chbgvx(3S)

NAME
     CHBGVX - 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 CHBGVX( JOBZ, RANGE, UPLO, N, KA, KB, AB, LDAB, BB, LDBB, Q,
                        LDQ, VL, VU, IL, IU, ABSTOL, M, W, Z, LDZ, WORK,
                        RWORK, IWORK, IFAIL, INFO )

         CHARACTER      JOBZ, RANGE, UPLO

         INTEGER        IL, INFO, IU, KA, KB, LDAB, LDBB, LDQ, LDZ, M, N

         REAL           ABSTOL, VL, VU

         INTEGER        IFAIL( * ), IWORK( * )

         REAL           RWORK( * ), W( * )

         COMPLEX        AB( LDAB, * ), BB( LDBB, * ), Q( LDQ, * ), 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
     CHBGVX 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.  Eigenvalues and eigenvectors can be
     selected by specifying either all eigenvalues, a range of values or a
     range of indices for the desired eigenvalues.


ARGUMENTS
     JOBZ    (input) CHARACTER*1
             = 'N':  Compute eigenvalues only;
             = 'V':  Compute eigenvalues and eigenvectors.

     RANGE   (input) CHARACTER*1
             = 'A': all eigenvalues will be found;
             = 'V': all eigenvalues in the half-open interval (VL,VU] will be
             found; = 'I': the IL-th through IU-th eigenvalues will be found.

     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 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 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 CPBSTF.

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

     Q       (output) COMPLEX array, dimension (LDQ, N)
             If JOBZ = 'V', the n-by-n matrix used in the reduction of A*x =
             (lambda)*B*x to standard form, i.e. C*x = (lambda)*x, and
             consequently C to tridiagonal form.  If JOBZ = 'N', the array Q
             is not referenced.
     LDQ     (input) INTEGER
             The leading dimension of the array Q.  If JOBZ = 'N', LDQ >= 1.
             If JOBZ = 'V', LDQ >= max(1,N).

     VL      (input) REAL
             VU      (input) REAL If RANGE='V', the lower and upper bounds of
             the interval to be searched for eigenvalues. VL < VU.  Not
             referenced if RANGE = 'A' or 'I'.

     IL      (input) INTEGER
             IU      (input) INTEGER If RANGE='I', the indices (in ascending
             order) of the smallest and largest eigenvalues to be returned.  1
             <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.  Not
             referenced if RANGE = 'A' or 'V'.

     ABSTOL  (input) REAL
             The absolute error tolerance for the eigenvalues.  An approximate
             eigenvalue is accepted as converged when it is determined to lie
             in an interval [a,b] of width less than or equal to

             ABSTOL + EPS *   max( |a|,|b| ) ,

             where EPS is the machine precision.  If ABSTOL is less than or
             equal to zero, then  EPS*|T|  will be used in its place, where
             |T| is the 1-norm of the tridiagonal matrix obtained by reducing
             AP to tridiagonal form.

             Eigenvalues will be computed most accurately when ABSTOL is set
             to twice the underflow threshold 2*SLAMCH('S'), not zero.  If
             this routine returns with INFO>0, indicating that some
             eigenvectors did not converge, try setting ABSTOL to
             2*SLAMCH('S').

     M       (output) INTEGER
             The total number of eigenvalues found.  0 <= M <= N.  If RANGE =
             'A', M = N, and if RANGE = 'I', M = IU-IL+1.

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

     Z       (output) COMPLEX 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) COMPLEX array, dimension (N)

     RWORK   (workspace) REAL array, dimension (7*N)

     IWORK   (workspace) INTEGER array, dimension (5*N)

     IFAIL   (output) INTEGER array, dimension (N)
             If JOBZ = 'V', then if INFO = 0, the first M elements of IFAIL
             are zero.  If INFO > 0, then IFAIL contains the indices of the
             eigenvectors that failed to converge.  If JOBZ = 'N', then IFAIL
             is not referenced.

     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:  then i eigenvectors failed to converge.  Their indices are
             stored in array IFAIL.  > N:   if INFO = N + i, for 1 <= i <= N,
             then CPBSTF
             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)

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