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 );

• 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').

Для получения подробной информации о возможностях оптимизации компилятора обратитесь к нашему Уведомлению об оптимизации.