math781-computer-problems-2-16.md

Exercise2_16.m

A = [
    0.473 -0.115 0;
    0.731 -0.391 0.267;
    0     -0.782 0.979;
    ];

perturb = zeros(size(A));
perturb(3, 3) = - 10^-14;

fprintf(' part a)\n');
fprintf('platform eps (ie `u`) => %d\n\n', eps);
[A2, flag, pivots, cond] = Factor(A);
%assert(flag == 0, 'can not factor');

fprintf('flag => %i\n', flag);
fprintf('partially factored A =>\n');
disp(A2);
fprintf('condition number => %d\n', cond);

fprintf('the smallest pivot is not close to the platform eps\n');
fprintf('the condition number is really large\n');

fprintf('perturbing `A_{3,3}`:\n');

A = A + perturb;
[A3, flag, pivots, cond] = Factor(A);
assert(flag == 0, 'can not factor perturbed A');

fprintf('A factored =>\n');
disp(A);
fprintf('condition number => %d (still fairly large)\n', cond);

fprintf('\n part b) =>\n');
fprintf('solving for `z` =>\n');
b = [
    0.084;
    0.357;
    0.833;
    ];

zb = Solve(A3, pivots, b);
disp(zb);

fprintf('\n part c) =>\n');
fprintf('solving for `z` =>\n');
b = [
    0.566;
    0.404;
    0.178;
    ];

zc = Solve(A3, pivots, b);
disp(zc);

fprintf('\n part d) =>\n');
fprintf('for part b => r = b - A * z =\n');
r = b - A * zb;
disp(r);
n = norm(r);
disp(n);

fprintf('for part c => r = b - A * z =\n');
r = b - A * zc;
disp(r);
n = norm(r);
disp(n);

stdout

Exercise2_16
 part a)
platform eps (ie `u`) => 2.220446e-16

flag => 3
partially factored A =>
     7.310000000000000e-01    -3.910000000000000e-01     2.670000000000000e-01
    -6.470588235294118e-01    -7.820000000000000e-01     9.790000000000000e-01
                         0     1.764705882352941e-01                         0

condition number => 1.797693e+308
the smallest pivot is not close to the platform eps
the condition number is really large
perturbing `A_{3,3}`:
A factored =>
     4.730000000000000e-01    -1.150000000000000e-01                         0
     7.310000000000000e-01    -3.910000000000000e-01     2.670000000000000e-01
                         0    -7.820000000000000e-01     9.789999999999900e-01

condition number => 1799373260817256 (still fairly large)

 part b) =>
solving for `z` =>
    -7.663944938440499e-02
    -1.045656170076726e+00
     1.562500000000000e-02


 part c) =>
solving for `z` =>
    -5.757337690530235e+13
    -2.368018024018137e+14
    -1.891511843495608e+14


 part d) =>
for part b => r = b - A * z =
     4.820000000000000e-01
     4.700000000000004e-02
    -6.550000000000000e-01

     8.145906947663961e-01

for part c => r = b - A * z =
     3.499999999999948e-03
    -2.249999999999974e-03
    -9.500000000000008e-03

     1.037123425634575e-02

diary off