F08TQF (ZHPGVD) (PDF version)
F08 Chapter Contents
F08 Chapter Introduction
NAG Library Manual

NAG Library Routine Document

F08TQF (ZHPGVD)

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

F08TQF (ZHPGVD) 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. If eigenvectors are desired, it uses a divide-and-conquer algorithm.

2  Specification

SUBROUTINE F08TQF ( ITYPE, JOBZ, UPLO, N, AP, BP, W, Z, LDZ, WORK, LWORK, RWORK, LRWORK, IWORK, LIWORK, INFO)
INTEGER  ITYPE, N, LDZ, LWORK, LRWORK, IWORK(max(1,LIWORK)), LIWORK, INFO
REAL (KIND=nag_wp)  W(N), RWORK(max(1,LRWORK))
COMPLEX (KIND=nag_wp)  AP(*), BP(*), Z(LDZ,*), WORK(max(1,LWORK))
CHARACTER(1)  JOBZ, UPLO
The routine may be called by its LAPACK name zhpgvd.

3  Description

F08TQF (ZHPGVD) 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 F08TQF (ZHPGVD) is called.
Constraints:
  • if JOBZ='V', LDZmax1,N;
  • otherwise LDZ1.
10:   WORK(max1,LWORK) – COMPLEX (KIND=nag_wp) arrayWorkspace
11:   LWORK – INTEGERInput
On entry: the dimension of the array WORK as declared in the (sub)program from which F08TQF (ZHPGVD) is called.
If LWORK=-1, a workspace query is assumed; the routine only calculates the optimal sizes of the WORK, RWORK and IWORK arrays, returns these values as the first entries of the WORK, RWORK and IWORK arrays, and no error message related to LWORK, LRWORK or LIWORK is issued.
Constraints:
  • if N1, LWORK1;
  • if JOBZ='N' and N>1, LWORKN;
  • if JOBZ='V' and N>1, LWORK2×N.
12:   RWORK(max1,LRWORK) – REAL (KIND=nag_wp) arrayWorkspace
13:   LRWORK – INTEGERInput
On entry: the dimension of the array RWORK as declared in the (sub)program from which F08TQF (ZHPGVD) is called.
If LRWORK=-1, a workspace query is assumed; the routine only calculates the optimal sizes of the WORK, RWORK and IWORK arrays, returns these values as the first entries of the WORK, RWORK and IWORK arrays, and no error message related to LWORK, LRWORK or LIWORK is issued.
Constraints:
  • if N1, LRWORK1;
  • if JOBZ='N' and N>1, LRWORKN;
  • if JOBZ='V' and N>1, LRWORK1+5×N+2×N2.
14:   IWORK(max1,LIWORK) – INTEGER arrayWorkspace
15:   LIWORK – INTEGERInput
On entry: the dimension of the array IWORK as declared in the (sub)program from which F08TQF (ZHPGVD) is called.
If LIWORK=-1, a workspace query is assumed; the routine only calculates the optimal sizes of the WORK, RWORK and IWORK arrays, returns these values as the first entries of the WORK, RWORK and IWORK arrays, and no error message related to LWORK, LRWORK or LIWORK is issued.
Constraints:
  • if JOBZ='N' or N1, LIWORK1;
  • if JOBZ='V' and N>1, LIWORK3+5×N.
16:   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 F08GQF (ZHPEVD) returned an error code:
N if INFO=i, F08GQF (ZHPEVD) 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 F08TCF (DSPGVD).

9  Example

This example finds all the eigenvalues and eigenvectors of the generalized Hermitian eigenproblem ABz=λz , 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 F08TNF (ZHPGV) illustrates solving a generalized Hermitian eigenproblem of the form Az = λ Bz .

9.1  Program Text

Program Text (f08tqfe.f90)

9.2  Program Data

Program Data (f08tqfe.d)

9.3  Program Results

Program Results (f08tqfe.r)


F08TQF (ZHPGVD) (PDF version)
F08 Chapter Contents
F08 Chapter Introduction
NAG Library Manual

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