# ?heevd

Computes all eigenvalues and (optionally) all eigenvectors of a complex Hermitian matrix using divide and conquer algorithm.

## Syntax

FORTRAN 77:

call cheevd(jobz, uplo, n, a, lda, w, work, lwork, rwork, lrwork, iwork, liwork, info)

call zheevd(job z, uplo, n, a, lda, w, work, lwork, rwork, lrwork, iwork, liwork, info)

Fortran 95:

call heevd(a, w [,job] [,uplo] [,info])

## Description

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

Note that for most cases of complex Hermetian eigenvalue problems the default choice should be ?heevr function as its underlying algorithm is faster and uses less workspace. ?heevd requires more workspace but is faster in some cases, especially for large matrices.

## 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

  COMPLEX for cheevd

  DOUBLE COMPLEX for zheevd

  Array, DIMENSION (lda, *).

  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).

lda

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

work

  COMPLEX for cheevd

  DOUBLE COMPLEX for zheevd.

  Workspace array, DIMENSION `max(1, lwork)`.

lwork

  INTEGER.

  The dimension of the array work. Constraints:

  if `n ≤ 1`, then `lwork ≥ 1`;

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

  if `jobz = 'V'` and n > 1, then `lwork ≥ n2+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 cheevd

  DOUBLE PRECISION for zheevd

  Workspace array, DIMENSION at least lrwork.

lrwork

  INTEGER.

  The dimension of the array rwork. Constraints:

  if `n ≤ 1`, then `lrwork ≥ 1`;

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

  if `job = '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 cheevd

  DOUBLE PRECISION for zheevd

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

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

a

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

work`(1)`

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

rwork`(1)`

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

iwork`(1)`

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

info

  INTEGER.

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

  If `info = i`, and `jobz = 'N'`, then the algorithm failed to converge; i off-diagonal elements of an intermediate tridiagonal form did not converge to zero;

  if `info = i`, and `jobz = 'V'`, then the algorithm failed to compute an eigenvalue while working on the submatrix lying in rows and columns `info/(n+1)` through `mod(info, n+1)`.

  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 heevd interface are the following:

a

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

w

  Holds the vector of length (n).

jobz

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

uplo

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

## Application Notes

The computed eigenvalues and eigenvectors are exact for a matrix `A + E` such that `||E||2 = O(ε)*||A||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 ?syevd. See also ?hpevd for matrices held in packed storage, and ?hbevd for banded matrices.

Parent topic: Symmetric Eigenproblems