?hpgv

Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

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Date 6/24/2024

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?hpgv

Computes all eigenvalues and, optionally, eigenvectors of a complex generalized Hermitian positive-definite eigenproblem with matrices in packed storage.

Syntax

call chpgv(itype, jobz, uplo, n, ap, bp, w, z, ldz, work, rwork, info)

call zhpgv(itype, jobz, uplo, n, ap, bp, w, z, ldz, work, rwork, info)

call hpgv(ap, bp, w [,itype] [,uplo] [,z] [,info])

Include Files

mkl.fi, lapack.f90

Description

The routine computes all the 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, stored in packed format, 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 ap and bp store the upper triangles of A and B;

If uplo = 'L', arrays ap and bp store the lower triangles of A and B.

n

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

ap, bp, work

COMPLEX for chpgv

DOUBLE COMPLEX for zhpgv.

Arrays:

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

The dimension of ap must be at least max(1, n*(n+1)/2).

bp(*) contains the packed upper or lower triangle of the Hermitian matrix B, as specified by uplo.

The dimension of bp must be at least max(1, n*(n+1)/2).

work(*) is a workspace array, size at least max(1, 2n-1).

ldz

INTEGER. The leading dimension of the output array z; ldz≥ 1. If jobz = 'V', ldz≥ max(1, n).

rwork

REAL for chpgv

DOUBLE PRECISION for zhpgv.

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

Output Parameters

ap

On exit, the contents of ap are overwritten.

bp

On exit, contains the triangular factor U or L from the Cholesky factorization B = U^H*U or B = L*L^H, in the same storage format as B.

w

REAL for chpgv

DOUBLE PRECISION for zhpgv.

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

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

z

COMPLEX for chpgv

DOUBLE COMPLEX for zhpgv.

Array z(ldz,*).

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

If jobz = 'V', then if info = 0, z contains the matrix Z of eigenvectors. The eigenvectors are normalized as follows:

if itype = 1 or 2, Z^H*B*Z = I;

if itype = 3, Z^H*inv(B)*Z = I;

If jobz = 'N', then z is not referenced.

info

INTEGER.

If info = 0, the execution is successful.

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

If info > 0, cpptrf/zpptrf and chpev/zhpev returned an error code:

If info = i≤n, chpev/zhpev failed to converge, and i off-diagonal elements of an intermediate tridiagonal did 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 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 hpgv interface are the following:

ap

Holds the array A of size (n*(n+1)/2).

bp

Holds the array B of size (n*(n+1)/2).

w

Holds the vector with the number of elements n.

z

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

itype

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

uplo

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

jobz

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

jobz = 'V', if z is present,

jobz = 'N', if z is omitted.

Parent topic: Generalized Symmetric Definite Eigenvalue Problems: LAPACK Driver Routines

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Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

?hpgv

Syntax

Include Files

Description

Input Parameters

Output Parameters

LAPACK 95 Interface Notes