isStable

Determine stability of lag operator polynomial

Syntax

[indicator,eigenvalues] = isStable(A)


Description

[indicator,eigenvalues] = isStable(A) takes a lag operator polynomial object A and checks if it is stable. The stability condition requires that the magnitudes of all roots of the characteristic polynomial are less than 1 to within a small numerical tolerance.

Input Arguments

A	Lag operator polynomial object, as produced by LagOp.

Output Arguments

indicator	Boolean value for the stability test. true indicates that A(L) is stable and that the magnitude of all eigenvalues of its characteristic polynomial are less than one; false indicates that A(L) is unstable and that the magnitude of at least one of the eigenvalues of its characteristic polynomial is greater than or equal to one.
eigenvalues	Eigenvalues of the characteristic polynomial associated with A(L). The length of eigenvalues is the product of the degree and dimension of A(L).

Examples

Check a Lag Operator Polynomial for Stability

Divide two Lag Operator polynomial objects and check if the resulting polynomial is stable:

A = LagOp({1 -0.6 0.08});
B = LagOp({1 -0.5});
[indicator,eigenvalues]=isStable(A\B)

indicator = logical
   1

eigenvalues = 4×1 complex

   0.3531 + 0.0000i
  -0.0723 + 0.3003i
  -0.0723 - 0.3003i
  -0.3086 + 0.0000i

Tips

• Zero-degree polynomials are always stable.

• For polynomials of degree greater than zero, the presence of NaN-valued coefficients returns a false stability indicator and vector of NaNs in eigenvalues.

• When testing for stability, the comparison incorporates a small numerical tolerance. The indicator is true when the magnitudes of all eigenvalues are less than 1-10*eps, where eps is machine precision. Users who wish to incorporate their own tolerance (including 0) may simply ignore indicator and determine stability as follows:

  [~,eigenvalues] = isStable(A);
  indicator = all(abs(eigenvalues) < (1-tol));

  for some small, nonnegative tolerance tol.

References

[1] Hamilton, J. D. Time Series Analysis. Princeton, NJ: Princeton University Press, 1994.