zhbgvd(3S)

NAME
     ZHBGVD - compute all the eigenvalues, and optionally, the eigenvectors of
     a complex generalized Hermitian-definite banded eigenproblem, of the form
     A*x=(lambda)*B*x

SYNOPSIS
     SUBROUTINE ZHBGVD( JOBZ, UPLO, N, KA, KB, AB, LDAB, BB, LDBB, W, Z, LDZ,
                        WORK, LWORK, RWORK, LRWORK, IWORK, LIWORK, INFO )

         CHARACTER      JOBZ, UPLO

         INTEGER        INFO, KA, KB, LDAB, LDBB, LDZ, LIWORK, LRWORK, LWORK,
                        N

         INTEGER        IWORK( * )

         DOUBLE         PRECISION RWORK( * ), W( * )

         COMPLEX*16     AB( LDAB, * ), BB( LDBB, * ), WORK( * ), Z( LDZ, * )

IMPLEMENTATION
     These routines are part of the SCSL Scientific Library and can be loaded
     using either the -lscs or the -lscs_mp option.  The -lscs_mp option
     directs the linker to use the multi-processor version of the library.

     When linking to SCSL with -lscs or -lscs_mp, the default integer size is
     4 bytes (32 bits). Another version of SCSL is available in which integers
     are 8 bytes (64 bits).  This version allows the user access to larger
     memory sizes and helps when porting legacy Cray codes.  It can be loaded
     by using the -lscs_i8 option or the -lscs_i8_mp option. A program may use
     only one of the two versions; 4-byte integer and 8-byte integer library
     calls cannot be mixed.

PURPOSE
     ZHBGVD computes all the eigenvalues, and optionally, the eigenvectors of
     a complex generalized Hermitian-definite banded eigenproblem, of the form
     A*x=(lambda)*B*x. Here A and B are assumed to be Hermitian and banded,
     and B is also positive definite.  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.


ARGUMENTS
     JOBZ    (input) CHARACTER*1
             = 'N':  Compute eigenvalues only;
             = 'V':  Compute eigenvalues and eigenvectors.
     UPLO    (input) CHARACTER*1
             = 'U':  Upper triangles of A and B are stored;
             = 'L':  Lower triangles of A and B are stored.

     N       (input) INTEGER
             The order of the matrices A and B.  N >= 0.

     KA      (input) INTEGER
             The number of superdiagonals of the matrix A if UPLO = 'U', or
             the number of subdiagonals if UPLO = 'L'. KA >= 0.

     KB      (input) INTEGER
             The number of superdiagonals of the matrix B if UPLO = 'U', or
             the number of subdiagonals if UPLO = 'L'. KB >= 0.

     AB      (input/output) COMPLEX*16 array, dimension (LDAB, N)
             On entry, the upper or lower triangle of the Hermitian band
             matrix A, stored in the first ka+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(ka+1+i-j,j) = A(i,j) for max(1,j-
             ka)<=i<=j; if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for
             j<=i<=min(n,j+ka).

             On exit, the contents of AB are destroyed.

     LDAB    (input) INTEGER
             The leading dimension of the array AB.  LDAB >= KA+1.

     BB      (input/output) COMPLEX*16 array, dimension (LDBB, N)
             On entry, the upper or lower triangle of the Hermitian band
             matrix B, stored in the first kb+1 rows of the array.  The j-th
             column of B is stored in the j-th column of the array BB as
             follows:  if UPLO = 'U', BB(kb+1+i-j,j) = B(i,j) for max(1,j-
             kb)<=i<=j; if UPLO = 'L', BB(1+i-j,j)    = B(i,j) for
             j<=i<=min(n,j+kb).

             On exit, the factor S from the split Cholesky factorization B =
             S**H*S, as returned by ZPBSTF.

     LDBB    (input) INTEGER
             The leading dimension of the array BB.  LDBB >= KB+1.

     W       (output) DOUBLE PRECISION array, dimension (N)
             If INFO = 0, the eigenvalues in ascending order.

     Z       (output) COMPLEX*16 array, dimension (LDZ, N)
             If JOBZ = 'V', then if INFO = 0, Z contains the matrix Z of
             eigenvectors, with the i-th column of Z holding the eigenvector
             associated with W(i). The eigenvectors are normalized so that
             Z**H*B*Z = I.  If JOBZ = 'N', then Z is not referenced.

     LDZ     (input) INTEGER
             The leading dimension of the array Z.  LDZ >= 1, and if JOBZ =
             'V', LDZ >= N.

     WORK    (workspace/output) COMPLEX*16 array, dimension (LWORK)
             On exit, if INFO=0, WORK(1) returns the optimal LWORK.

     LWORK   (input) INTEGER
             The dimension of the array WORK.  If N <= 1,               LWORK
             >= 1.  If JOBZ = 'N' and N > 1, LWORK >= N.  If JOBZ = 'V' and N
             > 1, LWORK >= 2*N**2.

             If LWORK = -1, then a workspace query is assumed; the routine
             only calculates the optimal size of the WORK array, returns this
             value as the first entry of the WORK array, and no error message
             related to LWORK is issued by XERBLA.

     RWORK   (workspace/output) DOUBLE PRECISION array, dimension (LRWORK)
             On exit, if INFO=0, RWORK(1) returns the optimal LRWORK.

     LRWORK  (input) INTEGER
             The dimension of array RWORK.  If N <= 1,               LRWORK >=
             1.  If JOBZ = 'N' and N > 1, LRWORK >= N.  If JOBZ = 'V' and N >
             1, LRWORK >= 1 + 5*N + 2*N**2.

             If LRWORK = -1, then a workspace query is assumed; the routine
             only calculates the optimal size of the RWORK array, returns this
             value as the first entry of the RWORK array, and no error message
             related to LRWORK is issued by XERBLA.

     IWORK   (workspace/output) INTEGER array, dimension (LIWORK)
             On exit, if INFO=0, IWORK(1) returns the optimal LIWORK.

     LIWORK  (input) INTEGER
             The dimension of array IWORK.  If JOBZ = 'N' or N <= 1, LIWORK >=
             1.  If JOBZ = 'V' and N > 1, LIWORK >= 3 + 5*N.

             If LIWORK = -1, then a workspace query is assumed; the routine
             only calculates the optimal size of the IWORK array, returns this
             value as the first entry of the IWORK array, and no error message
             related to LIWORK is issued by XERBLA.

     INFO    (output) INTEGER
             = 0:  successful exit
             < 0:  if INFO = -i, the i-th argument had an illegal value
             > 0:  if INFO = i, and i is:
             <= N:  the algorithm failed to converge:  i off-diagonal elements
             of an intermediate tridiagonal form did not converge to zero; >
             N:   if INFO = N + i, for 1 <= i <= N, then ZPBSTF
             returned INFO = i: B is not positive definite.  The factorization
             of B could not be completed and no eigenvalues or eigenvectors
             were computed.
FURTHER DETAILS
     Based on contributions by
        Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA


SEE ALSO
     INTRO_LAPACK(3S), INTRO_SCSL(3S)

     This man page is available only online.