# ?sbgvd

Computes all eigenvalues and, optionally, eigenvectors of a real generalized symmetric definite eigenproblem with banded matrices. If eigenvectors are desired, it uses a divide and conquer method.

## Syntax

FORTRAN 77:

call ssbgvd(jobz, uplo, n, ka, kb, ab, ldab, bb, ldbb, w, z, ldz, work, lwork, iwork, liwork, info)

call dsbgvd(jobz, uplo, n, ka, kb, ab, ldab, bb, ldbb, w, z, ldz, work, lwork, iwork, liwork, info)

Fortran 95:

call sbgvd(a, b, w [,uplo] [,z] [,info])

## Description

This routine computes all the 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.

If eigenvectors are desired, it uses a divide and conquer algorithm.

## 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.

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 ssbgvd

  DOUBLE PRECISION for dsbgvd

  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, its dimension `max(1, lwork)`.

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.

ldz

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

lwork

  INTEGER.

  The dimension of the array work.

  Constraints:

  If `n ≤ 1`, `lwork ≥ 1`;

  If `jobz = 'N'` and `n>1`, lwork ≥ 3n;

  If `jobz = 'V'` and `n>1`, `lwork ≥ 2n2+5n+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`, `liwork ≥ 1`;

  If `jobz = 'N'` and `n>1`, `liwork ≥ 1`;

  If `jobz = 'V'` and `n>1`, `liwork ≥ 5n+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

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.

w, z

  REAL for ssbgvd

  DOUBLE PRECISION for dsbgvd

  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.

work`(1)`

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

iwork`(1)`

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

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 sbgvd 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).

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

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 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.

Parent topic: Generalized Symmetric Definite Eigenproblems