SYGV_2STAGE - 5.2 English - 68552

SYGV_2STAGE computes all the eigenvalues, the eigenvectors

of a real generalized symmetric-definite eigenproblem.

Purpose:

    SYGV_2STAGE 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.
    This routine use the 2stage technique for the reduction to tridiagonal
    which showed higher performance on recent architecture and for large
    sizes N>2000.

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. Not available in this release.

• 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 positive definite 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 length of the array WORK. LWORK >= 1, when N <= 1;

  otherwise

  If JOBZ = ‘N’ and N > 1, LWORK must be queried. LWORK = MAX(1, dimension) where dimension = fla_max(stage1,stage2) + (KD+1)*N + 2*N = N*KD + N*max(KD+1,FACTOPTNB)

  • fla_max(2*KD*KD, KD*NTHREADS)

  • (KD+1)*N + 2*N where KD is the blocking size of the reduction, FACTOPTNB is the blocking used by the QR or LQ algorithm, usually FACTOPTNB=128 is a good choice NTHREADS is the number of threads used when openMP compilation is enabled, otherwise =1.

    If JOBZ = ‘V’ and N > 1, LWORK must be queried. Not yet available

    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.

• INFO – [out]

  INFO is INTEGER

  = 0: successful exit

  < 0: if INFO = -i, the i-th argument had an illegal value

  > 0: SPOTRF or SSYEV returned an error code:

  <= N: if INFO = i, SSYEV 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 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.