The Generalized Routh-Hurwitz Criteria for Quadratic and Cubic Polynomials

Quadratic polynomials

Consider the polynomial

with real.

Phase velocity direction

The number of roots of F (w) = 0 with positive real part is given by the number of sign changes
in the sequence

The number of roots of F (w)=0 with negative real part is given by the number of sign changes
in the sequence

Phase velocity direction: real coefficients

If all the coefficients in the polynomial are real (i.e., if B₁ = B₂ = 0), then we get the following
result:

If A₁ > 0 and A₂ > 0 then both roots have negative real part.
If A₁ > 0 and A₂ < 0 then one root has positive real part and one root has negative real part.
If A₁ < 0 and A₂ > 0 then both roots have positive real part.
If A₁< 0 and A₂ < 0 then one root has positive real part and one root has negative real part.

Instability

The number of unstable roots of F(w) (i.e., the number with positive imaginary part) is given
by the number of sign changes in the sequence

The number of stable roots of F(w) (i.e., the number with negative imaginary part) is given by
the number of sign changes in the sequence

Cubic polynomials

Consider the polynomial

with real.

Phase velocity direction

The number of roots of F(w) = 0 with positive real part is given by the number of sign changes
in the sequence

The number of roots of F(w)=0 with negative real part is given by the number of sign changes
in the sequence

Instability

The number of unstable roots of F(w) (i.e., the number with positive imaginary part) is given
by the number of sign changes in the sequence

The number of stable roots of F(w) (i.e., the number with negative imaginary part) is given by
the number of sign changes in the sequence