%% Evaluating Polynomials and Subtractive Cancellation
%
% Script:  Zoom4
% Plots (x-1)^6 near x=1 with smaller and smaller scale.
% The evaluation is done two ways: one is expanded out, the 
% uses the form above.  The expanded form leads to severe 
% subtractive cancellation error.

%% Expanded form of polynomomial 
% Evaluate the following form and zoom in to see what it does.
% 
% $$ y = x^6 - 6x^5 + 15x^4 - 20x^3 + 15x^2 - 6x +1 $$
%
close all
k=0;
n=100;
for delta = [.1 .01 .008 .007 .005  .003 ]
   x = linspace(1-delta,1+delta,n)';
   y = x.^6 - 6*x.^5 + 15*x.^4 - 20*x.^3 + 15*x.^2  - 6*x + ones(n,1);
   k = k+1;
   subplot(2,3,k)
   plot(x,y,x,zeros(1,n))
   axis([1-delta 1+delta -max(abs(y)) max(abs(y))])
end
%%
% The function is very jagged and noisy looking at when we zoom in.
% This is clearly not what is expected of a polynomial when evaluated 
% exactly!
%
%% Compact form of polynomial
% Now use
%
% $$ y = (x-1)^6 $$
%
%%
% and see what the zoomed in plots show.
%

k=0;  figure;
for delta = [.1 .01 .008 .007 .005  .003 ]
   x = linspace(1-delta,1+delta,n)';
   y = (x-ones(n,1)).^6 ;
   k = k+1;
   subplot(2,3,k)
   plot(x,y,x,zeros(1,n))
   axis([1-delta 1+delta -max(abs(y)) max(abs(y))])
end
%%
% Much better now!  We avoided the subtractions that appear in the expanded
% form, and the function performs much better!
%