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Chapter Contents
Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_lapack_dopgtr (f08gf)


    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example


nag_lapack_dopgtr (f08gf) generates the real orthogonal matrix Q, which was determined by nag_lapack_dsptrd (f08ge) when reducing a symmetric matrix to tridiagonal form.


[q, info] = f08gf(uplo, n, ap, tau)
[q, info] = nag_lapack_dopgtr(uplo, n, ap, tau)


nag_lapack_dopgtr (f08gf) is intended to be used after a call to nag_lapack_dsptrd (f08ge), which reduces a real symmetric matrix A to symmetric tridiagonal form T by an orthogonal similarity transformation: A=QTQT. nag_lapack_dsptrd (f08ge) 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.


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


Compulsory Input Parameters

1:     uplo – string (length ≥ 1)
This must be the same argument uplo as supplied to nag_lapack_dsptrd (f08ge).
Constraint: uplo='U' or 'L'.
2:     n int64int32nag_int scalar
n, the order of the matrix Q.
Constraint: n0.
3:     ap: – double array
The dimension of the array ap must be at least max1,n×n+1/2
Details of the vectors which define the elementary reflectors, as returned by nag_lapack_dsptrd (f08ge).
4:     tau: – double array
The dimension of the array tau must be at least max1,n-1
Further details of the elementary reflectors, as returned by nag_lapack_dsptrd (f08ge).

Optional Input Parameters


Output Parameters

1:     qldq: – double array
The first dimension of the array q will be max1,n.
The second dimension of the array q will be max1,n.
The n by n orthogonal matrix Q.
2:     info int64int32nag_int scalar
info=0 unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

If info=-i, parameter i had an illegal value on entry. The parameters are numbered as follows:
1: uplo, 2: n, 3: ap, 4: tau, 5: q, 6: ldq, 7: work, 8: info.
It is possible that info refers to a parameter that is omitted from the MATLAB interface. This usually indicates that an error in one of the other input parameters has caused an incorrect value to be inferred.


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

Further Comments

The total number of floating-point operations is approximately 43n3.
The complex analogue of this function is nag_lapack_zupgtr (f08gt).


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 nag_lapack_dsptrd (f08ge). The program then calls nag_lapack_dopgtr (f08gf) to form Q, and passes this matrix to nag_lapack_dsteqr (f08je) which computes the eigenvalues and eigenvectors of A.
function f08gf_example

fprintf('f08gf example results\n\n');

% Symmetric matrix A stored in symmetric packed format (Lower)
uplo = 'L';
n = int64(4);
ap = [2.07;     3.87;     4.2;     -1.15;
               -0.21;     1.87;     0.63;
                          1.15;     2.06;
% Reduce A to tridiagonal form
[apf, d, e, tau, info] = f08ge( ...
                                uplo, n, ap);

% Form Q
[q, info] = f08gf( ...
                   uplo, n, apf, tau);

% Calculate eigenvalues and eigenvectors
compz = 'Vectors';
[w, ~, z, info] = f08je( ...
                         compz, d, e, 'z', q);


% Normalize eigenvectors: largest element positive
for j = 1:n
  [~,k] = max(abs(z(:,j)));
  if z(k,j) < 0;
    z(:,j) = -z(:,j);


f08gf example results

   -5.0034   -1.9987    0.2013    8.0008

    0.5658   -0.2328   -0.3965    0.6845
   -0.3478    0.7994   -0.1780    0.4564
   -0.4740   -0.4087    0.5381    0.5645
    0.5781    0.3737    0.7221    0.0676

PDF version (NAG web site, 64-bit version, 64-bit version)
Chapter Contents
Chapter Introduction
NAG Toolbox

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