# ?hpevd

Uses divide and conquer algorithm to compute all eigenvalues and (optionally) all eigenvectors of a complex Hermitian matrix held in packed storage.

## Syntax

FORTRAN 77:

call chpevd(job z, uplo, n, ap, w, z, ldz, work, lwork, rwork, lrwork, iwork, liwork, info)

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

Fortran 95:

call hpevd(a, w [,uplo] [,z] [,info])

## Description

This routine computes all the eigenvalues, and optionally all the eigenvectors, of a complex Hermitian matrix A (held in packed storage). In other words, it can compute the spectral factorization of A as: `A = Z*Λ*ZH`.

Here Λ is a real diagonal matrix whose diagonal elements are the eigenvalues λi, and Z is the (complex) unitary matrix whose columns are the eigenvectors zi. Thus,

`A*zi = λi*zi` for `i = 1, 2, ..., n`.

If the eigenvectors are requested, then this routine uses a divide and conquer algorithm to compute eigenvalues and eigenvectors. However, if only eigenvalues are required, then it uses the Pal-Walker-Kahan variant of the QL or QR algorithm.

## Input Parameters

jobz

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

  If `jobz = 'N'`, then only eigenvalues are computed.

  If `jobz = 'V'`, then eigenvalues and eigenvectors are computed.

uplo

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

  If `uplo = 'U'`, ap stores the packed upper triangular part of A.

  If `uplo = 'L'`, ap stores the packed lower triangular part of A.

n

  INTEGER. The order of the matrix A (`n ≥ 0`).

ap, work

  COMPLEX for chpevd

  DOUBLE COMPLEX for zhpevd

  Arrays:

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

  The dimension of ap 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.

  Constraints:

  if `jobz = 'N'`, then `ldz ≥ 1`;

  if `jobz = 'V'`, then `ldz ≥ max(1, n)`.

lwork

  INTEGER.

  The dimension of the array work.

  Constraints:

  if `n ≤ 1`, then `lwork ≥ 1`;

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

  if `jobz = 'V'` and `n > 1`, then `lwork ≥ 2*n`.

  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 chpevd

  DOUBLE PRECISION for zhpevd

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

lrwork

  INTEGER.

  The dimension of the array rwork. Constraints:

  if `n ≤ 1`, then `lrwork ≥ 1`;

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

  if `jobz = 'V'` and `n > 1`, then `lrwork ≥ 2*n2 + 5*n + 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`, then `liwork ≥ 1`;

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

  if `jobz = 'V'` and `n > 1`, then `liwork ≥ 5*n+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

w

  REAL for chpevd

  DOUBLE PRECISION for zhpevd

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

  If `info = 0`, contains the eigenvalues of the matrix A in ascending order. See also info.

z

  COMPLEX for chpevd

  DOUBLE COMPLEX for zhpevd

  Array, DIMENSION (ldz,*).

  The second dimension of z must be:

  at least 1 if `jobz = 'N'`;

  at least max(1, n) if `jobz = 'V'`.

  If `jobz = 'V'`, then this array is overwritten by the unitary matrix Z which contains the eigenvectors of A.

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

ap

  On exit, this array is overwritten by the values generated during the reduction to tridiagonal form. The elements of the diagonal and the off-diagonal of the tridiagonal matrix overwrite the corresponding elements of A.

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`, then the algorithm failed to converge; i indicates the number of elements of an intermediate tridiagonal form which did not converge to zero.

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

## 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 hpevd interface are the following:

a

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

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.

## Application Notes

The computed eigenvalues and eigenvectors are exact for a matrix `T + E` such that `||E||2 = O(ε)*||T||2`, where ε is the machine precision.

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.

The real analogue of this routine is ?spevd.

See also ?heevd for matrices held in full storage, and ?hbevd for banded matrices.

Parent topic: Symmetric Eigenproblems