DSBGVD  - 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 DSBGVD( 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( * )

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


PURPOSE

       DSBGVD  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) DOUBLE PRECISION 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)  DOUBLE PRECISION 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 DPBSTF.

       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) DOUBLE PRECISION 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).

               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 >=
               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 DPBSTF
               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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