# ?hbgvx

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

## Syntax

call chbgvx(jobz, range, uplo, n, ka, kb, ab, ldab, bb, ldbb, q, ldq, vl, vu, il, iu, abstol, m, w, z, ldz, work, rwork, iwork, ifail, info)

call zhbgvx(jobz, range, uplo, n, ka, kb, ab, ldab, bb, ldbb, q, ldq, vl, vu, il, iu, abstol, m, w, z, ldz, work, rwork, iwork, ifail, info)

call hbgvx(ab, bb, w [,uplo] [,z] [,vl] [,vu] [,il] [,iu] [,m] [,ifail] [,q] [,abstol] [,info])

## Include Files

• mkl.fi, lapack.f90

## Description

The routine computes selected 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 assumed to be Hermitian 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.

## 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.

range

CHARACTER*1. Must be 'A' or 'V' or 'I'.

If range = 'A', the routine computes all eigenvalues.

If range = 'V', the routine computes eigenvalues lambda(i) in the half-open interval:

vl< lambda(i)vu.

If range = 'I', the routine computes eigenvalues with indices il to iu.

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 chbgvx

DOUBLE COMPLEX for zhbgvx

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, size 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.

vl, vu

REAL for chbgvx

DOUBLE PRECISION for zhbgvx.

If range = 'V', the lower and upper bounds of the interval to be searched for eigenvalues.

Constraint: vl< vu.

If range = 'A' or 'I', vl and vu are not referenced.

il, iu

INTEGER.

If range = 'I', the indices in ascending order of the smallest and largest eigenvalues to be returned.

Constraint: 1 iliun, if n > 0; il=1 and iu=0

if n = 0.

If range = 'A' or 'V', il and iu are not referenced.

abstol

REAL for chbgvx

DOUBLE PRECISION for zhbgvx.

The absolute error tolerance for the eigenvalues. See Application Notes for more information.

ldz

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

ldq

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

rwork

REAL for chbgvx

DOUBLE PRECISION for zhbgvx.

Workspace array, size at least max(1, 7n).

iwork

INTEGER.

Workspace array, size at least max(1, 5n).

## 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.

m

INTEGER. The total number of eigenvalues found,

0 mn. If range = 'A', m = n, and if range = 'I',

m = iu-il+1.

w

REAL for chbgvx

DOUBLE PRECISION for zhbgvx.

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

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

z, q

COMPLEX for chbgvx

DOUBLE COMPLEX for zhbgvx

Arrays:

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.

q(ldq,*).

The second dimension of q must be at least max(1, n).

If jobz = 'V', then q contains the n-by-n matrix used in the reduction of Ax = λBx to standard form, that is, Cx = λx and consequently C to tridiagonal form.

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

ifail

INTEGER.

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

If jobz = 'V', then if info = 0, the first m elements of ifail are zero; if info > 0, the ifail contains the indices of the eigenvectors that failed to converge.

If jobz = 'N', then ifail 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 hbgvx 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).

ifail

Holds the vector with the number of elements n.

q

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

uplo

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

vl

Default value for this element is vl = -HUGE(vl).

vu

Default value for this element is vu = HUGE(vl).

il

Default value for this argument is il = 1.

iu

Default value for this argument is iu = n.

abstol

Default value for this element is abstol = 0.0_WP.

jobz

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

jobz = 'V', if z is present,

jobz = 'N', if z is omitted.

Note that there will be an error condition if ifail or q is present and z is omitted.

range

Restored based on the presence of arguments vl, vu, il, iu as follows:

range = 'V', if one of or both vl and vu are present,

range = 'I', if one of or both il and iu are present,

range = 'A', if none of vl, vu, il, iu is present,

Note that there will be an error condition if one of or both vl and vu are present and at the same time one of or both il and iu are present.

## Application Notes

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+ε*max(|a|,|b|), where ε is the machine precision.

If abstol is less than or equal to zero, then ε*||T||1 will be used in its place, where T is 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*?lamch('S'), not zero.

If this routine returns with info > 0, indicating that some eigenvectors did not converge, try setting abstol to 2*?lamch('S').