F08SSF (ZHEGST) (PDF version)
F08 Chapter Contents
F08 Chapter Introduction
NAG Library Manual

NAG Library Routine Document

F08SSF (ZHEGST)

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

F08SSF (ZHEGST) reduces a complex Hermitian-definite generalized eigenproblem Az=λBz, ABz=λz or BAz=λz to the standard form Cy=λy, where A is a complex Hermitian matrix and B has been factorized by F07FRF (ZPOTRF).

2  Specification

SUBROUTINE F08SSF ( ITYPE, UPLO, N, A, LDA, B, LDB, INFO)
INTEGER  ITYPE, N, LDA, LDB, INFO
COMPLEX (KIND=nag_wp)  A(LDA,*), B(LDB,*)
CHARACTER(1)  UPLO
The routine may be called by its LAPACK name zhegst.

3  Description

To reduce the complex Hermitian-definite generalized eigenproblem Az=λBz, ABz=λz or BAz=λz to the standard form Cy=λy, F08SSF (ZHEGST) must be preceded by a call to F07FRF (ZPOTRF) which computes the Cholesky factorization of B; B must be positive definite.
The different problem types are specified by the parameter ITYPE, as indicated in the table below. The table shows how C is computed by the routine, and also how the eigenvectors z of the original problem can be recovered from the eigenvectors of the standard form.
ITYPE Problem UPLO B C z
1 Az=λBz 'U'
'L'
UHU 
LLH
U-HAU-1 
L-1AL-H
U-1y 
L-Hy
2 ABz=λz 'U'
'L'
UHU 
LLH
UAUH 
LHAL
U-1y 
L-Hy
3 BAz=λz 'U'
'L'
UHU 
LLH
UAUH 
LHAL
UHy 
Ly

4  References

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: indicates how the standard form is computed.
ITYPE=1
  • if UPLO='U', C=U-HAU-1;
  • if UPLO='L', C=L-1AL-H.
ITYPE=2 or 3
  • if UPLO='U', C=UAUH;
  • if UPLO='L', C=LHAL.
Constraint: ITYPE=1, 2 or 3.
2:     UPLO – CHARACTER(1)Input
On entry: indicates whether the upper or lower triangular part of A is stored and how B has been factorized.
UPLO='U'
The upper triangular part of A is stored and B=UHU.
UPLO='L'
The lower triangular part of A is stored and B=LLH.
Constraint: UPLO='U' or 'L'.
3:     N – INTEGERInput
On entry: n, the order of the matrices A and B.
Constraint: N0.
4:     A(LDA,*) – COMPLEX (KIND=nag_wp) arrayInput/Output
Note: the second dimension of the array A must be at least max1,N.
On entry: the n by n Hermitian matrix A.
  • If UPLO='U', the upper triangular part of A must be stored and the elements of the array below the diagonal are not referenced.
  • If UPLO='L', the lower triangular part of A must be stored and the elements of the array above the diagonal are not referenced.
On exit: the upper or lower triangle of A is overwritten by the corresponding upper or lower triangle of C as specified by ITYPE and UPLO.
5:     LDA – INTEGERInput
On entry: the first dimension of the array A as declared in the (sub)program from which F08SSF (ZHEGST) is called.
Constraint: LDAmax1,N.
6:     B(LDB,*) – COMPLEX (KIND=nag_wp) arrayInput
Note: the second dimension of the array B must be at least max1,N.
On entry: the Cholesky factor of B as specified by UPLO and returned by F07FRF (ZPOTRF).
7:     LDB – INTEGERInput
On entry: the first dimension of the array B as declared in the (sub)program from which F08SSF (ZHEGST) is called.
Constraint: LDBmax1,N.
8:     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.

7  Accuracy

Forming the reduced matrix C is a stable procedure. However it involves implicit multiplication by B-1 (if ITYPE=1) or B (if ITYPE=2 or 3). When F08SSF (ZHEGST) is used as a step in the computation of eigenvalues and eigenvectors of the original problem, there may be a significant loss of accuracy if B is ill-conditioned with respect to inversion. See the document for F08SNF (ZHEGV) for further details.

8  Further Comments

The total number of real floating point operations is approximately 4n3.
The real analogue of this routine is F08SEF (DSYGST).

9  Example

This example computes all the eigenvalues of Az=λBz, where
A = -7.36+0.00i 0.77-0.43i -0.64-0.92i 3.01-6.97i 0.77+0.43i 3.49+0.00i 2.19+4.45i 1.90+3.73i -0.64+0.92i 2.19-4.45i 0.12+0.00i 2.88-3.17i 3.01+6.97i 1.90-3.73i 2.88+3.17i -2.54+0.00i
and
B = 3.23+0.00i 1.51-1.92i 1.90+0.84i 0.42+2.50i 1.51+1.92i 3.58+0.00i -0.23+1.11i -1.18+1.37i 1.90-0.84i -0.23-1.11i 4.09+0.00i 2.33-0.14i 0.42-2.50i -1.18-1.37i 2.33+0.14i 4.29+0.00i .
Here B is Hermitian positive definite and must first be factorized by
F07FRF (ZPOTRF). The program calls F08SSF (ZHEGST) to reduce the problem to the standard form Cy=λy; then F08FSF (ZHETRD) to reduce C to tridiagonal form, and F08JFF (DSTERF) to compute the eigenvalues.

9.1  Program Text

Program Text (f08ssfe.f90)

9.2  Program Data

Program Data (f08ssfe.d)

9.3  Program Results

Program Results (f08ssfe.r)


F08SSF (ZHEGST) (PDF version)
F08 Chapter Contents
F08 Chapter Introduction
NAG Library Manual

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