% Used to produce the plots for the function approximation example

% Example 4.5

function [] = optfcnap()

p = 20;  % number of basis functions

% Define the basis functions for plotting purposes
x = 0:0.01:2*pi; x=x(:);
n = length(x);
zeta = zeros(n, p);
for i=1:n
  zeta(i,:) = makezeta(x(i), p)';
end

%% plot the basis functions
figure(1)
plot(x, zeta); hold on;

% define random sample points
rand('seed', 100);  % make sure we can repeat the results
samp = rand(50,1)*2*pi;
plot(samp, 0.6*ones(length(samp), 1), 'o');

xlabel('x');
axis([0 2*pi, 0 0.8]); hold off



%%%% for the sin fcn
yi = sin(samp);  % define the function outputs at the sample points
ysin = sin(x);

% Initialize the estimate
A0 = zeros(p, 1);

% Solve the ODE for the estimates
[t, A] = ode45('optAderiv', [0 1], A0, [], yi, samp, p);

nt = length(t);
Afin = A(nt,:);
zsin = Afin*zeta';

figure(2);
plot(x,ysin); hold on
plot(x,zsin, '--'); hold off
xlabel('x');
axis([0 2*pi, -1.1 1.1]);


%%%% for the cos^2 fcn
yi = cos(samp).^2;  % define the function outputs at the sample points
ycos = cos(x).^2;

% Re-initialize the estimate
A0 = zeros(p, 1);

% Solve the ODE for the estimates
[t, A] = ode45('optAderiv', [0 1], A0, [], yi, samp, p);

nt = length(t);
Afin = A(nt,:);
zcos = Afin*zeta';

figure(3);
plot(x,ycos); hold on
plot(x,zcos, '--'); hold off
xlabel('x');
axis([0 2*pi, 0 1.2]);




%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Define the basis functions
function [zeta] = makezeta(x, p)

sigma = 2;
cmax = 2*pi;
c = 1:p; c = (c(:) - 1)*cmax/(p-1);  % Gaussian centers

zeta = zeros(p,1);
for i=1:p
  zeta(i) = exp(-sigma^2*(x-c(i))^2);    
end
sz = sum(zeta);
zeta = zeta / sz;