SGBVX - 5.2 English - 68552

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

Purpose:

    SBGVX computes selected eigenvalues, and optionally, 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.  Eigenvalues and
    eigenvectors can be selected by specifying either all eigenvalues,
    a range of values or a range of indices for the desired eigenvalues.

Parameters:

• JOBZ – [in]

  JOBZ is CHARACTER*1

  = ‘N’: Compute eigenvalues only;

  = ‘V’: Compute eigenvalues and eigenvectors.

• RANGE – [in]

  RANGE is CHARACTER*1

  = ‘A’: all eigenvalues will be found.

  = ‘V’: all eigenvalues in the half-open interval (VL,VU] will be found.

  = ‘I’: the IL-th through IU-th eigenvalues will be found.

• 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.

• Q – [out]

  Q is REAL array, dimension (LDQ, N)

  If JOBZ = ‘V’, the n-by-n matrix used in the reduction of A*x = (lambda)*B*x to standard form, i.e. C*x = (lambda)*x, and consequently C to tridiagonal form.

  If JOBZ = ‘N’, the array Q is not referenced.

• LDQ – [in]

  LDQ is INTEGER

  The leading dimension of the array Q. If JOBZ = ‘N’, LDQ >= 1. If JOBZ = ‘V’, LDQ >= fla_max(1,N).

• VL – [in]

  VL is REAL

  If RANGE=’V’, the lower bound of the interval to be searched for eigenvalues. VL < VU. Not referenced if RANGE = ‘A’ or ‘I’.

• VU – [in]

  VU is REAL

  If RANGE=’V’, the upper bound of the interval to be searched for eigenvalues. VL < VU. Not referenced if RANGE = ‘A’ or ‘I’.

• IL – [in]

  IL is INTEGER

  If RANGE=’I’, the index of the smallest eigenvalue to be returned.

  1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.

  Not referenced if RANGE = ‘A’ or ‘V’.

• IU – [in]

  IU is INTEGER

  If RANGE=’I’, the index of the largest eigenvalue to be returned.

  1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.

  Not referenced if RANGE = ‘A’ or ‘V’.

• ABSTOL – [in]

  ABSTOL is REAL

  The absolute error tolerance for the eigenvalues. An approximate eigenvalue is accepted as converged when it is determined to lie in an interval [a,b] of width less than or equal to

  ABSTOL + EPS * fla_max( |a|,|b|) ,

  where EPS is the machine precision. If ABSTOL is less than or equal to zero, then EPS*|T| will be used in its place, where |T| is the 1-norm of the tridiagonal matrix obtained by reducing A to tridiagonal form.

  Eigenvalues will be computed most accurately when ABSTOL is set to twice the underflow threshold 2*SLAMCH(‘S’), not zero. If this routine returns with INFO>0, indicating that some eigenvectors did not converge, try setting ABSTOL to 2*SLAMCH(‘S’).

• M – [out]

  M is INTEGER

  The total number of eigenvalues found. 0 <= M <= N.

  If RANGE = ‘A’, M = N, and if RANGE = ‘I’, M = IU-IL+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 (7*N)

• IWORK – [out] IWORK is INTEGER array, dimension (5*N)

• IFAIL – [out]

  IFAIL is INTEGER array, dimension (M)

  If JOBZ = ‘V’, then if INFO = 0, the first M elements of IFAIL are zero. If INFO > 0, then IFAIL contains the indices of the eigenvalues that failed to converge.

  If JOBZ = ‘N’, then IFAIL is not referenced.

• INFO – [out]

  INFO is INTEGER

  = 0: successful exit

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

  <= N: if INFO = i, then i eigenvectors failed to converge. Their indices are stored in IFAIL.

  > N: SPBSTF returned an error code; i.e., 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.