OpenBLAS/lapack-netlib/SRC/zhbevd.f

*> \brief <b> ZHBEVD computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices</b>
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            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.
*>
*> 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] 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 complex16OTHEReigen
*
*  =====================================================================
      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 ) = 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 ) = LRWMIN
      IWORK( 1 ) = LIWMIN
      RETURN
*
*     End of ZHBEVD
*
      END