# ?sbevx

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

## Syntax

FORTRAN 77:

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)

Fortran 95:

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

## Description

This 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 `lambda(i)` in the half-open interval: `vl<lambda(i)≤vu`.

  If `range = 'I'`, the routine computes eigenvalues 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:

  ab (ldab,*) 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.

  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, DIMENSION at least max(1, 5n).

## Output Parameters

q

  REAL for ssbevx DOUBLE PRECISION for dsbevx.

  Array, DIMENSION (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`, and if `range = 'I'`, `m = iu-il+1`.

w, z

  REAL for ssbevx

  DOUBLE PRECISION for dsbevx

  Arrays:

  w(*), DIMENSION 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, DIMENSION 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.

## Fortran 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 Fortran 95 Interface Conventions.

Specific details for the routine sbevx interface are the following:

a

  Stands for argument ab in Fortan 77 interface. Holds the array A of size (kd+1,n).

w

  Holds the vector of length (n).

z

  Holds the matrix Z of size (n, n), where the values n and m are significant.

ifail

  Holds the vector of length (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 Eigenproblems