F08TAF (DSPGV) (PDF version)
F08 Chapter Contents
F08 Chapter Introduction
NAG Library Manual

NAG Library Routine Document

F08TAF (DSPGV)

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

F08TAF (DSPGV) computes all the eigenvalues and, optionally, all the eigenvectors of a real generalized symmetric-definite eigenproblem, of the form
Az=λBz ,   ABz=λz   or   BAz=λz ,
where A and B are symmetric, stored in packed format, and B is also positive definite.

2  Specification

SUBROUTINE F08TAF ( ITYPE, JOBZ, UPLO, N, AP, BP, W, Z, LDZ, WORK, INFO)
INTEGER  ITYPE, N, LDZ, INFO
REAL (KIND=nag_wp)  AP(*), BP(*), W(N), Z(LDZ,*), WORK(3*N)
CHARACTER(1)  JOBZ, UPLO
The routine may be called by its LAPACK name dspgv.

3  Description

F08TAF (DSPGV) first performs a Cholesky factorization of the matrix B as B=UTU , when UPLO='U' or B=LLT , when UPLO='L'. The generalized problem is then reduced to a standard symmetric eigenvalue problem
Cx=λx ,
which is solved for the eigenvalues and, optionally, the eigenvectors; the eigenvectors are then backtransformed to give the eigenvectors of the original problem.
For the problem Az=λBz , 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. For the problem A B z = λ z  we correspondingly have
Z-1 A Z-T = Λ   and   ZT B Z = I ,
and for B A z = λ z  we have
ZT A Z = Λ   and   ZT B-1 Z = I .

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
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore

5  Parameters

1:     ITYPE – INTEGERInput
On entry: specifies the problem type to be solved.
ITYPE=1
Az=λBz.
ITYPE=2
ABz=λz.
ITYPE=3
BAz=λz.
2:     JOBZ – CHARACTER(1)Input
On entry: if JOBZ='N', compute eigenvalues only.
If JOBZ='V', compute eigenvalues and eigenvectors.
Constraint: JOBZ='N' or 'V'.
3:     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'.
4:     N – INTEGERInput
On entry: n, the order of the matrices A and B.
Constraint: N0.
5:     AP(*) – REAL (KIND=nag_wp) arrayInput/Output
Note: the dimension of the array AP must be at least max1,N×N+1/2.
On entry: the n by n symmetric matrix A, packed by columns.
More precisely,
  • if UPLO='U', the upper triangle of A must be stored with element Aij in APi+jj-1/2 for ij;
  • if UPLO='L', the lower triangle of A must be stored with element Aij in APi+2n-jj-1/2 for ij.
On exit: the contents of AP are destroyed.
6:     BP(*) – REAL (KIND=nag_wp) arrayInput/Output
Note: the dimension of the array BP must be at least max1,N×N+1/2.
On entry: the upper or lower triangle of the symmetric matrix B, packed by columns.
More precisely,
  • if UPLO='U', the upper triangle of B must be stored with element Bij in BPi+jj-1/2 for ij;
  • if UPLO='L', the lower triangle of B must be stored with element Bij in BPi+2n-jj-1/2 for ij.
On exit: the triangular factor U or L from the Cholesky factorization B=UTU or B=LLT, in the same storage format as B.
7:     W(N) – REAL (KIND=nag_wp) arrayOutput
On exit: if INFO=0, the eigenvalues in ascending order.
8:     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. The eigenvectors are normalized as follows:
  • if ITYPE=1 or 2, ZTBZ=I;
  • if ITYPE=3, ZTB-1Z=I.
If JOBZ='N', Z is not referenced.
9:     LDZ – INTEGERInput
On entry: the first dimension of the array Z as declared in the (sub)program from which F08TAF (DSPGV) is called.
Constraints:
  • if JOBZ='V', LDZmax1,N;
  • otherwise LDZ1.
10:   WORK(3×N) – REAL (KIND=nag_wp) arrayWorkspace
11:   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
F07GDF (DPPTRF) or F08GAF (DSPEV) returned an error code:
N if INFO=i, F08GAF (DSPEV) failed to converge; i off-diagonal elements of an intermediate tridiagonal form did not converge to zero;
>N if INFO=N+i, for 1iN, then the leading minor of order i of 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.
The example program below illustrates the computation of approximate error bounds.

8  Further Comments

The total number of floating point operations is proportional to n3 .
The complex analogue of this routine is F08TNF (ZHPGV).

9  Example

This example finds all the eigenvalues and eigenvectors of the generalized symmetric eigenproblem Az = λ Bz , where
A = 0.24 0.39 0.42 -0.16 0.39 -0.11 0.79 0.63 0.42 0.79 -0.25 0.48 -0.16 0.63 0.48 -0.03   and   B = 4.16 -3.12 0.56 -0.10 -3.12 5.03 -0.83 1.09 0.56 -0.83 0.76 0.34 -0.10 1.09 0.34 1.18 ,
together with an estimate of the condition number of B, and approximate error bounds for the computed eigenvalues and eigenvectors.
The example program for F08TCF (DSPGVD) illustrates solving a generalized symmetric eigenproblem of the form ABz=λz .

9.1  Program Text

Program Text (f08tafe.f90)

9.2  Program Data

Program Data (f08tafe.d)

9.3  Program Results

Program Results (f08tafe.r)


F08TAF (DSPGV) (PDF version)
F08 Chapter Contents
F08 Chapter Introduction
NAG Library Manual

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