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

expand all

Check a Lag Operator Polynomial for Stability

Open Live Script

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.