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The Generalized RouthHurwitz Criteria for Quadratic and Cubic Polynomials
The Generalized RouthHurwitz 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 B_{1} = B_{2} = 0), then
we get the following
result:
If A_{1} > 0 and A_{2} > 0 then both roots have negative real part.
If A_{1} > 0 and A_{2} < 0 then one root has positive real part and one root has
negative real part.
If A_{1} < 0 and A_{2} > 0 then both roots have positive real part.
If A_{1}< 0 and A_{2} < 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
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