# ?sbev

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

## Syntax

FORTRAN 77:

call ssbev(jobz, uplo, n, kd, ab, ldab, w, z, ldz, work, info)

call dsbev(jobz, uplo, n, kd, ab, ldab, w, z, ldz, work, info)

Fortran 95:

call sbev(a, w [,uplo] [,z] [,info])

## Description

This routine computes all eigenvalues and, optionally, eigenvectors of a real symmetric band matrix A.

## 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 ssbev

  DOUBLE PRECISION for dsbev.

  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, 3n-2).

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) .

## Output Parameters

w, z

  REAL for ssbev

  DOUBLE PRECISION for dsbev

  Arrays:

  w(*), DIMENSION at least max(1, n).

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

  z(ldz,*).

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

  If `jobz = 'V'`, then if `info = 0`, z contains the orthonormal eigenvectors of the matrix A, with the i-th column of z holding the eigenvector associated with w(i).

  If `jobz = 'N'`, then z is not referenced.

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.

info

  INTEGER.

  If `info = 0`, the execution is successful.

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

  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.

## 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 sbev 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).

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.

Parent topic: Symmetric Eigenproblems