?sbevd

Developer Reference for Intel® oneAPI Math Kernel Library for C

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ID 766684

Date 12/16/2022

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?sbevd

Computes all eigenvalues and, optionally, all eigenvectors of a real symmetric band matrix using divide and conquer algorithm.

Syntax

lapack_int LAPACKE_ssbevd (int matrix_layout, char jobz, char uplo, lapack_int n, lapack_int kd, float* ab, lapack_int ldab, float* w, float* z, lapack_int ldz);

lapack_int LAPACKE_dsbevd (int matrix_layout, char jobz, char uplo, lapack_int n, lapack_int kd, double* ab, lapack_int ldab, double* w, double* z, lapack_int ldz);

Include Files

mkl.h

Description

The routine computes all the eigenvalues, and optionally all the eigenvectors, of a real symmetric band matrix A. In other words, it can compute the spectral factorization of A as:

A = Z*Λ*Z^T

Here Λ is a diagonal matrix whose diagonal elements are the eigenvalues λ_i, and Z is the orthogonal matrix whose columns are the eigenvectors z_i. Thus,

A*z_i = λ_i*z_i for i = 1, 2, ..., n.

If the eigenvectors are requested, then this routine uses a divide and conquer algorithm to compute eigenvalues and eigenvectors. However, if only eigenvalues are required, then it uses the Pal-Walker-Kahan variant of the QL or QR algorithm.

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.

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 symmetric 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.

ldz

The leading dimension of the output array z.

Constraints:

if jobz = 'N', then ldz≥ 1;

if jobz = 'V', then ldz≥ max(1, n) .

Output Parameters

w, z

Arrays:

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

If info = 0, contains the eigenvalues of the matrix A in ascending order. See also info.

z(size max(1, ldz*n if job = 'V' and at least 1 if job = 'N').

If job = 'V', then this array is overwritten by the orthogonal matrix Z which contains the eigenvectors of A. The i-th column of Z contains the eigenvector which corresponds to the eigenvalue w[i - 1].

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

ab

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

Return Values

This function returns a value info.

If info=0, the execution is successful.

If info = i, then the algorithm failed to converge; i indicates the number of elements of an intermediate tridiagonal form which did not converge to zero.

If info = -i, the i-th parameter had an illegal value.

Application Notes

The computed eigenvalues and eigenvectors are exact for a matrix A+E such that ||E||₂=O(ε)*||A||₂, where ε is the machine precision.

The complex analogue of this routine is hbevd.

See also syevd for matrices held in full storage, and spevd for matrices held in packed storage.

Parent topic: Symmetric Eigenvalue Problems: LAPACK Driver Routines

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