F08UAF (DSBGV) (PDF version)
F08 Chapter Contents
F08 Chapter Introduction
NAG Library Manual

NAG Library Routine Document

F08UAF (DSBGV)

Note:  before using this routine, please read the Users' Note for your implementation to check the interpretation of bold italicised terms and other implementation-dependent details.

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

F08UAF (DSBGV) computes all the eigenvalues and, optionally, the eigenvectors of a real generalized symmetric-definite banded eigenproblem, of the form
Az=λBz ,
where A and B are symmetric and banded, and B is also positive definite.

2  Specification

SUBROUTINE F08UAF ( JOBZ, UPLO, N, KA, KB, AB, LDAB, BB, LDBB, W, Z, LDZ, WORK, INFO)
INTEGER  N, KA, KB, LDAB, LDBB, LDZ, INFO
REAL (KIND=nag_wp)  AB(LDAB,*), BB(LDBB,*), W(N), Z(LDZ,*), WORK(3*N)
CHARACTER(1)  JOBZ, UPLO
The routine may be called by its LAPACK name dsbgv.

3  Description

The generalized symmetric-definite band problem
Az = λ Bz
is first reduced to a standard band symmetric problem
Cx = λx ,
where C is a symmetric band matrix, using Wilkinson's modification to Crawford's algorithm (see
Crawford (1973) and Wilkinson (1977)). The symmetric eigenvalue problem is then solved for the eigenvalues and the eigenvectors, if required, which are then backtransformed to the eigenvectors of the original problem.
The eigenvectors are normalized so that the matrix of eigenvectors, Z, satisfies
ZT A Z = Λ   and   ZT B Z = I ,
where Λ  is the diagonal matrix whose diagonal elements are the eigenvalues.

4  References

Anderson E, Bai Z, Bischof C, Blackford S, Demmel J, Dongarra J J, Du Croz J J, Greenbaum A, Hammarling S, McKenney A and Sorensen D (1999) LAPACK Users' Guide (3rd Edition) SIAM, Philadelphia http://www.netlib.org/lapack/lug
Crawford C R (1973) Reduction of a band-symmetric generalized eigenvalue problem Comm. ACM 16 41–44
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore
Wilkinson J H (1977) Some recent advances in numerical linear algebra The State of the Art in Numerical Analysis (ed D A H Jacobs) Academic Press

5  Parameters

1:     JOBZ – CHARACTER(1)Input
On entry: if JOBZ='N', compute eigenvalues only.
If JOBZ='V', compute eigenvalues and eigenvectors.
Constraint: JOBZ='N' or 'V'.
2:     UPLO – CHARACTER(1)Input
On entry: if UPLO='U', the upper triangles of A and B are stored.
If UPLO='L', the lower triangles of A and B are stored.
Constraint: UPLO='U' or 'L'.
3:     N – INTEGERInput
On entry: n, the order of the matrices A and B.
Constraint: N0.
4:     KA – INTEGERInput
5:     KB – INTEGERInput
6:     AB(LDAB,*) – REAL (KIND=nag_wp) arrayInput/Output
Note: the second dimension of the array AB must be at least max1,N.
On entry: the upper or lower triangle of the n by n symmetric band matrix A.
The matrix is stored in rows 1 to ka+1, more precisely,
  • if UPLO='U', the elements of the upper triangle of A within the band must be stored with element Aij in ABka+1+i-jj​ for ​max1,j-kaij;
  • if UPLO='L', the elements of the lower triangle of A within the band must be stored with element Aij in AB1+i-jj​ for ​jiminn,j+ka.
On exit: the contents of AB are destroyed.
7:     LDAB – INTEGERInput
On entry: the first dimension of the array AB as declared in the (sub)program from which F08UAF (DSBGV) is called.
Constraint: LDABKA+1.
8:     BB(LDBB,*) – REAL (KIND=nag_wp) arrayInput/Output
Note: the second dimension of the array BB must be at least max1,N.
On entry: the upper or lower triangle of the n by n symmetric band matrix B.
The matrix is stored in rows 1 to kb+1, more precisely,
  • if UPLO='U', the elements of the upper triangle of B within the band must be stored with element Bij in BBkb+1+i-jj​ for ​max1,j-kbij;
  • if UPLO='L', the elements of the lower triangle of B within the band must be stored with element Bij in BB1+i-jj​ for ​jiminn,j+kb.
On exit: the factor S from the split Cholesky factorization B=STS, as returned by F08UFF (DPBSTF).
9:     LDBB – INTEGERInput
On entry: the first dimension of the array BB as declared in the (sub)program from which F08UAF (DSBGV) is called.
Constraint: LDBBKB+1.
10:   W(N) – REAL (KIND=nag_wp) arrayOutput
On exit: if INFO=0, the eigenvalues in ascending order.
11:   Z(LDZ,*) – REAL (KIND=nag_wp) arrayOutput
Note: the second dimension of the array Z must be at least max1,N if JOBZ='V', and at least 1 otherwise.
On exit: if JOBZ='V', then if INFO=0, Z contains the matrix Z of eigenvectors, with the ith column of Z holding the eigenvector associated with Wi. The eigenvectors are normalized so that ZTBZ=I.
If JOBZ='N', Z is not referenced.
12:   LDZ – INTEGERInput
On entry: the first dimension of the array Z as declared in the (sub)program from which F08UAF (DSBGV) is called.
Constraints:
  • if JOBZ='V', LDZmax1,N;
  • otherwise LDZ1.
13:   WORK(3×N) – REAL (KIND=nag_wp) arrayWorkspace
14:   INFO – INTEGEROutput

6  Error Indicators and Warnings

Errors or warnings detected by the routine:
INFO<0
If INFO=-i, argument i had an illegal value. An explanatory message is output, and execution of the program is terminated.
INFO>0
If INFO=i and iN, the algorithm failed to converge: i off-diagonal elements of an intermediate tridiagonal form did not converge to zero.
If INFO=i and i>N, if INFO=N+i, for 1iN, then F08UFF (DPBSTF) returned INFO=i: B is not positive definite. The factorization of B could not be completed and no eigenvalues or eigenvectors were computed.

7  Accuracy

If B is ill-conditioned with respect to inversion, then the error bounds for the computed eigenvalues and vectors may be large, although when the diagonal elements of B differ widely in magnitude the eigenvalues and eigenvectors may be less sensitive than the condition of B would suggest. See Section 4.10 of Anderson et al. (1999) for details of the error bounds.

8  Further Comments

The total number of floating point operations is proportional to n3  if JOBZ='V' and, assuming that nka , is approximately proportional to n2 ka  otherwise.
The complex analogue of this routine is F08UNF (ZHBGV).

9  Example

This example finds all the eigenvalues of the generalized band symmetric eigenproblem Az = λ Bz , where
A = 0.24 0.39 0.42 0.00 0.39 -0.11 0.79 0.63 0.42 0.79 -0.25 0.48 0.00 0.63 0.48 -0.03   and   B = 2.07 0.95 0.00 0.00 0.95 1.69 -0.29 0.00 0.00 -0.29 0.65 -0.33 0.00 0.00 -0.33 1.17 .

9.1  Program Text

Program Text (f08uafe.f90)

9.2  Program Data

Program Data (f08uafe.d)

9.3  Program Results

Program Results (f08uafe.r)


F08UAF (DSBGV) (PDF version)
F08 Chapter Contents
F08 Chapter Introduction
NAG Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2011