we are going to solve the system ax = b for different b’s, where a =   1.8 0.1 0.1 −1.0 −2.0 0.2 1.0 −1.3 1.3   we know that the relative error of the right hand side is at most 0.0120 in 1-norm in matlab

We can use MATLAB to solve the system ax=b for different values of b with the given matrix a and maximum relative error of 0.0120 in 1-norm. One way to do this is to use the \ operator to perform the matrix division x=a\b, which will give us the solution x for the given right-hand side b.

To introduce the maximum relative error, we can use the norm function to calculate the 1-norm of the error e=b-a*x and compare it to the 1-norm of b multiplied by the maximum relative error. If the error is too large, we can adjust b and try again.

Here's some MATLAB code to implement this approach:

main.m
% define the matrix and maximum relative error
a = [1.8, 0.1, 0.1; -1.0, -2.0, 0.2; 1.0, -1.3, 1.3];
max_rel_error = 0.0120;

% loop over different right-hand side vectors
for i = 1:10  % for example, try 10 different b vectors
  % generate a random b vector with norm 1 and maximum relative error
  b = rand(3,1);
  b = b / norm(b,1);  % normalize b to have 1-norm of 1
  b = b + max_rel_error*randn(3,1);  % add random noise with maximum relative error
  b = b / norm(b,1);  % renormalize to ensure 1-norm of 1
  
  % solve the system and check the error
  x = a \ b;
  e = b - a*x;
  if norm(e,1) > max_rel_error*norm(b,1)
    disp(['Error too large for b = ', num2str(i)]);
  end
  
  % do something with the solution x
  disp(['Solution for b = ', num2str(i)]);
  disp(x);
end
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This code loops over 10 different randomly generated right-hand side vectors, each with 1-norm of 1 and maximum relative error 0.0120. It solves the system for each b using the \ operator, checks the error using the norm function, and displays the solution x for each b.

Note that the actual values of b that we get will depend on the problem we are trying to solve, and the way we generate them will also depend on the specific requirements of the problem.