% Linear system from -u''=x
%%%%%%%%%%%%%%%%%%%%%%%%%%%
clear all
n=25; h=1/n; n1=n-1; v=ones(n1-1,1);
A=2*eye(n1)-diag(v,1)-diag(v,-1); A=n*n*A;
cd=1/rcond(A)
b=zeros(n1,1); xe=b; t=b;
for i=1:n1
  t(i)=i*h;
  xe(i)=1/pi^2*sin(pi*i*h);
  b(i) =sin(pi*i*h);
end
x=A\b;
eta=back_comp(A,b,x)
etae=norm(A*(x-xe),inf)/norm(b,inf)
delta=2*eta/rcond(A)
deltae=norm(xe-x,inf)/norm(x,inf)
plot(t,xe,t,x)
pause
% It's wrong: we compare the solution of the continuous problem to the computed solution!!!
% In principle, the exact solution is not known.
% Assume it is given by xe(i).
b=A*xe;
x=A\b;
eta=back_comp(A,b,x)
etae=norm(A*(x-xe),inf)/norm(b,inf)
delta=2*eta/rcond(A)
deltae=norm(xe-x,inf)/norm(x,inf)
plot(t,xe,t,x)