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

## Syntax

lapack_int LAPACKE_ssbgv (int matrix_layout, char jobz, char uplo, lapack_int n, lapack_int ka, lapack_int kb, float* ab, lapack_int ldab, float* bb, lapack_int ldbb, float* w, float* z, lapack_int ldz);

lapack_int LAPACKE_dsbgv (int matrix_layout, char jobz, char uplo, lapack_int n, lapack_int ka, lapack_int kb, double* ab, lapack_int ldab, double* bb, lapack_int ldbb, double* w, double* z, lapack_int ldz);

• mkl.h

## Description

The routine computes all the 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.

## 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.

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.

ldz

The leading dimension of the output array z; ldz 1. If jobz = 'V', ldz 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 = ST*S, as returned by pbstf/pbstf.

w, z

Arrays:

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

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

z (size at least max(1, ldz*n)) .

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 ZT*B*Z = I.

If jobz = 'N', then z 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 in, 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 in, 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.

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