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: $a (lda :)$ – double array: The first dimension of the array a must be at least $\max (1, n)$ .
The second dimension of the array a must be at least $\max (1, n)$ .

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 $\max (1, n - 1)$

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: $n \geq 0$ .

Output Parameters

1: $a (lda :)$ – double array: The first dimension of the array a will be $\max (1, n)$ .
The second dimension of the array a will be $\max (1, n)$ .

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

{‖E‖}_{2} = O (ε),

where

ε

is the machine precision.

Further Comments

The total number of floating-point operations is approximately

\frac{4}{3} n^{3}

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 = (\begin{array}{r} 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 \end{array}) .

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

Open in the MATLAB editor: f08ff_example

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

PDF version (NAG web site, 64-bit version, 64-bit version)

Chapter Contents

Chapter Introduction

NAG Toolbox