# ?hegvx

Computes selected eigenvalues and, optionally, eigenvectors of a complex generalized Hermitian positive-definite eigenproblem.

## Syntax

call chegvx(itype, jobz, range, uplo, n, a, lda, b, ldb, vl, vu, il, iu, abstol, m, w, z, ldz, work, lwork, rwork, iwork, ifail, info)

call zhegvx(itype, jobz, range, uplo, n, a, lda, b, ldb, vl, vu, il, iu, abstol, m, w, z, ldz, work, lwork, rwork, iwork, ifail, info)

call hegvx(a, b, w [,itype] [,uplo] [,z] [,vl] [,vu] [,il] [,iu] [,m] [,ifail] [,abstol] [,info])

## Include Files

• mkl.fi, lapack.f90

## Description

The routine computes selected eigenvalues, and optionally, the eigenvectors of a complex generalized Hermitian positive-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. Eigenvalues and eigenvectors can be selected by specifying either a range of values or a range of indices for the desired eigenvalues.

## 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 = λ*B*x`;

if `itype = 2`, the problem type is `A*B*x = λ*x`;

if `itype = 3`, the problem type is `B*A*x = λ*x`.

jobz

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

If `jobz = 'N'`, then compute eigenvalues only.

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

range

CHARACTER*1. Must be 'A' or 'V' or 'I'.

If `range = 'A'`, the routine computes all eigenvalues.

If `range = 'V'`, the routine computes eigenvalues `lambda(i)` in the half-open interval:

`vl<`lambda(i) vu.

If `range = 'I'`, the routine computes eigenvalues with indices il to iu.

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 chegvx

DOUBLE COMPLEX for zhegvx.

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 leading dimension of a; at least max(1, n).

ldb

INTEGER. The leading dimension of b; at least max(1, n).

vl, vu

REAL for chegvx

DOUBLE PRECISION for zhegvx.

If `range = 'V'`, the lower and upper bounds of the interval to be searched for eigenvalues.

Constraint: `vl< vu`.

If `range = 'A'` or 'I', vl and vu are not referenced.

il, iu

INTEGER.

If `range = 'I'`, the indices in ascending order of the smallest and largest eigenvalues to be returned.

Constraint: `1 ≤ il ≤ iu ≤ n`, if `n > 0`; `il=1` and `iu=0`

if `n = 0`.

If `range = 'A'` or 'V', il and iu are not referenced.

abstol

REAL for chegvx

DOUBLE PRECISION for zhegvx.

The absolute error tolerance for the eigenvalues. See Application Notes for more information.

ldz

INTEGER. The leading dimension of the output array z. Constraints:

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

lwork

INTEGER.

The dimension of the array work; `lwork ≥ max(1, 2n)`.

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 chegvx

DOUBLE PRECISION for zhegvx.

Workspace array, size at least max(1, 7n).

iwork

INTEGER.

Workspace array, size at least max(1, 5n).

## Output Parameters

a

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

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`.

m

INTEGER. The total number of eigenvalues found,

`0 ≤ m ≤ n`. If `range = 'A'`, `m = n`, and if `range = 'I'`,

`m = iu-il+1`.

w

REAL for chegvx

DOUBLE PRECISION for zhegvx.

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

The first m elements of w contain the selected eigenvalues in ascending order.

z

COMPLEX for chegvx

DOUBLE COMPLEX for zhegvx.

Array z(ldz,*). The second dimension of z must be at least max(1, m).

If `jobz = 'V'`, then if `info = 0`, the first m columns of z contain the orthonormal eigenvectors of the matrix A corresponding to the selected eigenvalues, with the i-th column of z holding the eigenvector associated with w(i). 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 z is not referenced.

If an eigenvector fails to converge, then that column of z contains the latest approximation to the eigenvector, and the index of the eigenvector is returned in ifail.

Note: you must ensure that at least max(1,m) columns are supplied in the array z; if `range = 'V'`, the exact value of m is not known in advance and an upper bound must be used.

work(1)

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

ifail

INTEGER.

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

If `jobz = 'V'`, then if `info = 0`, the first m elements of ifail are zero; if `info > 0`, the ifail contains the indices of the eigenvectors that failed to converge.

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

info

INTEGER.

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

If `info = -i`, the ith argument had an illegal value.

If `info > 0`, cpotrf/zpotrf and cheevx/zheevx returned an error code:

If `info = i ≤ n`, cheevx/zheevx failed to converge, and i eigenvectors failed to converge. Their indices are stored in the array ifail;

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.

## LAPACK 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 LAPACK 95 Interface Conventions.

Specific details for the routine hegvx 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.

z

Holds the matrix Z of size (n, n), where the values n and m are significant.

ifail

Holds the vector of length n.

itype

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

uplo

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

vl

Default value for this element is vl = -HUGE(vl).

vu

Default value for this element is vu = HUGE(vl).

il

Default value for this argument is `il = 1`.

iu

Default value for this argument is `iu = n`.

abstol

Default value for this element is `abstol = 0.0_WP`.

jobz

Restored based on the presence of the argument z as follows:

`jobz = 'V'`, if z is present,

`jobz = 'N'`, if z is omitted.

Note that there will be an error condition if ifail is present and z is omitted.

range

Restored based on the presence of arguments vl, vu, il, iu as follows:

`range = 'V'`, if one of or both vl and vu are present,

`range = 'I'`, if one of or both il and iu are present,

`range = 'A'`, if none of vl, vu, il, iu is present,

Note that there will be an error condition if one of or both vl and vu are present and at the same time one of or both il and iu are present.

## Application Notes

An approximate eigenvalue is accepted as converged when it is determined to lie in an interval [a,b] of width less than or equal to `abstol+ε*max(|a|,|b|)`, where ε is the machine precision.

If abstol is less than or equal to zero, then `ε*||T||1` will be used in its place, where T is the tridiagonal matrix obtained by reducing C to tridiagonal form, where C is the symmetric matrix of the standard symmetric problem to which the generalized problem is transformed. Eigenvalues will be computed most accurately when abstol is set to twice the underflow threshold 2*?lamch('S'), not zero.

If this routine returns with `info > 0`, indicating that some eigenvectors did not converge, try setting abstol to 2*?lamch('S').

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.

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