function poly_interp( n, icase )
%function poly_interp( n, icase )
%
% Matlab function to study polynomial interpolation to various test functions
% (determined by switch 'icase').
%
% Using equi-spaced vs Chebyshev vs modified Chebyshev nodes on [-1,1]
% for the input value of 'n'.
%
% C.G. (11-96)
%

m = 1000 ;  t = [ 0 : m ]' * 2 / m - 1 ;   % number/vector of evaluation points

%
% Generate data for plotting true function for this test case.
%

if     ( icase == 1 )
  func  = ' f(x) = sin( pi * x ) ,  ' ;            % smooth function: sin(pi*x)
  fplot = sin( pi * t ) ;
elseif ( icase == 2 )
  func  = ' f(x) = 1 / ( 1 + 16 x^2 ) ,   ' ;           % Runge's nasty example
  fplot = 1 ./ ( 1 + 16 * t.^2 ) ;
end

%
% Put the different node sets in the columns of the array tau.
%

tau = zeros( n + 1, 3 ) ;        % initialize an array of the proper dimensions

tau(:,1) = [ 0 : n ]' * 2 / n - 1 ;            % equally spaced nodes on [-1,1]
tau(:,2) = - cos( ( 2 * [0:n]' + 1 ) * pi / ( 2 * n + 2 ) ) ; % Chebyshev nodes
tau(:,3) = - cos( [ 0 : n ]' * pi / n ) ;            % modified Chebyshev nodes

%
% Generate the interpolant for this test case and plot against true function.
%

P = zeros( m + 1, 3 ) ;          % initialize an array of the proper dimensions

for k = 1 : 3

  if     ( icase == 1 )
    fdata = sin( pi * tau(:,k) ) ;                 % smooth function: sin(pi*x)
  elseif ( icase == 2 )
    fdata = 1 ./ ( 1 + 16 * tau(:,k).^2 ) ;             % Runge's nasty example
  end

  P(:,k) = polyval( polyfit( tau(:,k), fdata, n ), t ) ;

end

clf ,  plot( t, [ fplot, P ] )

title( [ 'Chebyshev vs Equi-Spaced Interpolation,  ', ...
          func, '  {\it{n}} = ', num2str(n) ] )

xlabel( '\it{x}' ) ,  ylabel( '\it{y}' )