% This notebook demonstrate some of the linear algebra
% features of EULER.
% 
% Let us first show some easy things. We solve a linear
% system Ax=b.
>A=[0,1;1,2]
>b=[1;1]
>x=A\b
% The test.
>A.x-b
% Let us try a huge problem. We generate a 100x100 matrix
% A with normally distributed entries. Then we set b equal
% to the sum of all rows of A. The solution of Ax=b is
% the vector 1. We only check the first three coefficients
% of the computes solution x.
>A=normal(100,100); b=sum(A); x=A\b; x[1:3]
% We can check, how good this solution is.
>max(abs(x-1)')
% The ill-conditioned Hilbert matrix yields bad results
% even for small sizes.
>H=hilbert(12); b=sum(H); x=H\b; max(abs(x-1)')
% However, we can compute the residuum r=A.x-b with maximal
% accuracy and solve the problem A.x=r. This yields an
% improved solution x-r.
>r=residuum(H,x,b); x=x-H\r; max(abs(x-1)')
% Another iteration makes the result even better.
>r=residuum(H,x,b); x=x-H\r; max(abs(x-1)')
% The function xlgs does exactly this until a good solution
% is derived.
>x=xlgs(H,b); max(abs(x-1)')
% The INTERVAL.E file contains an optimal interval solver,
% using the Krawzyk method.
>load "interval"
% Now we get good inclusions.
>ilgs(H,b)
>