?sbevd

Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

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Date 3/31/2023

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?sbevd

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

Syntax

call ssbevd(jobz, uplo, n, kd, ab, ldab, w, z, ldz, work, lwork, iwork, liwork, info)

call dsbevd(jobz, uplo, n, kd, ab, ldab, w, z, ldz, work, lwork, iwork, liwork, info)

call sbevd(ab, w [,uplo] [,z] [,info])

Include Files

mkl.fi, lapack.f90

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

jobz

CHARACTER*1. Must be 'N' or 'V'.

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

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

uplo

CHARACTER*1. 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

INTEGER. The order of the matrix A (n≥ 0).

kd

INTEGER. The number of super- or sub-diagonals in A

(kd≥ 0).

ab, work

REAL for ssbevd

DOUBLE PRECISION for dsbevd.

Arrays:

ab(lda,*) is an array containing either upper or lower triangular part of the symmetric matrix A (as specified by uplo) in band storage format.

The second dimension of ab must be at least max(1, n).

work is a workspace array, its dimension max(1, lwork).

ldab

INTEGER. The leading dimension of ab; must be at least kd+1.

ldz

INTEGER. The leading dimension of the output array z.

Constraints:

if jobz = 'N', then ldz≥ 1;

if jobz = 'V', then ldz≥ max(1, n) .

lwork

INTEGER.

The dimension of the array work.

Constraints:

if n≤ 1, then lwork≥ 1;

if jobz = 'N' and n > 1, then lwork≥ 2n;

if jobz = 'V' and n > 1, then lwork≥ 2*n² + 5*n + 1.

If lwork = -1, then a workspace query is assumed; the routine only calculates the optimal size of the work and iwork arrays, returns these values as the first entries of the work and iwork arrays, and no error message related to lwork or liwork is issued by xerbla. See Application Notes for details.

iwork

INTEGER. Workspace array, its dimension max(1, liwork).

liwork

INTEGER.

The dimension of the array iwork. Constraints: if n≤ 1, then liwork < 1; if job = 'N' and n > 1, then liwork < 1; if job = 'V' and n > 1, then liwork < 5*n+3.

If liwork = -1, then a workspace query is assumed; the routine only calculates the optimal size of the work and iwork arrays, returns these values as the first entries of the work and iwork arrays, and no error message related to lwork or liwork is issued by xerbla. See Application Notes for details.

Output Parameters

w, z

REAL for ssbevd

DOUBLE PRECISION for dsbevd

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(ldz,*).

The second dimension of z must be:

at least 1 if job = 'N';

at least max(1, n) if job = 'V'.

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).

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.

work(1)

On exit, if lwork > 0, then work(1) returns the required minimal size of lwork.

iwork(1)

On exit, if liwork > 0, then iwork(1) returns the required minimal size of liwork.

info

INTEGER.

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.

LAPACK 95 Interface Notes

Routines in Fortran 95 interface have fewer arguments in the calling sequence than their FORTRAN 77 counterparts. For general conventions applied to skip redundant or restorable arguments, see LAPACK 95 Interface Conventions.

Specific details for the routine sbevd interface are the following:

ab

Holds the array A of size (kd+1,n).

w

Holds the vector with the number of elements n.

z

Holds the matrix Z of size (n, n).

uplo

Must be 'U' or 'L'. The default value is 'U'.

jobz

Restored based on the presence of the argument z as follows:

jobz = 'V', if z is present,

jobz = 'N', if z is omitted.

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.

If it is not clear how much workspace to supply, use a generous value of lwork (or liwork) for the first run or set lwork = -1 (liwork = -1).

If any of admissible lwork (or liwork) has any of admissible sizes, which is no less than the minimal value described, the routine completes the task, though probably not so fast as with a recommended workspace, and provides the recommended workspace in the first element of the corresponding array (work, iwork) on exit. Use this value (work(1), iwork(1)) for subsequent runs.

If lwork = -1 (liwork = -1), the routine returns immediately and provides the recommended workspace in the first element of the corresponding array (work, iwork). This operation is called a workspace query.

Note that if work (liwork) is less than the minimal required value and is not equal to -1, the routine returns immediately with an error exit and does not provide any information on the recommended workspace.

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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