%% Strang exercise 2.3/17. Fit a quadratic thru 3 points
%
% All rights reserved, Matthias Kawski, June 2007.
% http://math.asu.edu/~kawski
%
% Copy and paste the commands into your command window.
%
% Desired viewpoint (to be much refined over time): 
% Functions are really long column vectors of their values
%
% More advanced ways would define sub functions so that there
% is no need for retyping the definitions of e.g. f1 and F1.
% However, here the emphasis is on functions as column vectors!
% 
x=[1 2 3]'
y=[4 8 14]'
f1=ones(size(x))
f2=x
f3=x.*x
A=[f1 f2 f3]
c=A\y
%
% for fun:
%
plot(x,y,'ro')
axis([0,5,0,20])
%
xx=[0:0.2:4]'
F1=ones(size(xx))
F2=xx
F3=xx.*xx
AA=[F1 F2 F3]
yy=AA*c
%
hold on
plot(xx,yy,'b-')
%
% Much wilder: There is nothing special about polynomials,
% indeed what matters is the linear parameterization only.
% Here are a few random points, and some different functions!
%
x=[1:5]'
y=floor(rand(size(x))*10)
clf
plot(x,y,'ro')
axis([0,6,-2,12])
%
f1=ones(size(x))
f2=1./(1+4*(x-2).*(x-2));
f3=1./(1+4*(x-4).*(x-4));
f4=sin(x);
f5=sin(2*x);
A=[f1 f2 f3 f4 f5]
c=A\y
%
xx=[0:0.1:6]';
F1=ones(size(xx));
F2=1./(1+4*(xx-2).*(xx-2));
F3=1./(1+4*(xx-4).*(xx-4));
F4=sin(xx);
F5=sin(2*xx);
AA=[F1 F2 F3 F4 F5];
size(AA)
yy=AA*c;
%
hold on
plot(xx,yy,'b-')
plot(xx,F1,'k-',xx,F2,'r-',xx,F3,'m-',xx,F4,'g-.',xx,F5,'c:')