SBGVD - 5.2 English - 68552

SBGVD computes all the eigenvalues, and optionally, the eigenvectors.

Purpose:

    SBGVD 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 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:

• 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.

• KA – [in]

  KA is INTEGER

  The number of superdiagonals of the matrix A if UPLO = ‘U’, or the number of subdiagonals if UPLO = ‘L’. KA >= 0.

• KB – [in]

  KB is INTEGER

  The number of superdiagonals of the matrix B if UPLO = ‘U’, or the number of subdiagonals if UPLO = ‘L’. KB >= 0.

• AB – [inout]

  AB is REAL array, dimension (LDAB, N)

  On entry, the upper or lower triangle of the symmetric 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 fla_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 – [in]

  LDAB is INTEGER

  The leading dimension of the array AB. LDAB >= KA+1.

• BB – [inout]

  BB is REAL array, dimension (LDBB, N)

  On entry, the upper or lower triangle of the symmetric 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 fla_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 factorization B = S**T*S, as returned by SPBSTF.

• LDBB – [in]

  LDBB is INTEGER

  The leading dimension of the array BB. LDBB >= KB+1.

• W – [out]

  W is REAL array, dimension (N)

  If INFO = 0, the eigenvalues in ascending order.

• Z – [out]

  Z is REAL 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 – [in]

  LDZ is INTEGER

  The leading dimension of the array Z. LDZ >= 1, and if JOBZ = ‘V’, LDZ >= fla_max(1,N).

• 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 >= 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 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 LIWORK > 0, IWORK(1) returns the optimal LIWORK.

• LIWORK – [in]

  LIWORK is 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 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: 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 SPBSTF returned INFO = i: B is not positive definite. The factorization of B could not be completed and no eigenvalues or eigenvectors were computed.