SYGVD - 5.2 English - 68552

SYGVD computes all the eigenvalues, the eigenvectors

of a real generalized symmetric-definite eigenproblem.

Purpose:

    SYGVD 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 assumptions 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
    without guard digits, but we know of none.

Parameters:

• ITYPE – [in]

  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 – [in]

  JOBZ is CHARACTER*1

  = ‘N’: Compute eigenvalues only;

  = ‘V’: Compute eigenvalues and eigenvectors.

• UPLO – [in]

  UPLO is CHARACTER*1

  = ‘U’: Upper triangles of A and B are stored;

  = ‘L’: Lower triangles of A and B are stored.

• N – [in]

  N is INTEGER

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

• A – [inout]

  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 – [in]

  LDA is INTEGER

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

• B – [inout]

  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 – [in]

  LDB is INTEGER

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

• W – [out]

  W is REAL array, dimension (N)

  If INFO = 0, the eigenvalues in ascending order.

• WORK – [out]

  WORK is REAL array, dimension (MAX(1,LWORK))

  On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

• LWORK – [in]

  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 – [out]

  IWORK is INTEGER array, dimension (MAX(1,LIWORK))

  On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.

• LIWORK – [in]

  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 – [out]

  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 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.