?sbgvx

Developer Reference for Intel® oneAPI Math Kernel Library for C

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ID 766684

Date 3/22/2024

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?sbgvx

Computes selected eigenvalues and, optionally, eigenvectors of a real generalized symmetric definite eigenproblem with banded matrices.

Syntax

lapack_int LAPACKE_ssbgvx (int matrix_layout, char jobz, char range, char uplo, lapack_int n, lapack_int ka, lapack_int kb, float* ab, lapack_int ldab, float* bb, lapack_int ldbb, float* q, lapack_int ldq, float vl, float vu, lapack_int il, lapack_int iu, float abstol, lapack_int* m, float* w, float* z, lapack_int ldz, lapack_int* ifail);

lapack_int LAPACKE_dsbgvx (int matrix_layout, char jobz, char range, char uplo, lapack_int n, lapack_int ka, lapack_int kb, double* ab, lapack_int ldab, double* bb, lapack_int ldbb, double* q, lapack_int ldq, double vl, double vu, lapack_int il, lapack_int iu, double abstol, lapack_int* m, double* w, double* z, lapack_int ldz, lapack_int* ifail);

Include Files

mkl.h

Description

The routine computes selected eigenvalues, and optionally, the eigenvectors of a real generalized symmetric-definite banded eigenproblem, of the form A*x = λ*B*x. Here A and B are assumed to be symmetric and banded, and B is also positive definite. Eigenvalues and eigenvectors can be selected by specifying either all eigenvalues, 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 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 in range il to iu.

uplo

Must be 'U' or 'L'.

If uplo = 'U', arrays ab and bb store the upper triangles of A and B;

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

n

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

ka

The number of super- or sub-diagonals in A

(ka≥ 0).

kb

The number of super- or sub-diagonals in B (kb≥ 0).

ab, bb

Arrays:

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

bb(size at least max(1, ldbb*n) for column major layout and max(1, ldbb*(kb + 1)) for row major layout) is an array containing either upper or lower triangular part of the symmetric matrix B (as specified by uplo) in band storage format.

ldab

The leading dimension of the array ab; must be at least ka+1 for column major layout and at least max(1, n) for row major layout.

ldbb

The leading dimension of the array bb; must be at least kb+1 for column major layout and at least max(1, 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 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).

ldq

The leading dimension of the output array q; ldq < 1.

If jobz = 'V', ldq < max(1, n).

Output Parameters

ab

On exit, the contents of ab are overwritten.

bb

On exit, contains the factor S from the split Cholesky factorization B = S^T*S, as returned by pbstf/pbstf.

m

The total number of eigenvalues found,

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

m = iu-il+1.

w, z, q

Arrays:

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

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

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

If jobz = 'V', then if info = 0, z contains the matrix Z of eigenvectors, with the i-th column of z holding the eigenvector associated with w(i). The eigenvectors are normalized so that Z^T*B*Z = I.

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

q (size max(1, ldq*n)) .

If jobz = 'V', then q contains the n-by-n matrix used in the reduction of A*x = lambda*B*x to standard form, that is, C*x= lambda*x and consequently C to tridiagonal form.

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

ifail

Array, size m.

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, and

if i≤n, the algorithm 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 pbstf/pbstf returned info = i and 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').

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

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Developer Reference for Intel® oneAPI Math Kernel Library for C

?sbgvx

Syntax

Include Files

Description

Input Parameters

Output Parameters

Return Values

Application Notes