% 1. polynomials: coefficient vectors
c = [ 3 -2.23 -5.1 9.8]

c =

    3.0000   -2.2300   -5.1000    9.8000

y = polyval(c,0)

y =

    9.8000

x = -1:0.01:+1;
y = polyval(c,x);
size(y)

ans =

     1   201

plot(x,y)
% 2. Polynomials defined by roots
% we can compute the roots of a polynomial:
r = roots(c)

r =

  -1.5985          
   1.1709 + 0.8200i
   1.1709 - 0.8200i

% let us check that r contains the roots:
residual = polyval(c,r)

residual =

  1.0e-013 *

  -0.3197          
   0.1066 - 0.0178i
   0.1066 + 0.0178i

% via poly we can define a polynomial,
% its coefficient vector given the roots
q = poly(r)

q =

    1.0000   -0.7433   -1.7000    3.2667

q = c(1)*q

q =

    3.0000   -2.2300   -5.1000    9.8000

c

c =

    3.0000   -2.2300   -5.1000    9.8000

format long
q

q =

  Columns 1 through 2

   3.000000000000000  -2.229999999999994

  Columns 3 through 4

  -5.100000000000011   9.800000000000001

c

c =

  Columns 1 through 2

   3.000000000000000  -2.230000000000000

  Columns 3 through 4

  -5.100000000000000   9.800000000000001

% we can make a plot of the roots
plot(r,'bx')
c = randn(200);

size(c)

ans =

   200   200

r = roots(c(:,1));
plot(r,'r+')
% 3. curve fitting
c = [3  -2.23 -5.1 9.8];
x = -2:0.01:+2;
y = polyval(c,x);
% we just sampled the polynomial in the 
% range [-2,+2], with space 0.01
% (x,y) are the exact data
noise = randn(1,size(y,2));
noise = noise/norm(noise);
ey = y + noise;
figure;
plot(x,y)
hold on
plot(x,ey,'r+')
hold off
plot(x,y);
hold on
plot(x,ey,'r');
% we would like to recognize if the noisy
% data comes from a cubic or a quadric
ec = polyfit(x,ey,3)

ec =

  Columns 1 through 2

   3.005585921349974  -2.227866664172377

  Columns 3 through 4

  -5.112285076466999   9.797437655746483

% we see a coefficient vector close to
% the original one appearing
fy = polyval(ec,x);
plot(x,fy)
figure
subplot(3,1,1)
plot(x,y);
subplot(3,1,2)
plot(x,ey);
subplot(3,1,3)
plot(x,fy)
diary off