Developer Reference for Intel® oneAPI Math Kernel Library for C

ID 766684
Date 3/22/2024
Public

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?hpgvx

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

Syntax

lapack_int LAPACKE_chpgvx( int matrix_layout, lapack_int itype, char jobz, char range, char uplo, lapack_int n, lapack_complex_float* ap, lapack_complex_float* bp, float vl, float vu, lapack_int il, lapack_int iu, float abstol, lapack_int* m, float* w, lapack_complex_float* z, lapack_int ldz, lapack_int* ifail );

lapack_int LAPACKE_zhpgvx( int matrix_layout, lapack_int itype, char jobz, char range, char uplo, lapack_int n, lapack_complex_double* ap, lapack_complex_double* bp, double vl, double vu, lapack_int il, lapack_int iu, double abstol, lapack_int* m, double* w, lapack_complex_double* z, lapack_int ldz, lapack_int* ifail );

Include Files

  • mkl.h

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, stored in packed format, 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

matrix_layout

Specifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR) or column major (LAPACK_COL_MAJOR).

itype

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

Must be 'N' or 'V'.

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

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

range

Must be 'A' or 'V' or 'I'.

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

If range = 'V', the routine computes eigenvalues w[i] in the half-open interval:

vl<w[i]vu.

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

uplo

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

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

ap, bp

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).

vl, vu

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

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

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

if n = 0.

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

abstol

The absolute error tolerance for the eigenvalues.

See Application Notes for more information.

ldz

The leading dimension of the output array z; ldz 1. If jobz = 'V', ldz max(1, n) for column major layout and ldz max(1, m) for row major layout.

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 = UH*U or B = L*LH, in the same storage format as B.

m

The total number of eigenvalues found,

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

m = iu-il+1.

w

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

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

z

Array z(size at least max(1, ldz*m) for column major layout and max(1, ldz*n) for row major layout).

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.

ifail

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.

Return Values

This function returns a value info.

If info=0, the execution is successful.

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

If info > 0, cpptrf/zpptrf and chpevx/zhpevx returned an error code:

If info = in, chpevx/zhpevx failed to converge, and i eigenvectors failed to converge. Their indices are stored in the array ifail;

If info = n + i, for 1 in, 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.

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 is used as tolerance, where T is the tridiagonal matrix obtained by reducing A to tridiagonal form. 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').