# ?heev

Computes all eigenvalues and, optionally, eigenvectors of a Hermitian matrix.

## Syntax

FORTRAN 77:

call cheev(jobz, uplo, n, a, lda, w, work, lwork, rwork, info)

call zheev(jobz, uplo, n, a, lda, w, work, lwork, rwork, info)

Fortran 95:

call heev(a, w [,jobz] [,uplo] [,info])

## Description

This routine computes all eigenvalues and, optionally, eigenvectors of a complex Hermitian matrix A.

Note that for most cases of complex Hermitian eigenvalue problems the default choice should be ?heevr function as its underlying algorithm is faster and uses less workspace.

## 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'`, a stores the upper triangular part of A.

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

n

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

a, work

  COMPLEX for cheev

  DOUBLE COMPLEX for zheev

  Arrays:

  a(lda,*) is an array containing either upper or lower triangular part of the Hermitian matrix A, as specified by uplo.

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

  work is a workspace array, its dimension `max(1, lwork)`.

lda

  INTEGER. The first dimension of the array a. Must be at least max(1, n).

lwork

  INTEGER.

  The dimension of the array work. C

  onstraint: `lwork ≥ max(1, 2n-1)`.

  If `lwork = -1`, then a workspace query is assumed; the routine only calculates the optimal size of the work array, returns this value as the first entry of the work array, and no error message related to lwork is issued by xerbla.

  See Application Notes for the suggested value of lwork.

rwork

  REAL for cheev

  DOUBLE PRECISION for zheev.

  Workspace array, DIMENSION at least max(1, 3n-2).

## Output Parameters

a

  On exit, if `jobz = 'V'`, then if `info = 0`, array a contains the orthonormal eigenvectors of the matrix A.

  If `jobz = 'N'`, then on exit the lower triangle

  (if `uplo = 'L'`) or the upper triangle (if `uplo = 'U'`) of A, including the diagonal, is overwritten.

w

  REAL for cheev

  DOUBLE PRECISION for zheev

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

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

work(1)

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

info

  INTEGER.

  If `info = 0`, the execution is successful.

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

  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.

## 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 heev interface are the following:

a

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

w

  Holds the vector of length (n).

job

  Must be 'N' or 'V'. The default value is 'N'.

uplo

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

## Application Notes

For optimum performance use

`lwork ≥ (nb+1)*n`,

where nb is the blocksize for ?hetrd returned by ilaenv.

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

If you choose the first option and set any of admissible lwork 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 on exit. Use this value (`work(1)`) for subsequent runs.

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

Note that if you set lwork 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: Symmetric Eigenproblems