nag_dorgtr (f08ffc) (PDF version)
f08 Chapter Contents
f08 Chapter Introduction
NAG C Library Manual

NAG Library Function Document

nag_dorgtr (f08ffc)

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

nag_dorgtr (f08ffc) generates the real orthogonal matrix Q, which was determined by nag_dsytrd (f08fec) when reducing a symmetric matrix to tridiagonal form.

2  Specification

#include <nag.h>
#include <nagf08.h>
void  nag_dorgtr (Nag_OrderType order, Nag_UploType uplo, Integer n, double a[], Integer pda, const double tau[], NagError *fail)

3  Description

nag_dorgtr (f08ffc) is intended to be used after a call to nag_dsytrd (f08fec), which reduces a real symmetric matrix A to symmetric tridiagonal form T by an orthogonal similarity transformation: A=QTQT. nag_dsytrd (f08fec) represents the orthogonal matrix Q as a product of n-1 elementary reflectors.
This function 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  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:     uploNag_UploTypeInput
On entry: this must be the same argument uplo as supplied to nag_dsytrd (f08fec).
Constraint: uplo=Nag_Upper or Nag_Lower.
3:     nIntegerInput
On entry: n, the order of the matrix Q.
Constraint: n0.
4:     a[dim]doubleInput/Output
Note: the dimension, dim, of the array a must be at least max1,pda×n.
On entry: details of the vectors which define the elementary reflectors, as returned by nag_dsytrd (f08fec).
On exit: the n by n orthogonal matrix Q.
If order=Nag_ColMajor, the i,jth element of the matrix is stored in a[j-1×pda+i-1].
If order=Nag_RowMajor, the i,jth element of the matrix is stored in a[i-1×pda+j-1].
5:     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.
6:     tau[dim]const doubleInput
Note: the dimension, dim, of the array tau must be at least max1,n-1.
On entry: further details of the elementary reflectors, as returned by nag_dsytrd (f08fec).
7:     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.
NE_INT_2
On entry, pda=value and n=value.
Constraint: pdamax1,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

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 function is nag_zungtr (f08ftc).

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 .
Here A is symmetric and must first be reduced to tridiagonal form by nag_dsytrd (f08fec). The program then calls nag_dorgtr (f08ffc) to form Q, and passes this matrix to nag_dsteqr (f08jec) which computes the eigenvalues and eigenvectors of A.

9.1  Program Text

Program Text (f08ffce.c)

9.2  Program Data

Program Data (f08ffce.d)

9.3  Program Results

Program Results (f08ffce.r)


nag_dorgtr (f08ffc) (PDF version)
f08 Chapter Contents
f08 Chapter Introduction
NAG C Library Manual

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