# ?hegvd

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

## Syntax

FORTRAN 77:

call chegvd(itype, jobz, uplo, n, a, lda, b, ldb, w, work, lwork, rwork, lrwork, iwork, liwork, info)

call zhegvd(itype, jobz, uplo, n, a, lda, b, ldb, w, work, lwork, rwork, lrwork, iwork, liwork, info)

Fortran 95:

call hegvd(a, b, w [,itype] [,jobz] [,uplo] [,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 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 a and b store the upper triangles of A and B;

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

n

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

a, b, work

  COMPLEX for chegvd

  DOUBLE COMPLEX for zhegvd.

  Arrays:

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

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

  b(ldb,*) contains the upper or lower triangle of the Hermitian positive definite matrix B, as specified by uplo.

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

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

lda

  INTEGER. The first dimension of a; at least max(1, n).

ldb

  INTEGER. The first dimension of b; at least 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`+1;

  If `jobz = 'V'` and `n>1`, `lwork ≥ n2+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 chegvd

  DOUBLE PRECISION for zhegvd.

  Workspace array, 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, 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

a

  On exit, if `jobz = 'V'`, then if `info = 0`, a 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 on exit the upper triangle (if `uplo = 'U'`) or the lower triangle (if `uplo = 'L'`) of A, including the diagonal, is destroyed.

b

  On exit, if `info ≤ n`, the part of b containing the matrix is overwritten by the triangular factor U or L from the Cholesky factorization `B = UH*U` or `B = L*LH`.

w

  REAL for chegvd

  DOUBLE PRECISION for zhegvd.

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

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

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 = 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 = 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 hegvd interface are the following:

a

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

b

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

w

  Holds the vector of length (n).

itype

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

jobz

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

uplo

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

## 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