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Code Listing by Feng Cheng Chang

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The MATLAB script file M_polyroots.m is to compute all the roots with multiplicities of any given polynomials.

Let a given polynomial p(x) be expressed as

p(x) = x^9 +7x^8 +12x^7 -12x^6 -42x^5 -6x^4 +44x^3 +20x^2 -15x -9

= (x +3)^2 * (x +1)^4 * (x -1)^3

= (x^2 +2x -3)^2 * (x +1)^3 * (x^2 -1)

It does give, respectively,

c = [ +1 +7 +12 -12 -42 -6 +44 +20 -15 -9 ]

...



A given polynomial p(x) is transformed into a rational function r(x). The poles and residues of the derived rational function are found to be equivalent to the roots and multiplicities of the original polynomial.
p(x) = Given polynomial



For given
p(x) = PROD[i=1,m]{SUM[j=2,n+2]{(A(i,j)*x^(j-2))^A(i,1)}}
we shall get
p(x) = SUM[s=1,N+1]{p(s)^(N+1-s)}

For example
If
p(x) = (x-4)^5 * (3x^6-7x^3+5x+2)^2 * (x^3+8)^3 * x^2
or
A = [...



Polynomial division by convolution.

Calculate inverse Z-transform -- (Polynomial division) - Up to K terms,

q(z) = b(z)/a(z),
where
b(z)=b(0)+...+b(k)/z^k +...+b(n)/z^n.
a(z)=a(0)+...+a(k)/z^k...