F08TNF (ZHPGV) (PDF version)
F08 Chapter Contents
F08 Chapter Introduction
NAG Library Manual

NAG Library Routine Document

F08TNF (ZHPGV)

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

F08TNF (ZHPGV) computes all the eigenvalues and, optionally, all the eigenvectors of a complex generalized Hermitian-definite eigenproblem, of the form
Az=λBz ,   ABz=λz   or   BAz=λz ,
where A and B are Hermitian, stored in packed format, and B is also positive definite.

2  Specification

SUBROUTINE F08TNF ( ITYPE, JOBZ, UPLO, N, AP, BP, W, Z, LDZ, WORK, RWORK, INFO)
INTEGER  ITYPE, N, LDZ, INFO
REAL (KIND=nag_wp)  W(N), RWORK(3*N-2)
COMPLEX (KIND=nag_wp)  AP(*), BP(*), Z(LDZ,*), WORK(2*N-1)
CHARACTER(1)  JOBZ, UPLO
The routine may be called by its LAPACK name zhpgv.

3  Description

F08TNF (ZHPGV) first performs a Cholesky factorization of the matrix B as B=UHU , when UPLO='U' or B=LLH , 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
ZH A Z = Λ   and   ZH 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-H = Λ   and   ZH B Z = I ,
and for B A z = λ z  we have
ZH A Z = Λ   and   ZH 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(*) – COMPLEX (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 Hermitian 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(*) – COMPLEX (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 Hermitian 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=UHU or B=LLH, 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,*) – COMPLEX (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, ZHBZ=I;
  • if ITYPE=3, ZHB-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 F08TNF (ZHPGV) is called.
Constraints:
  • if JOBZ='V', LDZmax1,N;
  • otherwise LDZ1.
10:   WORK(2×N-1) – COMPLEX (KIND=nag_wp) arrayWorkspace
11:   RWORK(3×N-2) – REAL (KIND=nag_wp) arrayWorkspace
12:   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
F07GRF (ZPPTRF) or F08GNF (ZHPEV) returned an error code:
N if INFO=i, F08GNF (ZHPEV) 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 real analogue of this routine is F08TAF (DSPGV).

9  Example

This example finds all the eigenvalues and eigenvectors of the generalized Hermitian eigenproblem Az = λ Bz , where
A = -7.36i+0.00 0.77-0.43i -0.64-0.92i 3.01-6.97i 0.77+0.43i 3.49i+0.00 2.19+4.45i 1.90+3.73i -0.64+0.92i 2.19-4.45i 0.12i+0.00 2.88-3.17i 3.01+6.97i 1.90-3.73i 2.88+3.17i -2.54i+0.00
and
B = 3.23i+0.00 1.51-1.92i 1.90+0.84i 0.42+2.50i 1.51+1.92i 3.58i+0.00 -0.23+1.11i -1.18+1.37i 1.90-0.84i -0.23-1.11i 4.09i+0.00 2.33-0.14i 0.42-2.50i -1.18-1.37i 2.33+0.14i 4.29i+0.00 ,
together with an estimate of the condition number of B, and approximate error bounds for the computed eigenvalues and eigenvectors.
The example program for F08TQF (ZHPGVD) illustrates solving a generalized symmetric eigenproblem of the form ABz=λz .

9.1  Program Text

Program Text (f08tnfe.f90)

9.2  Program Data

Program Data (f08tnfe.d)

9.3  Program Results

Program Results (f08tnfe.r)


F08TNF (ZHPGV) (PDF version)
F08 Chapter Contents
F08 Chapter Introduction
NAG Library Manual

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