) along with their moduli, in descending order of magnitude. Thus to check for stationarity or invertibility you should check whether the modulus of the largest eigenvalue is less than one.

References

Wei W W S (1990) Time Series Analysis: Univariate and Multivariate Methods Addison–Wesley

Parameters

Compulsory Input Parameters

1: $k$ – int64int32nag_int scalar: $k$ , the dimension of the multivariate time series.

Constraint: $k \geq 1$ .
2: $ip$ – int64int32nag_int scalar: The number of AR (or MA) parameter matrices, $p$ (or $q$ ).

Constraint: $ip \geq 1$ .
3: $par (ip \times k \times k)$ – double array: The AR (or MA) parameter matrices read in row by row in the order $ϕ_{1}, ϕ_{2}, \dots, ϕ_{p}$ (or $θ_{1}, θ_{2}, \dots, θ_{q}$ ). That is, $par ((l - 1) \times k \times k + (i - 1) \times k + j)$ must be set equal to the $(i, j)$ th element of $ϕ_{l}$ , for $l = 1, 2, \dots, p$ (or the $(i, j)$ th element of $θ_{l}$ , for $l = 1, 2, \dots, q$ ).

Optional Input Parameters

None.

Output Parameters

1: $rr (k \times ip)$ – double array: The real parts of the eigenvalues.
2: $ri (k \times ip)$ – double array: The imaginary parts of the eigenvalues.
3: $rmod (k \times ip)$ – double array: The moduli of the eigenvalues.
4: $ifail$ – int64int32nag_int scalar: $ifail = 0$ unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

Errors or warnings detected by the function:

$ifail = 1$

On entry,	$k < 1$ ,
or	$ip < 1$ .

$ifail = 2$: An excessive number of iterations are needed to evaluate the eigenvalues of $A (ϕ)$ (or $B (θ)$ ). This is an unlikely exit. All output arguments are undefined.

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.

$ifail = - 999$: Dynamic memory allocation failed.

Accuracy

The accuracy of the results depends on the original matrix and the multiplicity of the roots.

Further Comments

The time taken is approximately proportional to

k p^{3}

(or

k q^{3}

Example

This example finds the eigenvalues of

A (ϕ)

where

k = 2

and

p = 1

and

ϕ_{1} = [\begin{array}{r} 0.802 & 0.065 \\ 0.000 & 0.575 \end{array}]

Open in the MATLAB editor: g13dx_example

function g13dx_example


fprintf('g13dx example results\n\n');

k   = int64(2);
ip  = int64(1);
par = [0.802;     0.065;     0;     0.575];

% Calculate zeros
[rr, ri, rmod, ifail] = g13dx( ...
                               k, ip, par);

% Display results
fprintf('%20s%13s\n', 'Eigenvalues', 'Moduli');
fprintf('%20s%13s\n', '-----------', '------');
for i = 1:k*ip
  if ri(i)>=0
    fprintf(' %10.3f  + %6.3f i  %8.3f\n', rr(i), ri(i), rmod(i));
  else
    fprintf(' %10.3f  - %6.3f i  %8.3f\n', rr(i), -ri(i), rmod(i));
  end
end

g13dx example results

         Eigenvalues       Moduli
         -----------       ------
      0.802  +  0.000 i     0.802
      0.575  +  0.000 i     0.575

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

Chapter Contents

Chapter Introduction

NAG Toolbox