?hbgv

Computes all eigenvalues and, optionally, eigenvectors of a complex generalized Hermitian positive-definite eigenproblem with banded matrices.

Syntax

call chbgv(jobz, uplo, n, ka, kb, ab, ldab, bb, ldbb, w, z, ldz, work, rwork, info)

call zhbgv(jobz, uplo, n, ka, kb, ab, ldab, bb, ldbb, w, z, ldz, work, rwork, info)

call hbgv(ab, bb, w [,uplo] [,z] [,info])

Include Files

• mkl.fi, lapack.f90

Description

The routine computes all the eigenvalues, and optionally, the eigenvectors of a complex generalized Hermitian positive-definite banded eigenproblem, of the form A*x = λ*B*x. Here A and B are Hermitian and banded matrices, and matrix B is also positive definite.

Input Parameters

jobz

CHARACTER*1. Must be 'N' or 'V'.

If jobz = 'N', then compute eigenvalues only.

If jobz = 'V', then compute eigenvalues and eigenvectors.

uplo

CHARACTER*1. Must be 'U' or 'L'.

If uplo = 'U', arrays ab and bb store the upper triangles of A and B;

If uplo = 'L', arrays ab and bb store the lower triangles of A and B.

n

INTEGER. The order of the matrices A and B (n 0).

ka

INTEGER. The number of super- or sub-diagonals in A

(ka 0).

kb

INTEGER. The number of super- or sub-diagonals in B (kb 0).

ab, bb, work

COMPLEX for chbgv

DOUBLE COMPLEX for zhbgv

Arrays:

ab(ldab,*) is an array containing either upper or lower triangular part of the Hermitian matrix A (as specified by uplo) in band storage format.

The second dimension of the array ab must be at least max(1, n).

bb(ldbb,*) is an array containing either upper or lower triangular part of the Hermitian matrix B (as specified by uplo) in band storage format.

The second dimension of the array bb must be at least max(1, n).

work(*) is a workspace array, dimension at least max(1, n).

ldab

INTEGER. The leading dimension of the array ab; must be at least ka+1.

ldbb

INTEGER. The leading dimension of the array bb; must be at least kb+1.

ldz

INTEGER. The leading dimension of the output array z; ldz 1. If jobz = 'V', ldz max(1, n).

rwork

REAL for chbgv

DOUBLE PRECISION for zhbgv.

Workspace array, size at least max(1, 3n).

Output Parameters

ab

On exit, the contents of ab are overwritten.

bb

On exit, contains the factor S from the split Cholesky factorization B = SH*S, as returned by pbstf/pbstf.

w

REAL for chbgv

DOUBLE PRECISION for zhbgv.

Array, size at least max(1, n).

If info = 0, contains the eigenvalues in ascending order.

z

COMPLEX for chbgv

DOUBLE COMPLEX for zhbgv

Array z(ldz,*).

The second dimension of z must be at least max(1, 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 that ZH*B*Z = I.

If jobz = 'N', then z is not referenced.

info

INTEGER.

If info = 0, the execution is successful.

If info = -i, the i-th argument had an illegal value.

If info > 0, and

if in, the algorithm failed to converge, and i off-diagonal elements of an intermediate tridiagonal did not converge to zero;

if info = n + i, for 1 in, then pbstf/pbstf returned info = i and B is not positive-definite. The factorization of B could not be completed and no eigenvalues or eigenvectors were computed.

LAPACK 95 Interface Notes

Routines in Fortran 95 interface have fewer arguments in the calling sequence than their FORTRAN 77 counterparts. For general conventions applied to skip redundant or restorable arguments, see LAPACK 95 Interface Conventions.

Specific details for the routine hbgv interface are the following:

ab

Holds the array A of size (ka+1,n).

bb

Holds the array B of size (kb+1,n).

w

Holds the vector with the number of elements n.

z

Holds the matrix Z of size (n, n).

uplo

Must be 'U' or 'L'. The default value is 'U'.

jobz

Restored based on the presence of the argument z as follows:

jobz = 'V', if z is present,

jobz = 'N', if z is omitted.