Fraction-Free Unit Circle Stability Tests

The paper considers five fraction-free (FF) tests to determine whether a complex or a real polynomial has all its zeros inside the unit-circle. A FF test for complex polynomial is applicable to both complex and real polynomials where the FF property means that when it is applied to Gaussian or real integer coefficients, it can be completed over the respective ring of integers, i.e., it is Gaussian integers preserving (GIP) for Gaussian integer coefficient polynomials and integer preserving (IP) for integer coefficient polynomials. Three of the FF tests are for both complex and real polynomials. Two of the FF tests are specific for real polynomials and are IP for a polynomial with integer coefficients. Two GIP tests and two corresponding IP tests are immittance-type (stem from the Bistritz test). The third GIP test, a modified test proposed by Jury, is scattering-type (stems from the Schur–Cohn test). The two real immittance-type IP tests are significantly more efficient as integer algorithms than using any of the GIP tests for a real integer polynomial. The focus of the paper is on stability conditions. However, assuming normal conditions, stability conditions are always embedded in rules for counting also zeros outside the unit-circle.