% fn.m

function y = fn(x)

% on input x is independent variable
% return function values for function to be interpolated

%y = sin(2*pi*x);

%y = exp(-10.*(x-.5).^2);

n = length(x);
y = rand(n,1);

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% interpV.m

function c = interpV(x,y)

% input:
%    x: column vector with distinct entries
%    y: column vector of data
%  output:
%    c: column vector of interp. poly coefficients s.t. p(x(i)) = y(i)

n = length(x);
vandermonde
c = v\y;

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% poly.m

% polynomial evaluation at a point z
% on input: c = vector of coeffs of polynomial 

% nested multiplication
n = length(c);
pval = 0;
for i=n:-1:1
   pval = z*pval + c(i);
end

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%polyTest.m

%driver to illustrate polynomial interpolation

% set up points to be interpolated from a known function
x=0:.1:1;
x=x';
y=fn(x);

%set up matrix and get interpolating coefficients
vandermonde  
c=interpV(x,y);

%evaluate polynomial at many more points
zx=0:.01:1;  
zx=zx';
vpoly   

plot(zx,pval,'c',x,y,'m',x,y,'m+');

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% vandermonde.m

% set up vandermonde matrix
% on input, given x = column vector of interpolating points

n = length(x);
v = ones(n,n);
for j=2:n
   v(:,j) = x.*v(:,j-1);
end


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% vpoly.m

% polynomial evaluation for entire vector zx
% on input: c = vector of coeffs of polynomial 

% nested multiplication
n = length(c);
m = length(zx);
pval = zeros(m,1);
for i=n:-1:1
   pval = zx.*pval + c(i);
end