# ?hegv

Computes all eigenvalues and, optionally, eigenvectors of a complex generalized Hermitian definite eigenproblem.

## Syntax

FORTRAN 77:

call chegv(itype, jobz, uplo, n, a, lda, b, ldb, w, work, lwork, rwork, info)

call zhegv(itype, jobz, uplo, n, a, lda, b, ldb, w, work, lwork, rwork, info)

Fortran 95:

call hegv(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.

## 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 chegv

  DOUBLE COMPLEX for zhegv.

  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; `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 chegv

  DOUBLE PRECISION for zhegv.

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

## 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 chegv

  DOUBLE PRECISION for zhegv.

  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.

info

  INTEGER.

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

  If `info = -i`, the i-th argument has an illegal value.

  If `info > 0`, cpotrf/zpotrf and cheev/zheev return an error code:

  If `info = i ≤ n`, cheev/zheev fails to converge, and i off-diagonal elements of an intermediate tridiagonal do 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 can not be completed and no eigenvalues or eigenvectors are 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 hegv 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

For optimum performance use lwork ≥ (nb+1)*n, where nb is the blocksize for chetrd/zhetrd 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: Generalized Symmetric Definite Eigenproblems