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:     order Nag_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 3.3.1.3 in How to Use the NAG Library and its Documentation for a more detailed explanation of the use of this argument.
Constraint: order=Nag_RowMajor or Nag_ColMajor.
2:     comp_type Nag_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:     uplo Nag_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:     n IntegerInput
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:     pda IntegerInput
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:     pdb IntegerInput
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:     fail NagError *Input/Output
The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

6
Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
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.
See Section 2.7.6 in How to Use the NAG Library and its Documentation for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.

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
Parallelism and Performance

nag_zhegst (f08ssc) 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 function. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

9
Further Comments

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

10
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.

10.1
Program Text

Program Text (f08ssce.c)

10.2
Program Data

Program Data (f08ssce.d)

10.3
Program Results

Program Results (f08ssce.r)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017