# ?hbgv

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

## Syntax

FORTRAN 77:

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)

Fortran 95:

call hbgv(a, b, w [,uplo] [,z] [,info])

## Description

This routine computes all the eigenvalues, and optionally, the eigenvectors of a complex generalized Hermitian-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 first dimension of the array ab; must be at least ka+1.

ldbb

  INTEGER. The first 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, DIMENSION 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 = S`H*S, as returned by cpbstf/zpbstf.

w

  REAL for chbgv

  DOUBLE PRECISION for zhbgv.

  Array, DIMENSION 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 `i ≤ n`, 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 ≤ i ≤ n`, then cpbstf/zpbstf returned `info = i` and B is not positive-definite. The factorization of B could not be completed and no eigenvalues or eigenvectors were computed.

## Fortran 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 Fortran 95 Interface Conventions.

Specific details for the routine hbgv interface are the following:

a

  Stands for argument ab in Fortan 77 interface. Holds the array A of size (ka+1,n).

b

  Stands for argument bb in Fortan 77 interface. Holds the array B of size (kb+1,n).

w

  Holds the vector of length (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.

Parent topic: Generalized Symmetric Definite Eigenproblems