% Fit a combination of e^x, sin(x), gamma(x) to the function f(x)=1/x.
% Tref&Bau problem 11.2(a).  ELB 11/8/03

m=400; % number of subintervals in discretization.
x=linspace(1,2,m+1)';  A=[exp(x) sin(x) gamma(x)];  b=1./x;
disp('coefficients of approximation [of exp(x) sin(x) gamma(x), respectively]:')
c=A\b; format long g, format compact, disp(c')
err=A*c-1./x; errl=err(1:m); % errl has left endpt values
format short e, disp('estimated L^1, L^2, and L^infty error:')
disp([sum(abs(err))/m    sqrt(sum(err.^2)/m)   max(abs(err))])

figure(1), clf, plot(x,A*c,x,1./x), axis([1 2 .5 1])
title('f(x)=1/x  superimposed on linear combination of  exp(x), sin(x), gamma(x)')
figure(2), clf, plot(x,err), grid on
title('signed error in approximation')