?hbevx

Developer Reference for Intel® oneAPI Math Kernel Library for C

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ID 766684

Date 3/31/2023

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?hbevx

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

Syntax

lapack_int LAPACKE_chbevx( int matrix_layout, char jobz, char range, char uplo, lapack_int n, lapack_int kd, lapack_complex_float* ab, lapack_int ldab, lapack_complex_float* q, lapack_int ldq, 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_zhbevx( int matrix_layout, char jobz, char range, char uplo, lapack_int n, lapack_int kd, lapack_complex_double* ab, lapack_int ldab, lapack_complex_double* q, lapack_int ldq, 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 band matrix A. 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).

jobz

Must be 'N' or 'V'.

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

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

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', ab stores the upper triangular part of A.

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

n

The order of the matrix A (n≥ 0).

kd

The number of super- or sub-diagonals in A

(kd≥ 0).

ab

ab (size at least max(1, ldab*n) for column major layout and at least max(1, ldab*(kd + 1)) for row major layout) is an array containing either upper or lower triangular part of the Hermitian matrix A (as specified by uplo) in band storage format.

ldab

The leading dimension of ab; must be at least kd +1 for column major layout and n for row major layout.

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 ≤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

The absolute error tolerance to which each eigenvalue is required. See Application notes for details on error tolerance.

ldq, ldz

The leading dimensions of the output arrays q and z, respectively.

Constraints:

ldq≥ 1, ldz≥ 1;

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

Output Parameters

q

Array, size max(1, ldz*n).

If jobz = 'V', the n-by-n unitary matrix is used in the reduction to tridiagonal form.

If jobz = 'N', the array q is not referenced.

m

The total number of eigenvalues found,

0 ≤m≤n.

If range = 'A', m = n, if range = 'I', m = iu-il+1, and if range = 'V', the exact value of m is not known in advance..

w

Array, size at least max(1, n). The first m elements contain the selected eigenvalues of the matrix A 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 - 1].

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.

ab

On exit, this array is overwritten by the values generated during the reduction to tridiagonal form.

If uplo = 'U', the first superdiagonal and the diagonal of the tridiagonal matrix T are returned in rows kd and kd+1 of ab, and if uplo = 'L', the diagonal and first subdiagonal of T are returned in the first two rows of ab.

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 = 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||₁ will be used in its place, 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').

Parent topic: Symmetric Eigenvalue Problems: LAPACK Driver Routines

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Developer Reference for Intel® oneAPI Math Kernel Library for C

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