?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

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)

call sbgvd(ab, bb, w [,uplo] [,z] [,info])

Include Files

• mkl.fi, lapack.f90

Description

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

ldbb

INTEGER. The leading 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 pbstf/pbstf.

w, z

REAL for ssbgvd

DOUBLE PRECISION for dsbgvd

Arrays:

w(*), size 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 pbstf/pbstf returned `info = i` and B is not positive-definite. The factorization of B could not be completed and no eigenvalues or eigenvectors were computed.

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 sbgvd interface are the following:

ab

Holds the array A of size (ka+1,n).

bb

Holds the array B of size (kb+1,n).

w

Holds the vector with the number of elements 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.

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