
         ELSE IF( lrwork.LT.lrwmin .AND. .NOT.lquery ) THEN
            info = -13
         ELSE IF( liwork.LT.liwmin .AND. .NOT.lquery ) THEN
            info = -15
         END IF
      END IF
*
      IF( info.NE.0 ) THEN
         CALL xerbla( 'ZHBEVD', -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( ab( 1, 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 = zlanhb( 'M', uplo, n, kd, ab, ldab, 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
         IF( lower ) THEN
            CALL zlascl( 'B', kd, kd, one, sigma, n, n, ab, ldab,
     $                   info )
         ELSE
            CALL zlascl( 'Q', kd, kd, one, sigma, n, n, ab, ldab,
     $                   info )
         END IF
      END IF
*
*     Call ZHBTRD to reduce Hermitian band matrix to tridiagonal form.
*
      inde = 1
      indwrk = inde + n
      indwk2 = 1 + n*n
      llwk2 = lwork - indwk2 + 1
      llrwk = lrwork - indwrk + 1
      CALL zhbtrd( jobz, uplo, n, kd, ab, ldab, w, rwork( inde ), z,
     $             ldz, work, iinfo )
*
*     For eigenvalues only, call DSTERF.  For eigenvectors, call ZSTEDC.
*
      IF( .NOT.wantz ) THEN
         CALL dsterf( n, w, rwork( inde ), info )
      ELSE
         CALL zstedc( 'I', n, w, rwork( inde ), work, n,
     $                work( indwk2 ),
     $                llwk2, rwork( indwrk ), llrwk, iwork, liwork,
     $                info )
         CALL zgemm( 'N', 'N', n, n, n, cone, z, ldz, work, n, czero,
     $               work( indwk2 ), n )
         CALL zlacpy( 'A', n, n, work( indwk2 ), n, z, ldz )
      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 ) = real( lrwmin )
      iwork( 1 ) = liwmin
      RETURN
*
*     End of ZHBEVD
*
      SUBROUTINE zhbevd( JOBZ, UPLO, N, KD, AB, LDAB, W, Z, LDZ, …
      END