% Lecture 12, question 2
% to get a plot for part a

clear;
close all;

m = 10;
n = 5;

dx = 2/(n-1);
x = -1:dx:1;

dy = 2/(m-1);
y = -1:dy:1;

% let f(x) = abs(x);
f = abs(x);
f = f';

% create vandermonde matrix
  V = vander(x);
% put it into a least squares formulation
  V = fliplr(V); 
% get coefficients of polynomial, Ac = f
  c = V\f; 
% the polynomial interpolant has the form
% p(x) = c(1) + c(2)x + c(3)x^2 + ...
% form polynomial interpolant
  p = 0;
  for i = 1:n
    p = p + c(i)*y.^(i-1);
  end

% to generate the data points directly via multiplication
% by an m x n matrix A.

%  Y = fliplr(vander(y))
%  Y = Y(:,1:n);
%  A = Y * inv(V);
%  p = A*f;
  
  subplot(1,2,1);
  plot(x,f,'r-',x,f,'ro',y,p,'b-');  
  legend('actual function','data points','interpolant');
  title('n = 10');
  xlabel('x');
  ylabel('f(x)');
  

  
m = 100;
n = 20;

dx = 2/(n-1);
x = -1:dx:1;

dy = 2/(m-1);
y = -1:dy:1;

% let f(x) = abs(x);
f = abs(x);
f = f';
%f = ones(n,1);
%f(2) = 1+ 1e-8;

% create vandermonde matrix
  V = vander(x);
% put it into a least squares formulation
  V = fliplr(V); 
% get coefficients of polynomial, Ac = f
  c = V\f; 
% the polynomial interpolant has the form
% p(x) = c(1) + c(2)x + c(3)x^2 + ...
% form polynomial interpolant
  p = 0;
  for i = 1:n
    p = p + c(i)*y.^(i-1);
  end

  subplot(1,2,2);
  plot(x,f,'r-',x,f,'ro',y,p,'b-');  
  legend('actual function','data points','interpolant');
  title('n = 20');
  xlabel('x');
  ylabel('f(x)');