?sbevx

Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

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ID 766686

Date 3/31/2023

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?sbevx

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

Syntax

call ssbevx(jobz, range, uplo, n, kd, ab, ldab, q, ldq, vl, vu, il, iu, abstol, m, w, z, ldz, work, iwork, ifail, info)

call dsbevx(jobz, range, uplo, n, kd, ab, ldab, q, ldq, vl, vu, il, iu, abstol, m, w, z, ldz, work, iwork, ifail, info)

call sbevx(ab, w [,uplo] [,z] [,vl] [,vu] [,il] [,iu] [,m] [,ifail] [,q] [,abstol] [,info])

Include Files

mkl.fi, lapack.f90

Description

The routine computes selected eigenvalues and, optionally, eigenvectors of a real symmetric 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

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.

range

CHARACTER*1. 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 in range il to iu.

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 ssbevx

DOUBLE PRECISION for dsbevx.

Arrays:

Array ab(lda,*) contains 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.

The dimension of work must be at least max(1, 7n).

ldab

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

vl, vu

REAL for ssbevx

DOUBLE PRECISION for dsbevx.

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

INTEGER.

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

REAL for chpevx

DOUBLE PRECISION for zhpevx

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

ldq, ldz

INTEGER. 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) .

iwork

INTEGER. Workspace array, size at least max(1, 5n).

Output Parameters

q

REAL for ssbevxDOUBLE PRECISION for dsbevx.

Array, size (ldz,n).

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

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

m

INTEGER. 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, z

REAL for ssbevx

DOUBLE PRECISION for dsbevx

Arrays:

w(*), size at least max(1, n). The first m elements of w contain the selected eigenvalues of the matrix A in ascending order.

z(ldz,*).

The second dimension of z must be at least max(1, m).

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).

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.

Note: you must ensure that at least max(1,m) columns are supplied in the array z; if range = 'V', the exact value of m is not known in advance and an upper bound must be used.

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

INTEGER.

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 the eigenvectors that failed to converge.

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

info

INTEGER.

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.

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 sbevx 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), where the values n and m are significant.

ifail

Holds the vector with the number of elements n.

q

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

uplo

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

vl

Default value for this element is vl = -HUGE(vl).

vu

Default value for this element is vu = HUGE(vl).

il

Default value for this argument is il = 1.

iu

Default value for this argument is iu = n.

abstol

Default value for this element is abstol = 0.0_WP.

jobz

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

jobz = 'V', if z is present,

jobz = 'N', if z is omitted

Note that there will be an error condition if either ifail or q is present and z is omitted.

range

Restored based on the presence of arguments vl, vu, il, iu as follows:

range = 'V', if one of or both vl and vu are present,

range = 'I', if one of or both il and iu are present,

range = 'A', if none of vl, vu, il, iu is present,

Note that there will be an error condition if one of or both vl and vu are present and at the same time one of or both il and iu are present.

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: Symmetric Eigenvalue Problems: LAPACK Driver Routines

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Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

?sbevx