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NAG Toolbox

NAG Toolbox: nag_lapack_dorgtr (f08ff)

 Contents

    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example

Purpose

nag_lapack_dorgtr (f08ff) generates the real orthogonal matrix Q, which was determined by nag_lapack_dsytrd (f08fe) when reducing a symmetric matrix to tridiagonal form.

Syntax

[a, info] = f08ff(uplo, a, tau, 'n', n)
[a, info] = nag_lapack_dorgtr(uplo, a, tau, 'n', n)

Description

nag_lapack_dorgtr (f08ff) is intended to be used after a call to nag_lapack_dsytrd (f08fe), which reduces a real symmetric matrix A to symmetric tridiagonal form T by an orthogonal similarity transformation: A=QTQT. nag_lapack_dsytrd (f08fe) 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.

References

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

Parameters

Compulsory Input Parameters

1:     uplo – string (length ≥ 1)
This must be the same argument uplo as supplied to nag_lapack_dsytrd (f08fe).
Constraint: uplo='U' or 'L'.
2:     alda: – double array
The first dimension of the array a must be at least max1,n.
The second dimension of the array a must be at least max1,n.
Details of the vectors which define the elementary reflectors, as returned by nag_lapack_dsytrd (f08fe).
3:     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_dsytrd (f08fe).

Optional Input Parameters

1:     n int64int32nag_int scalar
Default: the first dimension of the array a and the second dimension of the array a.
n, the order of the matrix Q.
Constraint: n0.

Output Parameters

1:     alda: – double array
The first dimension of the array a will be max1,n.
The second dimension of the array a 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

   info=-i
If info=-i, parameter i had an illegal value on entry. The parameters are numbered as follows:
1: uplo, 2: n, 3: a, 4: lda, 5: tau, 6: work, 7: lwork, 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.

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.

Further Comments

The total number of floating-point operations is approximately 43n3.
The complex analogue of this function is nag_lapack_zungtr (f08ft).

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_lapack_dsytrd (f08fe). The program then calls nag_lapack_dorgtr (f08ff) to form Q, and passes this matrix to nag_lapack_dsteqr (f08je) which computes the eigenvalues and eigenvectors of A.
function f08ff_example


fprintf('f08ff example results\n\n');

uplo = 'L';
a = [ 2.07,  0,    0,     0;
      3.87, -0.21, 0,     0;
      4.20,  1.87, 1.15,  0;
     -1.15,  0.63, 2.06, -1.81];

% Reduce to tridiagonal form
[t, d, e, tau, info] = f08fe( ...
			      uplo, a);

% Form Q
[q, info] = f08ff( ...
		   uplo, t, tau);

% Compute eigenvalues and eigenvectors
[w, e, z, info] = f08je( ...
			 'V', d, e, 'z', q);

disp('Eigenvalues');
disp(w');
disp('Eigenvectors');
disp(z);


f08ff example results

Eigenvalues
   -5.0034   -1.9987    0.2013    8.0008

Eigenvectors
   -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


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