F08GFF (DOPGTR) (PDF version)
F08 Chapter Contents
F08 Chapter Introduction
NAG Library Manual

NAG Library Routine Document

F08GFF (DOPGTR)

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

F08GFF (DOPGTR) generates the real orthogonal matrix Q, which was determined by F08GEF (DSPTRD) when reducing a symmetric matrix to tridiagonal form.

2  Specification

SUBROUTINE F08GFF ( UPLO, N, AP, TAU, Q, LDQ, WORK, INFO)
INTEGER  N, LDQ, INFO
REAL (KIND=nag_wp)  AP(*), TAU(*), Q(LDQ,*), WORK(N-1)
CHARACTER(1)  UPLO
The routine may be called by its LAPACK name dopgtr.

3  Description

F08GFF (DOPGTR) is intended to be used after a call to F08GEF (DSPTRD), which reduces a real symmetric matrix A to symmetric tridiagonal form T by an orthogonal similarity transformation: A=QTQT. F08GEF (DSPTRD) represents the orthogonal matrix Q as a product of n-1 elementary reflectors.
This routine may be used to generate Q explicitly as a square matrix.

4  References

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

5  Parameters

1:     UPLO – CHARACTER(1)Input
Constraint: UPLO='U' or 'L'.
2:     N – INTEGERInput
On entry: n, the order of the matrix Q.
Constraint: N0.
3:     AP(*) – REAL (KIND=nag_wp) arrayInput
Note: the dimension of the array AP must be at least max1,N×N+1/2.
On entry: details of the vectors which define the elementary reflectors, as returned by F08GEF (DSPTRD).
4:     TAU(*) – REAL (KIND=nag_wp) arrayInput
Note: the dimension of the array TAU must be at least max1,N-1.
On entry: further details of the elementary reflectors, as returned by F08GEF (DSPTRD).
5:     Q(LDQ,*) – REAL (KIND=nag_wp) arrayOutput
Note: the second dimension of the array Q must be at least max1,N.
On exit: the n by n orthogonal matrix Q.
6:     LDQ – INTEGERInput
On entry: the first dimension of the array Q as declared in the (sub)program from which F08GFF (DOPGTR) is called.
Constraint: LDQmax1,N.
7:     WORK(N-1) – REAL (KIND=nag_wp) arrayWorkspace
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

The computed matrix Q differs from an exactly orthogonal matrix by a matrix E such that
E2 = Oε ,
where ε is the machine precision.

8  Further Comments

The total number of floating point operations is approximately 43n3.
The complex analogue of this routine is F08GTF (ZUPGTR).

9  Example

This example computes all the eigenvalues and eigenvectors of the matrix A, where
A = 2.07 3.87 4.20 -1.15 3.87 -0.21 1.87 0.63 4.20 1.87 1.15 2.06 -1.15 0.63 2.06 -1.81 ,
using packed storage. Here A is symmetric and must first be reduced to tridiagonal form by
F08GEF (DSPTRD). The program then calls F08GFF (DOPGTR) to form Q, and passes this matrix to F08JEF (DSTEQR) which computes the eigenvalues and eigenvectors of A.

9.1  Program Text

Program Text (f08gffe.f90)

9.2  Program Data

Program Data (f08gffe.d)

9.3  Program Results

Program Results (f08gffe.r)


F08GFF (DOPGTR) (PDF version)
F08 Chapter Contents
F08 Chapter Introduction
NAG Library Manual

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