NAG FL Interface
f08tnf (zhpgv)

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1 Purpose

f08tnf 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

Fortran Interface
Subroutine f08tnf ( itype, jobz, uplo, n, ap, bp, w, z, ldz, work, rwork, info)
Integer, Intent (In) :: itype, n, ldz
Integer, Intent (Out) :: info
Real (Kind=nag_wp), Intent (Out) :: w(n), rwork(3*n-2)
Complex (Kind=nag_wp), Intent (Inout) :: ap(*), bp(*), z(ldz,*)
Complex (Kind=nag_wp), Intent (Out) :: work(2*n-1)
Character (1), Intent (In) :: jobz, uplo
C Header Interface
#include <nag.h>
void  f08tnf_ (const Integer *itype, const char *jobz, const char *uplo, const Integer *n, Complex ap[], Complex bp[], double w[], Complex z[], const Integer *ldz, Complex work[], double rwork[], Integer *info, const Charlen length_jobz, const Charlen length_uplo)
The routine may be called by the names f08tnf, nagf_lapackeig_zhpgv or its LAPACK name zhpgv.

3 Description

f08tnf 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
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore

5 Arguments

1: itype Integer Input
On entry: specifies the problem type to be solved.
Constraint: itype=1, 2 or 3.
2: jobz Character(1) Input
On entry: indicates whether eigenvectors are computed.
Only eigenvalues are computed.
Eigenvalues and eigenvectors are computed.
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 Integer Input
On entry: n, the order of the matrices A and B.
Constraint: n0.
5: ap(*) Complex (Kind=nag_wp) array Input/Output
Note: the dimension of the array ap must be at least max(1,n×(n+1)/2).
On entry: the upper or lower triangle of the n×n Hermitian matrix A, packed by columns.
More precisely,
  • if uplo='U', the upper triangle of A must be stored with element Aij in ap(i+j(j-1)/2) for ij;
  • if uplo='L', the lower triangle of A must be stored with element Aij in ap(i+(2n-j)(j-1)/2) for ij.
On exit: the contents of ap are destroyed.
6: bp(*) Complex (Kind=nag_wp) array Input/Output
Note: the dimension of the array bp must be at least max(1,n×(n+1)/2).
On entry: the upper or lower triangle of the n×n Hermitian matrix B, packed by columns.
More precisely,
  • if uplo='U', the upper triangle of B must be stored with element Bij in bp(i+j(j-1)/2) for ij;
  • if uplo='L', the lower triangle of B must be stored with element Bij in bp(i+(2n-j)(j-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) array Output
On exit: the eigenvalues in ascending order.
8: z(ldz,*) Complex (Kind=nag_wp) array Output
Note: the second dimension of the array z must be at least max(1,n) if jobz='V', and at least 1 otherwise.
On exit: if jobz='V', 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 Integer Input
On entry: the first dimension of the array z as declared in the (sub)program from which f08tnf is called.
  • if jobz='V', ldz max(1,n) ;
  • otherwise ldz1.
10: work(2×n-1) Complex (Kind=nag_wp) array Workspace
11: rwork(3×n-2) Real (Kind=nag_wp) array Workspace
12: info Integer Output
On exit: info=0 unless the routine detects an error (see Section 6).

6 Error Indicators and Warnings

If info=-i, argument i had an illegal value. An explanatory message is output, and execution of the program is terminated.
The algorithm failed to converge; value off-diagonal elements of an intermediate tridiagonal form did not converge to zero.
If info=n+value, for 1valuen, then the leading minor of order value 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 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.
f08tnf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f08tnf makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

9 Further Comments

The total number of floating-point operations is proportional to n3 .
The real analogue of this routine is f08taf.

10 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 )  
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 illustrates solving a generalized symmetric eigenproblem of the form ABz=λz .

10.1 Program Text

Program Text (f08tnfe.f90)

10.2 Program Data

Program Data (f08tnfe.d)

10.3 Program Results

Program Results (f08tnfe.r)