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


SYNOPSIS

       SUBROUTINE SSYGVD( ITYPE, JOBZ, UPLO, N, A, LDA,  B,  LDB,
                          W, WORK, LWORK, IWORK, LIWORK, INFO )

           CHARACTER      JOBZ, UPLO

           INTEGER        INFO, ITYPE, LDA, LDB, LIWORK, LWORK, N

           INTEGER        IWORK( * )

           REAL           A( LDA, * ), B( LDB, * ), W( * ), WORK(
                          * )


PURPOSE

       SSYGVD  computes  all the eigenvalues, and optionally, the
       eigenvectors  of  a  real  generalized  symmetric-definite
       eigenproblem,     of     the     form    A*x=(lambda)*B*x,
       A*Bx=(lambda)*x, or B*A*x=(lambda)*x. Here  A  and  B  are
       assumed  to  be symmetric 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

       ITYPE   (input) INTEGER
               Specifies the problem type to be solved:
               = 1:  A*x = (lambda)*B*x
               = 2:  A*B*x = (lambda)*x
               = 3:  B*A*x = (lambda)*x

       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.

               On  entry, the symmetric matrix A.  If UPLO = 'U',
               the leading N-by-N upper triangular part of A con­
               tains  the  upper triangular part of the matrix A.
               If UPLO = 'L', the leading N-by-N lower triangular
               part  of  A  contains the lower triangular part of
               the matrix A.

               On exit, if JOBZ = 'V', then if INFO = 0,  A  con­
               tains the matrix Z of eigenvectors.  The eigenvec­
               tors are normalized as follows: if ITYPE = 1 or 2,
               Z**T*B*Z = I; if ITYPE = 3, Z**T*inv(B)*Z = I.  If
               JOBZ = 'N', then on exit the  upper  triangle  (if
               UPLO='U')  or  the lower triangle (if UPLO='L') of
               A, including the diagonal, is destroyed.

       LDA     (input) INTEGER
               The leading dimension of  the  array  A.   LDA  >=
               max(1,N).

       B       (input/output) REAL array, dimension (LDB, N)
               On  entry, the symmetric matrix B.  If UPLO = 'U',
               the leading N-by-N upper triangular part of B con­
               tains  the  upper triangular part of the matrix B.
               If UPLO = 'L', the leading N-by-N lower triangular
               part  of  B  contains the lower triangular part of
               the matrix B.

               On exit, if INFO <= N, the part  of  B  containing
               the matrix is overwritten by the triangular factor
               U or L from the Cholesky factorization B =  U**T*U
               or B = L*L**T.

       LDB     (input) INTEGER
               The  leading  dimension  of  the  array B.  LDB >=
               max(1,N).

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

       WORK    (workspace/output) REAL 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  >=
               2*N+1.   If JOBZ = 'V' and N > 1, LWORK >= 1 + 6*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


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

       LIWORK  (input) INTEGER
               The dimension of the array  IWORK.   If  N  <=  1,
               LIWORK  >= 1.  If JOBZ  = 'N' and 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:  SPOTRF or SSYEVD returned an error code:
               <=  N:   if INFO = i, SSYEVD failed to converge; i
               off-diagonal elements of an intermediate tridiago­
               nal  form did not converge to zero; > N:   if INFO
               = N + i, for 1 <= i <= N, then the  leading  minor
               of  order  i  of  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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