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: $n \geq 0$ .
3: $ap (:)$ – double array: The dimension of the array ap must be at least $\max (1, n \times (n + 1) / 2)$

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 $\max (1, n - 1)$

Further details of the elementary reflectors, as returned by nag_lapack_dsptrd (f08ge).

Optional Input Parameters

None.

Output Parameters

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

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_zupgtr (f08gt).

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}),

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

Open in the MATLAB editor: f08gf_example

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;
                                   -1.81];
% 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);

disp('Eigenvalues');
disp(w');

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

disp('Eigenvectors');
disp(z);

f08gf 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