# ?hpgvd

Computes all eigenvalues and, optionally, eigenvectors of a complex generalized Hermitian definite eigenproblem with matrices in packed storage. If eigenvectors are desired, it uses a divide and conquer method.

## Syntax

FORTRAN 77:

call chpgvd(itype, jobz, uplo, n, ap, bp, w, z, ldz, work, lwork, rwork, lrwork, iwork, liwork, info)

call zhpgvd(itype, jobz, uplo, n, ap, bp, w, z, ldz, work, lwork, rwork, lrwork, iwork, liwork, info)

Fortran 95:

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

## Description

This routine computes all the 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.

If eigenvectors are desired, it uses a divide and conquer algorithm.

## 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.

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 chpgvd

  DOUBLE COMPLEX for zhpgvd.

  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, its dimension `max(1, lwork)`.

ldz

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

lwork

  INTEGER.

  The dimension of the array work.

  Constraints:

  If `n ≤ 1`, `lwork ≥ 1`;

  If `jobz = 'N'` and `n>1`, `lwork ≥ n`;

  If `jobz = 'V'` and `n>1`, `lwork ≥ 2n`.

  If `lwork = -1`, then a workspace query is assumed; the routine only calculates the optimal size of the work, rwork and iwork arrays, returns these values as the first entries of the work, rwork and iwork arrays, and no error message related to lwork or lrwork or liwork is issued by xerbla. See Application Notes for details.

rwork

  REAL for chpgvd

  DOUBLE PRECISION for zhpgvd.

  Workspace array, its dimension `max(1, lrwork)`.

lrwork

  INTEGER.

  The dimension of the array rwork.

  Constraints:

  If `n ≤ 1`, `lrwork ≥ 1`;

  If `jobz = 'N'` and `n>1`, `lrwork ≥ n`;

  If `jobz = 'V'` and `n>1`, `lrwork ≥ 2n2+5n+1`.

  If `lrwork = -1`, then a workspace query is assumed; the routine only calculates the optimal size of the work, rwork and iwork arrays, returns these values as the first entries of the work, rwork and iwork arrays, and no error message related to lwork or lrwork or liwork is issued by xerbla. See Application Notes for details.

iwork

  INTEGER.

  Workspace array, its dimension `max(1, liwork)`.

liwork

  INTEGER.

  The dimension of the array iwork.

  Constraints:

  If `n ≤ 1`, `liwork ≥ 1`;

  If `jobz = 'N'` and `n>1`, `liwork ≥ 1`;

  If `jobz = 'V'` and `n>1`, `liwork ≥ 5n+3`.

  If `liwork = -1`, then a workspace query is assumed; the routine only calculates the optimal size of the work, rwork and iwork arrays, returns these values as the first entries of the work, rwork and iwork arrays, and no error message related to lwork or lrwork or liwork is issued by xerbla. See Application Notes for details.

## 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.

w

  REAL for chpgvd

  DOUBLE PRECISION for zhpgvd.

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

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

z

  COMPLEX for chpgvd

  DOUBLE COMPLEX for zhpgvd.

  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. 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.

work`(1)`

  On exit, if `info = 0`, then work`(1)` returns the required minimal size of lwork.

rwork`(1)`

  On exit, if `info = 0`, then rwork`(1)` returns the required minimal size of lrwork.

iwork`(1)`

  On exit, if `info = 0`, then iwork`(1)` returns the required minimal size of liwork.

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 chpevd/zhpevd returned an error code:

  If `info = i ≤ n`, chpevd/zhpevd 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 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 hpgvd 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).

itype

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

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.

## Application Notes

If you are in doubt how much workspace to supply, use a generous value of lwork (liwork or lrwork) for the first run or set `lwork = -1` (`liwork = -1`, `lrwork = -1`).

If you choose the first option and set any of admissible lwork (liwork or lrwork) sizes, which is no less than the minimal value described, the routine completes the task, though probably not so fast as with a recommended workspace, and provides the recommended workspace in the first element of the corresponding array (work, iwork, rwork) on exit. Use this value (`work(1)`, `iwork(1)`, `rwork(1)`) for subsequent runs.

If you set `lwork = -1` (`liwork = -1`, `lrwork = -1`), the routine returns immediately and provides the recommended workspace in the first element of the corresponding array (work, iwork, rwork). This operation is called a workspace query.

Note that if you set lwork (liwork, lrwork) to less than the minimal required value and not -1, the routine returns immediately with an error exit and does not provide any information on the recommended workspace.

Parent topic: Generalized Symmetric Definite Eigenproblems