?heevx

Developer Reference for Intel® oneAPI Math Kernel Library for C

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ID 766684

Date 3/31/2023

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?heevx

Computes selected eigenvalues and, optionally, eigenvectors of a Hermitian matrix.

Syntax

lapack_int LAPACKE_cheevx( int matrix_layout, char jobz, char range, char uplo, lapack_int n, lapack_complex_float* a, lapack_int lda, 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_zheevx( int matrix_layout, char jobz, char range, char uplo, lapack_int n, lapack_complex_double* a, lapack_int lda, 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, eigenvectors of a complex Hermitian matrix A. Eigenvalues and eigenvectors can be selected by specifying either a range of values or a range of indices for the desired eigenvalues.

Note that for most cases of complex Hermetian eigenvalue problems the default choice should be heevr function as its underlying algorithm is faster and uses less workspace. ?heevx is faster for a few selected eigenvalues.

Input Parameters

matrix_layout

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

jobz

Must be 'N' or 'V'.

If jobz = 'N', then only eigenvalues are computed.

If jobz = 'V', then eigenvalues and eigenvectors are computed.

range

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

If range = 'A', all eigenvalues will be found.

If range = 'V', all eigenvalues in the half-open interval (vl, vu] will be found.

If range = 'I', the eigenvalues with indices il through iu will be found.

uplo

Must be 'U' or 'L'.

If uplo = 'U', a stores the upper triangular part of A.

If uplo = 'L', a stores the lower triangular part of A.

n

The order of the matrix A (n ≥ 0).

a

a (size max(1, lda*n)) is an array containing either upper or lower triangular part of the Hermitian matrix A, as specified by uplo.

lda

The leading dimension of the array a. Must be at least max(1, n).

vl, vu

If range = 'V', the lower and upper bounds of the interval to be searched for eigenvalues; vl≤vu. Not referenced if range = 'A'or 'I'.

il, iu

If range = 'I', the indices of the smallest and largest eigenvalues to be returned. Constraints:

1 ≤il≤iu≤n, if n > 0;il = 1 and iu = 0, if n = 0. Not referenced if range = 'A'or 'V'.

abstol

ldz

The leading dimension of the output array z; ldz≥ 1.

If jobz = 'V', then ldz≥max(1, n) for column major layout and lda≥ max(1, m) for row major layout.

Output Parameters

a

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

m

The total number of eigenvalues found; 0 ≤m≤n.

If range = 'A', m = n, and if range = 'I', m = iu-il+1.

w

Array, size max(1, n). The first m elements contain the selected eigenvalues of the matrix A in ascending order.

z

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

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).

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.

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

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, then ifail contains the indices of the eigenvectors that failed to converge.

If jobz = 'V', 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 = i, then i eigenvectors failed to converge; their indices are stored in the array ifail.

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|| will be used in its place, where ||T|| is the 1-norm of 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').

Parent topic: Symmetric Eigenvalue Problems: LAPACK Driver Routines

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Developer Reference for Intel® oneAPI Math Kernel Library for C

?heevx