nag_zhegst (f08ssc) (PDF version)
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f08 Chapter Introduction
NAG C Library Manual

NAG Library Function Document

nag_zhegst (f08ssc)

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

nag_zhegst (f08ssc) 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 nag_zpotrf (f07frc).

2  Specification

#include <nag.h>
#include <nagf08.h>
void  nag_zhegst (Nag_OrderType order, Nag_ComputeType comp_type, Nag_UploType uplo, Integer n, Complex a[], Integer pda, const Complex b[], Integer pdb, NagError *fail)

3  Description

To reduce the complex Hermitian-definite generalized eigenproblem Az=λBz, ABz=λz or BAz=λz to the standard form Cy=λy, nag_zhegst (f08ssc) must be preceded by a call to nag_zpotrf (f07frc) which computes the Cholesky factorization of B; B must be positive definite.
The different problem types are specified by the argument comp_type, as indicated in the table below. The table shows how C is computed by the function, and also how the eigenvectors z of the original problem can be recovered from the eigenvectors of the standard form.
  order=Nag_ColMajor order=Nag_RowMajor
comp_type Problem uplo B C z B C z
1 Az=λBz Nag_Upper
Nag_Lower
UHU 
LLH
U-HAU-1 
L-1AL-H
U-1y 
L-Hy
UUH 
LHL
U-1AU-H 
L-HAL-1
U-Hy 
L-1y
2 ABz=λz Nag_Upper
Nag_Lower
UHU 
LLH
UAUH 
LHAL
U-1y 
L-Hy
UUH 
LHL
UHAU 
LALH
U-Hy 
L-1y
3 BAz=λz Nag_Upper
Nag_Lower
UHU 
LLH
UAUH 
LHAL
UHy 
Ly
UUH 
LHL
UHAU 
LALH
Uy 
LHy

4  References

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

5  Arguments

1:     orderNag_OrderTypeInput
On entry: the order argument specifies the two-dimensional storage scheme being used, i.e., row-major ordering or column-major ordering. C language defined storage is specified by order=Nag_RowMajor. See Section 3.2.1.3 in the Essential Introduction for a more detailed explanation of the use of this argument.
Constraint: order=Nag_RowMajor or Nag_ColMajor.
2:     comp_typeNag_ComputeTypeInput
On entry: indicates how the standard form is computed.
comp_type=Nag_Compute_1
  • if uplo=Nag_Upper, C=U-HAU-1 when order=Nag_ColMajor and C=U-1AU-H when order=Nag_RowMajor;
  • if uplo=Nag_Lower, C=L-1AL-H when order=Nag_ColMajor and C=L-HAL-1 when order=Nag_RowMajor.
comp_type=Nag_Compute_2 or Nag_Compute_3
  • if uplo=Nag_Upper, C=UAUH when order=Nag_ColMajor and C=UHAU when order=Nag_RowMajor;
  • if uplo=Nag_Lower, C=LHAL when order=Nag_ColMajor and C=LALH when order=Nag_RowMajor.
Constraint: comp_type=Nag_Compute_1, Nag_Compute_2 or Nag_Compute_3.
3:     uploNag_UploTypeInput
On entry: indicates whether the upper or lower triangular part of A is stored and how B has been factorized.
uplo=Nag_Upper
The upper triangular part of A is stored and B=UHU when order=Nag_ColMajor and B=UUH when order=Nag_RowMajor.
uplo=Nag_Lower
The lower triangular part of A is stored and B=LLH when order=Nag_ColMajor and B=LHL when order=Nag_RowMajor.
Constraint: uplo=Nag_Upper or Nag_Lower.
4:     nIntegerInput
On entry: n, the order of the matrices A and B.
Constraint: n0.
5:     a[dim]ComplexInput/Output
Note: the dimension, dim, of the array a must be at least max1,pda×n.
On entry: the n by n Hermitian matrix A.
If order=Nag_ColMajor, Aij is stored in a[j-1×pda+i-1].
If order=Nag_RowMajor, Aij is stored in a[i-1×pda+j-1].
If uplo=Nag_Upper, the upper triangular part of A must be stored and the elements of the array below the diagonal are not referenced.
If uplo=Nag_Lower, 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 comp_type and uplo.
6:     pdaIntegerInput
On entry: the stride separating row or column elements (depending on the value of order) of the matrix A in the array a.
Constraint: pdamax1,n.
7:     b[dim]const ComplexInput
Note: the dimension, dim, of the array b must be at least max1,pdb×n.
On entry: the Cholesky factor of B as specified by uplo and returned by nag_zpotrf (f07frc).
8:     pdbIntegerInput
On entry: the stride separating row or column elements (depending on the value of order) of the matrix B in the array b.
Constraint: pdbmax1,n.
9:     failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

6  Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
NE_BAD_PARAM
On entry, argument value had an illegal value.
NE_INT
On entry, n=value.
Constraint: n0.
On entry, pda=value.
Constraint: pda>0.
On entry, pdb=value.
Constraint: pdb>0.
NE_INT_2
On entry, pda=value and n=value.
Constraint: pdamax1,n.
On entry, pdb=value and n=value.
Constraint: pdbmax1,n.
NE_INTERNAL_ERROR
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.

7  Accuracy

Forming the reduced matrix C is a stable procedure. However it involves implicit multiplication by B-1 (if comp_type=Nag_Compute_1) or B (if comp_type=Nag_Compute_2 or Nag_Compute_3). When nag_zhegst (f08ssc) 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.

8  Further Comments

The total number of real floating point operations is approximately 4n3.
The real analogue of this function is nag_dsygst (f08sec).

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 nag_zpotrf (f07frc). The program calls nag_zhegst (f08ssc) to reduce the problem to the standard form Cy=λy; then nag_zhetrd (f08fsc) to reduce C to tridiagonal form, and nag_dsterf (f08jfc) to compute the eigenvalues.

9.1  Program Text

Program Text (f08ssce.c)

9.2  Program Data

Program Data (f08ssce.d)

9.3  Program Results

Program Results (f08ssce.r)


nag_zhegst (f08ssc) (PDF version)
f08 Chapter Contents
f08 Chapter Introduction
NAG C Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2012