demo_cond.m

DEMO_COND demonstrates the role of the condition number of a matrix (with respect to inversion)

%  DEMO_COND  demonstrates the role of the condition
%  number of a matrix (with respect to inversion)
%  in the role of linear system solving.
%
%  Matthias Heinkenschloss
%  Department of Computational and Applied Mathematics
%  Rice University
%  Feb 22, 2001
%
clear; clc;

set(0, 'defaultaxesfontsize',18,'defaultaxeslinewidth',1.2,...
       'defaultlinelinewidth',1.2,'defaultpatchlinewidth',1.2,...
       'defaulttextfontsize',18);


disp(' Demonstrate the role of the condition number of a matrix ')
disp(' (with respect to inversion) in the accuracy of the computed')
disp(' solution of a linear system ')
disp('                                           ')

disp(' Hit a key to continue '); pause

clear;clc;
disp('  Example 1                                ')
disp('  Linear System with Hilbert matrix        ')
disp('                                           ')


fprintf(1, '  n    cond_inf(H)    || x_ex - x ||_inf/ ||x_ex||_inf  \n' );
n1 = 4;
n2 = 10;
for n = n1:n2
    x_ex           = ones(n,1);
    H              = hilb(n);
    b              = H*x_ex;
    c(n-n1+1)      = cond(H,inf);
    x              = H\b;
    relerr(n-n1+1) = norm(x-x_ex,inf)/norm(x_ex,inf);
    
    fprintf(1, ' %2d   %12.6e     %12.6e \n', n,  c(n-n1+1), relerr(n-n1+1)  );
end


semilogy((n1:n2), c, 'k--'); hold on
semilogy((n1:n2), relerr, 'r-'); hold off
xlabel(' n ')
legend('cond_inf(H)', '|| x_{ex} - x ||_inf/ ||x_{ex}||_inf')

%print -depsc hilbert_cond.eps


disp('                                           ')
disp(' Hit a key to continue '); pause

clear;clc;
disp('  Example 2                                ')
disp('  Linear System with Vandermonde matrix    ')
disp('                                           ')


fprintf(1, '  n      cond_2(A)    || x_ex - x ||_2/ ||x_ex||_2 \n' );
n1 = 4;
n2 = 30;
for n = n1:n2
    t              = -1 + (2/(n-1))*(0:n-1)';
    A              = zeros(n,n);
    for j = 1:n
        A(:,j)     = t.^(j-1);
    end
    x_ex           = ones(n,1);
    b              = A*x_ex;
    c(n-n1+1)      = cond(A,2);
    x              = A\b;
    abserr(n-n1+1) = norm(x-x_ex,2);
    relerr(n-n1+1) = abserr(n-n1+1)/norm(x_ex,2);
    
    fprintf(1, ' %2d      %12.6e     %12.6e \n', n,  c(n-n1+1), relerr(n-n1+1)  );
end


semilogy((n1:n2), c, 'k--'); hold on
semilogy((n1:n2), abserr, 'r:'); hold on
semilogy((n1:n2), relerr, 'b-'); hold off
xlabel(' n ')
legend('cond_2(A)', '|| x_{ex} - x ||_2', '|| x_{ex} - x ||_2/ ||x_{ex}||_2')

%print -depsc vandermonde_cond.eps