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.

ITYPE

          ITYPE is 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

          JOBZ is CHARACTER*1
          = 'N':  Compute eigenvalues only;
          = 'V':  Compute eigenvalues and eigenvectors.

UPLO

          UPLO is CHARACTER*1
          = 'U':  Upper triangles of A and B are stored;
          = 'L':  Lower triangles of A and B are stored.

N

          N is INTEGER
          The order of the matrices A and B.  N >= 0.

A

          A is REAL array, dimension (LDA, N)
          On entry, the symmetric matrix A.  If UPLO = 'U', the
          leading N-by-N upper triangular part of A contains 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 contains the
          matrix Z of eigenvectors.  The eigenvectors 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

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

B

          B is REAL array, dimension (LDB, N)
          On entry, the symmetric matrix B.  If UPLO = 'U', the
          leading N-by-N upper triangular part of B contains 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

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

W

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

WORK

          WORK is REAL array, dimension (MAX(1,LWORK))
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

LWORK

          LWORK is 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 sizes of the WORK and IWORK
          arrays, returns these values as the first entries of the WORK
          and IWORK arrays, and no error message related to LWORK or
          LIWORK is issued by XERBLA.

IWORK

          IWORK is INTEGER array, dimension (MAX(1,LIWORK))
          On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.

LIWORK

          LIWORK is 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 sizes of the WORK and
          IWORK arrays, returns these values as the first entries of
          the WORK and IWORK arrays, and no error message related to
          LWORK or LIWORK is issued by XERBLA.

INFO

          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value
          > 0:  SPOTRF or SSYEVD returned an error code:
             <= N:  if INFO = i and JOBZ = 'N', then the algorithm
                    failed to converge; i off-diagonal elements of an
                    intermediate tridiagonal form did not converge to
                    zero;
                    if INFO = i and JOBZ = 'V', then the algorithm
                    failed to compute an eigenvalue while working on
                    the submatrix lying in rows and columns INFO/(N+1)
                    through mod(INFO,N+1);
             > N:   if INFO = N + i, for 1 <= i <= N, then the leading
                    principal minor of order i of B is not positive.
                    The factorization of B could not be completed and
                    no eigenvalues or eigenvectors were computed.

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

  Modified so that no backsubstitution is performed if SSYEVD fails to
  converge (NEIG in old code could be greater than N causing out of
  bounds reference to A - reported by Ralf Meyer).  Also corrected the
  description of INFO and the test on ITYPE. Sven, 16 Feb 05.

Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA