% 1. Polynomials and Roots
% We can define a polynomial by
%   (1) algebraically, give its coefficients;
%   (2) geometrically, give its roots.
c = [3 -2.23 -5.1 9.8];
polyval(c,0)

ans =

    9.8000

% to plot the polynomial with coefficients in c:
x = -1:0.1:1;
y = polyval(c,x);
plot(x,y)
% we can compute the roots of p via roots:
r = roots(c)

r =

  -1.5985          
   1.1709 + 0.8200i
   1.1709 - 0.8200i

polyval(c,r)

ans =

  1.0e-013 *

  -0.3197          
   0.1066 - 0.0178i
   0.1066 + 0.0178i

% Now  we may define the same polynomial,
% up to its leading constant via "poly" :
cr = poly(r)

cr =

    1.0000   -0.7433   -1.7000    3.2667

% the coefficient vector cr defines a cubic
% polynomial with the same roots as r,
% how would we check this ?
polyval(cr,r)

ans =

  1.0e-015 *

   0.4441          
        0 + 0.2220i
        0 - 0.2220i

p = c(1)*cr

p =

    3.0000   -2.2300   -5.1000    9.8000

c

c =

    3.0000   -2.2300   -5.1000    9.8000

norm(p-c)

ans =

  1.2909e-014

% we just computed the backward error:
% ans measures the change to c we should
% make for the polynomial defined by c
% to have the computed roots r
%
% 2. Curve Fitting
% Polynomial of low degree are often used
% in modeling.  We will generate some noise
% to the current polynomial defined by c.
% Given the samples in x and y
%  -- we used these samples for plotting.
rc = polyfit(x,y,3)

rc =

    3.0000   -2.2300   -5.1000    9.8000

% linear regression approximates the data
% with a linear function
lc = polyfit(x,y,1)

lc =

   -3.1260    8.9823

hold on
ly = polyval(lc,x);
plot(x,ly,'r')
% an application of polyfit is the
% removal of noise
% suppose that our samples "y" have
% errors on them
noise = randn(1,size(y,2));
ey = y + noise; % noisy samples
plot(x,ey,'g+')
plot(x,ey,'g')
% to remove the noise, we do a polyfit
cc = polyfit(x,ey,3)

cc =

    3.2885   -3.2438   -5.0382   10.2963

cy = polyval(cc,x); % for plotting
plot(x,cy,'r');
% we started by looking at polynomials
% given by their coefficient vectors,
% then by their roots, and finally by
% possibly noisy samples
diary off