Partial-Fraction Decomposition

This function is used for Partial-Fraction Decomposition.

% (1) For the fraction bellow：
% s+2
% F(s) = ----------------
% s(s+1)^2(s+3)
% Input：
% >> num_input = [1 2];
% >> den_input = [1 0;1 1;1 1;1 3];
% >> frac_decomp(num_input,den_input)
%
% Result is：
% roots and counts:
% [ 0, 1]
% [ -3, 1]
% [ -1, 2]
%
% symbolic Expression:
%
% 1. s + 2.
% --------------------
% 2
% s (s + 3.) (s + 1.)
% Final Expression:
%
% 0.667 0.0833 0.500 0.750
% ----- + ------ - --------- - ------
% s s + 3. 2 s + 1.
% (s + 1.)
%
% (2) For the following fraction：
% 1
% F(s) = ------------------
% (s+1)^2(s+2s+2)
%
% Input：
% >> num_input = [1];
% >> den_input = [0 1 1;0 1 1;1 2 2];
% >> frac_decomp(num_input,den_input)
%
% Results is:
% roots and counts:
% [ -1+i, 1]
% [ -1-i, 1]
% [ -1, 2]
%
% symbolic Expression:
%
% 1.
% -----------------------------------------
% 2
% (s + 1. - 1. I) (s + 1. + 1. I) (s + 1.)
% Final Expression:
%
% 0. - 0.500 I 0. + 0.500 I 1.00 - 0. I
% - ------------- - ------------- + -----------
% s + 1. - 1. I s + 1. + 1. I 2
% (s + 1.)