Lecture 31: Polynomials and Curve Fitting.
------------------------------------------

% saving/loading the data (revisited)
M = [1 2 3 4 5];
openvar('M', M);
save('h:\data');
clear
load('h:\data');
openvar('M', M);
diary off

% a polynomial is presented by the array of its coefficients
% c = 3x^3-2x^2-5x+11
c = [3 -2 -5 11]

c =

     3    -2    -5    11

polyval(c,3.3)

ans =

   80.5310

x = -1:0.1:1;
openvar('x', x);
y = polyval(c,x);
openvar('y', y);
plot(x,y);
r = roots(c);
openvar('r', r);
plot(r,'rx')
dc = polyder(c);
openvar('dc', dc);
hold on
plot(roots(dc),'bo')
openvar('dc', dc);
rdc = roots(dc)

rdc =

    1.0000
   -0.5556

cr = poly(r)

cr =

    1.0000   -0.6667   -1.6667    3.6667

c(1)*cr - c

ans =

  1.0e-013 *

         0    0.0044    0.0533   -0.2487

% curve fitting
% polyfit(x,y,d), where d is the degree of the curve
hold off
plot(x,y);
hold on
plot(x,y,'rx')
noise = randn(1,size(y,2));
noise = noise/norm(noise); % normalize the noise
ey = y + noise;
plot(x,ey,'go')
ec = polyfit(x,ey,3);
yec = polyval(ec,x);
plot(x,yec,'g--')
ec = polyfit(x,ey,5);
yec = polyval(ec,x);
plot(x,yec,'r--')
ec = polyfit(x,ey,20);
yec = polyval(ec,x);
plot(x,yec,'r--')

% Exe 2, p 153
f = randn(1,101);
openvar('f', f);
r = roots(f);
hold off
plot(r);
plot(r, 'bx');
diary off