SSBGVD  - compute all the eigenvalues, and optionally, the
       eigenvectors  of  a  real  generalized  symmetric-definite
       banded eigenproblem, of the form A*x=(lambda)*B*x


SYNOPSIS

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

           CHARACTER      JOBZ, UPLO

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

           INTEGER        IWORK( * )

           REAL           AB( LDAB, * ), BB( LDBB, * ), W(  *  ),
                          WORK( * ), Z( LDZ, * )


PURPOSE

       SSBGVD  computes  all the eigenvalues, and optionally, the
       eigenvectors  of  a  real  generalized  symmetric-definite
       banded  eigenproblem, of the form A*x=(lambda)*B*x. Here A
       and B are assumed to be symmetric 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  assump­
       tions  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 with­
       out 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.

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

       AB      (input/output) REAL array, dimension (LDAB, N)
               On entry, the upper or lower triangle of the  sym­
               metric  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) REAL array, dimension (LDBB, N)
               On  entry, the upper or lower triangle of the sym­
               metric 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(ka+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 fac­
               torization B = S**T*S, as returned by SPBSTF.

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

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

       Z       (output) REAL 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 Z**T*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 >= max(1,N).

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

               The  dimension  of  the  array  WORK.   If N <= 1,
               LWORK >= 1.  If JOBZ = 'N' and N  >  1,  LWORK  >=
               3*N.   If JOBZ = 'V' and N > 1, LWORK >= 1 + 5*N +
               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.

       IWORK   (workspace/output) INTEGER array, dimension
               (LIWORK)
               On exit, if LIWORK > 0, IWORK(1) returns the opti­
               mal LIWORK.

       LIWORK  (input) INTEGER
               The dimension of the 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 ille­
               gal 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 SPBSTF
               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  Ken­
       tucky, USA


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