%%%%%%% Lesson 3: Working with Complex Numbers and Polynomials
%
% 1) Complex Numbers: operations and special related functions
% 2) Polynomials: MATLAB representation and special related functions
%___________________________________________________________________________
% Complex numbers: MATLAB interprets the letter i to represent sqrt(-1)
%    		  PROVIDED THE USER HASN"T DEFINED IT AS SOMETHING ELSE
%___________________________________________________________________________
sqrt(-1)
ans =
        0 + 1.0000i

s1 = 3 + 4i	%Defining complex numbers
s1 =
   3.0000 + 4.0000i

s2 = 5 - 12j;	%Can use j also -- MATLAB caters to electrical engineers 8-)
%____________________________________________
		%MATLAB does "everything" with complex numbers as with real!!!
A = [s1 j; j s2]	%Generate matrix of complex entries
A =

   3.0000 + 4.0000i        0 + 1.0000i
        0 + 1.0000i   5.0000 -12.0000i

%can do   s1 + s2, s1-s2, etc..	
Ainv = inv(A)		%Matrix inverse 
Ainv =

   0.1176 - 0.1581i   0.0037 - 0.0147i
   0.0037 - 0.0147i   0.0294 + 0.0699i

%____________________________________________
%MATLAB also provides special functions for complex numbers

a = real([s1 s2]); b=imag([s1 s2]);%extract real and imaginary part 
a =                b=
     3     5         4   -12

r = abs([s1 s2])		%calculate magnitude -- element-wise
r =
     5    13
alpha = angle([s1; s2])		%calculate phase angle in rad -- element-wise
alpha =

    0.9273
   -1.1760

%_________________________________________________________
%%%%%%%%%%%%%%% Polynomials
%MATLAB represents polynomials by a vector of its coefficients
%in order of descending powers

p1 = [1 4 4]		%Represents the polynomial          s^2 + 4s + 4
p2 = [2 0 0 -1]		%Represents the polynomial   2s^3 +0s^2 + 0s - 1
%____________________________________________
%MATLAB also provides special functions for polynomials
r2 = roots(p2)		%three roots of third-order polynomial -- via "roots"
r2 =

  -0.3969 + 0.6874i
  -0.3969 - 0.6874i
   0.7937          

%____________________________________________
p3 = poly(r2)		%Coefficients of polynomial with given set of roots
p3 =

    1.0000    0.0000         0   -0.5000

a = polyval(p3,5)	%Evaluate polynomial  p3  at s = 5
a =

  124.5000
%____________________________________________
			%Coefficients of product of two polynomials
p4 = conv([1 1],[1 1])	%(s+1)*(s+1) = s^2 + 2s + 1

p4 =

     1     2     1

poly2str(p4,'s')	%display polynomial as equation in variable s
ans =

   s^2 + 2 s + 1

%____________________________________________
%We use polynomials primarily to express system functions

num = [1 0];
den = [1 1 5];
printsys(num,den,'s')		%Show system function in "pretty" form
 
num/den = 
 
         s
   ------------
   s^2 +  s + 5
%____________________________________________________________________________