# ?sbgvx

Computes selected eigenvalues and, optionally, eigenvectors of a real generalized symmetric definite eigenproblem with banded matrices.

## Syntax

FORTRAN 77:

call ssbgvx(jobz, range, uplo, n, ka, kb, ab, ldab, bb, ldbb, q, ldq, vl, vu, il, iu, abstol, m, w, z, ldz, work, iwork, ifail, info)

call dsbgvx(jobz, range, uplo, n, ka, kb, ab, ldab, bb, ldbb, q, ldq, vl, vu, il, iu, abstol, m, w, z, ldz, work, iwork, ifail, info)

Fortran 95:

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

## Description

This routine computes selected eigenvalues, and optionally, the eigenvectors of a real generalized symmetric-definite banded eigenproblem, of the form `A*x = λ*B*x`. Here A and B are assumed to be symmetric and banded, and B is also positive definite. Eigenvalues and eigenvectors can be selected by specifying either all eigenvalues, 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 compute eigenvalues only.

  If `jobz = 'V'`, then compute eigenvalues and eigenvectors.

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'`, arrays ab and bb store the upper triangles of A and B;

  If `uplo = 'L'`, arrays ab and bb store the lower triangles of A and B.

n

  INTEGER. The order of the matrices A and B (`n ≥ 0`).

ka

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

  (`ka ≥ 0`).

kb

  INTEGER. The number of super- or sub-diagonals in B (kb ≥ 0).

ab, bb, work

  REAL for ssbgvx

  DOUBLE PRECISION for dsbgvx

  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 the array ab must be at least max(1, n).

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

  The second dimension of the array bb must be at least max(1, n).

  work(*) is a workspace array, DIMENSION (7*n).

ldab

  INTEGER. The first dimension of the array ab; must be at least ka+1.

ldbb

  INTEGER. The first dimension of the array bb; must be at least kb+1.

vl, vu

  REAL for ssbgvx

  DOUBLE PRECISION for dsbgvx.

  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 ssbgvx

  DOUBLE PRECISION for dsbgvx.

  The absolute error tolerance for the eigenvalues. See Application Notes for more information.

ldz

  INTEGER. The leading dimension of the output array z; `ldz ≥ 1`. If `jobz = 'V'`, `ldz ≥ max(1, n)`.

ldq

  INTEGER. The leading dimension of the output array q; `ldq < 1`.

  If `jobz = 'V'`, `ldq < max(1, n)`.

iwork

  INTEGER.

  Workspace array, DIMENSION (5*n).

## Output Parameters

ab

  On exit, the contents of ab are overwritten.

bb

  On exit, contains the factor S from the split Cholesky factorization `B = ST*S`, as returned by spbstf/dpbstf.

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, q

  REAL for ssbgvx

  DOUBLE PRECISION for dsbgvx

  Arrays:

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

  If `info = 0`, contains the eigenvalues 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 matrix Z of eigenvectors , with the i-th column of z holding the eigenvector associated with w(i). The eigenvectors are normalized so that `ZT*B*Z = I`.

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

  q(ldq,*) .

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

  If `jobz = 'V'`, then q contains the n-by-n matrix used in the reduction of `A*x = lambda*B*x` to standard form, that is, `C*x= lambda*`x and consequently C to tridiagonal form.

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

ifail

  INTEGER.

  Array, DIMENSION (m).

  If `jobz = 'V'`, then if `info = 0`, the first m elements of ifail are zero; if `info > 0`, the ifail contains the indices of 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 argument had an illegal value.

  If `info > 0`, and

  if `i ≤ n`, the algorithm failed to converge, and i off-diagonal elements of an intermediate tridiagonal did not converge to zero;

  if `info = n + i`, for `1 ≤ i ≤ n`, then spbstf/dpbstf returned `info = i` and B is not positive-definite. The factorization of B could not be completed and no eigenvalues or eigenvectors were computed.

## 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 sbgvx interface are the following:

a

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

b

  Stands for argument bb in Fortan 77 interface. Holds the array B of size (kb+1,n).

w

  Holds the vector of length (n).

z

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

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 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: Generalized Symmetric Definite Eigenproblems