# ?hpgvx

Computes selected eigenvalues and, optionally, eigenvectors of a generalized Hermitian definite eigenproblem with matrices in packed storage.

## Syntax

FORTRAN 77:

call chpgvx(itype, jobz, range, uplo, n, ap, bp, vl, vu, il, iu, abstol, m, w, z, ldz, work, rwork, iwork, ifail, info)

call zhpgvx(itype, jobz, range, uplo, n, ap, bp, vl, vu, il, iu, abstol, m, w, z, ldz, work, rwork, iwork, ifail, info)

Fortran 95:

call hpgvx(a, b, w [,itype] [,uplo] [,z] [,vl] [,vu] [,il] [,iu] [,m] [,ifail] [,abstol] [,info])

## Description

This routine computes selected eigenvalues, and optionally, the eigenvectors of a complex generalized Hermitian-definite eigenproblem, of the form

`A*x = λ*B*x`, `A*B*x = λ*x`, or `B*A*x = λ*x`.

Here A and B are assumed to be Hermitian, stored in packed format, and B is also positive definite. Eigenvalues and eigenvectors can be selected by specifying either a range of values or a range of indices for the desired eigenvalues.

## Input Parameters

itype

  INTEGER. Must be 1 or 2 or 3. Specifies the problem type to be solved:

  if `itype = 1`, the problem type is `A*x = lambda*B*x`;

  if `itype = 2`, the problem type is `A*B*x = lambda*x`;

  if `itype = 3`, the problem type is `B*A*x = lambda*x`.

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 ap and bp store the upper triangles of A and B;

  If `uplo = 'L'`, arrays ap and bp store the lower triangles of A and B.

n

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

ap, bp, work

  COMPLEX for chpgvx

  DOUBLE COMPLEX for zhpgvx.

  Arrays:

  ap(*) contains the packed upper or lower triangle of the Hermitian matrix A, as specified by uplo.

  The dimension of ap must be at least max(1, n*(n+1)/2).

  bp(*) contains the packed upper or lower triangle of the Hermitian matrix B, as specified by uplo.

  The dimension of bp must be at least max(1, n*(n+1)/2).

  work(*) is a workspace array, DIMENSION at least max(1, 2n).

vl, vu

  REAL for chpgvx

  DOUBLE PRECISION for zhpgvx.

  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 ≤ il ≤ iu ≤ n`, 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 chpgvx

  DOUBLE PRECISION for zhpgvx.

  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)`.

rwork

  REAL for chpgvx

  DOUBLE PRECISION for zhpgvx.

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

iwork

  INTEGER.

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

## Output Parameters

ap

  On exit, the contents of ap are overwritten.

bp

  On exit, contains the triangular factor U or L from the Cholesky factorization `B = UH*U` or `B = L*LH`, in the same storage format as B.

m

  INTEGER. The total number of eigenvalues found,

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

  `m = iu-il+1`.

w

  REAL for chpgvx

  DOUBLE PRECISION for zhpgvx.

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

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

z

  COMPLEX for chpgvx

  DOUBLE COMPLEX for zhpgvx.

  Array z(ldz,*).

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

  If `jobz = 'V'`, then if `info = 0`, the first m columns of z contain the orthonormal eigenvectors of the matrix A corresponding to the selected eigenvalues, with the i-th column of z holding the eigenvector associated with w(i). The eigenvectors are normalized as follows:

  if `itype = 1` or `2`, `ZH*B*Z = I`;

  if `itype = 3`, `ZH*inv(B)*Z = I`;

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

  If an eigenvector fails to converge, then that column of z contains the latest approximation to the eigenvector, and the index of the eigenvector is returned in ifail.

  Note: you must ensure that at least max(1,m) columns are supplied in the array z; if `range = 'V'`, the exact value of m is not known in advance and an upper bound must be used.

ifail

  INTEGER.

  Array, DIMENSION 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`, cpptrf/zpptrf and chpevx/zhpevx returned an error code:

  If `info = i ≤ n`, chpevx/zhpevx failed to converge, and i eigenvectors failed to converge. Their indices are stored in the array ifail;

  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.

## 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 hpgvx interface are the following:

a

  Stands for argument ap in Fortan 77 interface. Holds the array A of size (n*(n+1)/2).

b

  Stands for argument bp in Fortan 77 interface. Holds the array B of size (n*(n+1)/2).

w

  Holds the vector of length (n).

z

  Holds the matrix Z of size (n, n), where the values n and m are significant.

ifail

  Holds the vector of length (n).

itype

  Must be 1, 2, or 3. The default value is 1.

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 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` is used as tolerance, 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').

Parent topic: Generalized Symmetric Definite Eigenproblems