math781-computer-problems-2-17.m

A = [ 
    0.172 0.013 0.144;
    0.368 0.681 0.271;
    0.099 0.510 0.329;
    ];

[A2, success, pivots, cond_num] = Factor(A);
assert(success == 0, 'failed to factor A');

b = [
    1.44 4.35 1.32 3.95;
    2.84 9.30 2.90 8.29;
    2.36 3.45 3.25 7.35;
    ];

[m,n] = size(b);
x = zeros(m, n);

for i = 1:n
    out = Solve(A2, pivots, b(:,i));
    x(:,i) = out;
end

fprintf(' part a)\n');
x
fprintf('norm(x, Inf) = %d\n', norm(x, Inf));
fprintf('condition number = %d\n', cond_num);

fprintf('\nAssuming exact data =>\n');
fprintf('u = %d\n', eps());

rel_err_bnd = (cond_num / (1 - cond_num * 10 * eps())) * (10 * eps());

fprintf('absolute error bound = %d\n', rel_err_bnd * norm(x, Inf));
fprintf('Remarks: very low error bound, much lower than the condition number.\n');

fprintf('\n');
fprintf('Assuming rounded data =>\n');
u_a = 10^(-2);
u_b = u_a;
fprintf('u_a = %d, u_b = %d\n', u_a, u_b);

rel_err_bnd = (cond_num / (1 - cond_num * 10 * u_a)) * (10 * u_a + 10 * u_b);
fprintf('absolute error bound = %d\n', rel_err_bnd * norm(x, Inf));
fprintf('Remarks: the absolute error bound is much\n\tgreater than the condition number.\n');

fprintf('\n part b)\n');

A_inv = zeros(m, m);

for i = 1:m
    b = zeros(m, 1);
    b(i, 1) = 1.0;
    
    out = Solve(A2, pivots, b);
    A_inv(:,i) = out;
end

A_inv

fprintf('exact cond(A) = %d\n', norm(A_inv, Inf) * norm(A, Inf));

stdout

format long e
Exercise2_17
 part a)

x =

     3.140783569018794e+00     2.237596248442763e+01     2.146024337702900e-01     7.050291746699174e+00
    -1.394314816153835e-02     1.860712214737706e-01     6.188937759548281e-01     8.285203102998583e-01
     6.249767271214358e+00     3.464691158328394e+00     8.854463627111791e+00     1.893457677454059e+01

norm(x, Inf) = 3.750350e+01
condition number = 6.061168e+00

Assuming exact data =>
u = 2.220446e-16
absolute error bound = 5.047407e-13
Remarks: very low error bound, much lower than the condition number.

Assuming rounded data =>
u_a = 1.000000e-02, u_b = 1.000000e-02
absolute error bound = 1.154226e+02
Remarks: the absolute error bound is much
	greater than the condition number.

 part b)

A_inv =

     2.781856394194422e+00     2.241423290015830e+00    -3.063869399265308e+00
    -3.054211863582577e+00     1.371888592353571e+00     2.067620055564530e-01
     3.897398989063425e+00    -2.801106649884160e+00     3.640955768065272e+00

exact cond(A) = 1.364809e+01
diary off