nag_dspgst (f08tec) (PDF version)
f08 Chapter Contents
f08 Chapter Introduction
NAG C Library Manual

NAG Library Function Document

nag_dspgst (f08tec)

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

nag_dspgst (f08tec) reduces a real symmetric-definite generalized eigenproblem Az=λBz, ABz=λz or BAz=λz to the standard form Cy=λy, where A is a real symmetric matrix and B has been factorized by nag_dpptrf (f07gdc), using packed storage.

2  Specification

#include <nag.h>
#include <nagf08.h>
void  nag_dspgst (Nag_OrderType order, Nag_ComputeType comp_type, Nag_UploType uplo, Integer n, double ap[], const double bp[], NagError *fail)

3  Description

To reduce the real symmetric-definite generalized eigenproblem Az=λBz, ABz=λz or BAz=λz to the standard form Cy=λy using packed storage, nag_dspgst (f08tec) must be preceded by a call to nag_dpptrf (f07gdc) 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.
comp_type Problem uplo B C z
Nag_Compute_1 Az=λBz Nag_Upper
Nag_Lower
UTU 
LLT
U-TAU-1 
L-1AL-T
U-1y 
L-Ty
Nag_Compute_2 ABz=λz Nag_Upper
Nag_Lower
UTU 
LLT
UAUT 
LTAL
U-1y 
L-Ty
Nag_Compute_3 BAz=λz Nag_Upper
Nag_Lower
UTU 
LLT
UAUT 
LTAL
UTy 
Ly

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-TAU-1;
  • if uplo=Nag_Lower, C=L-1AL-T.
comp_type=Nag_Compute_2 or Nag_Compute_3
  • if uplo=Nag_Upper, C=UAUT;
  • if uplo=Nag_Lower, C=LTAL.
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=UTU.
uplo=Nag_Lower
The lower triangular part of A is stored and B=LLT.
Constraint: uplo=Nag_Upper or Nag_Lower.
4:     nIntegerInput
On entry: n, the order of the matrices A and B.
Constraint: n0.
5:     ap[dim]doubleInput/Output
Note: the dimension, dim, of the array ap must be at least max1,n×n+1/2.
On entry: the upper or lower triangle of the n by n symmetric matrix A, packed by rows or columns.
The storage of elements Aij depends on the order and uplo arguments as follows:
  • if order=Nag_ColMajor and uplo=Nag_Upper,
              Aij is stored in ap[j-1×j/2+i-1], for ij;
  • if order=Nag_ColMajor and uplo=Nag_Lower,
              Aij is stored in ap[2n-j×j-1/2+i-1], for ij;
  • if order=Nag_RowMajor and uplo=Nag_Upper,
              Aij is stored in ap[2n-i×i-1/2+j-1], for ij;
  • if order=Nag_RowMajor and uplo=Nag_Lower,
              Aij is stored in ap[i-1×i/2+j-1], for ij.
On exit: the upper or lower triangle of ap is overwritten by the corresponding upper or lower triangle of C as specified by comp_type and uplo, using the same packed storage format as described above.
6:     bp[dim]const doubleInput
Note: the dimension, dim, of the array bp must be at least max1,n×n+1/2.
On entry: the Cholesky factor of B as specified by uplo and returned by nag_dpptrf (f07gdc).
7:     failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

6  Error Indicators and Warnings

NE_BAD_PARAM
On entry, argument value had an illegal value.
NE_INT
On entry, n=value.
Constraint: n0.
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_dspgst (f08tec) 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 floating point operations is approximately n3.
The complex analogue of this function is nag_zhpgst (f08tsc).

9  Example

This example computes all the eigenvalues of Az=λBz, where
A = 0.24 0.39 0.42 -0.16 0.39 -0.11 0.79 0.63 0.42 0.79 -0.25 0.48 -0.16 0.63 0.48 -0.03   and   B= 4.16 -3.12 0.56 -0.10 -3.12 5.03 -0.83 1.09 0.56 -0.83 0.76 0.34 -0.10 1.09 0.34 1.18 ,
using packed storage. Here B is symmetric positive definite and must first be factorized by nag_dpptrf (f07gdc). The program calls nag_dspgst (f08tec) to reduce the problem to the standard form Cy=λy; then nag_dsptrd (f08gec) to reduce C to tridiagonal form, and nag_dsterf (f08jfc) to compute the eigenvalues.

9.1  Program Text

Program Text (f08tece.c)

9.2  Program Data

Program Data (f08tece.d)

9.3  Program Results

Program Results (f08tece.r)


nag_dspgst (f08tec) (PDF version)
f08 Chapter Contents
f08 Chapter Introduction
NAG C Library Manual

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