Error Reduction using a Preconditioner for Linear System Ax =b

Hello,

I am trying to solve the Ax =b for a non linear system, my A matrix is 10x4 complex double and b matrix is 10x1 double

A matrix :

b matrix :

I already have the original values of x and they are

x_original = [800000000;200000000;10000;10000]; % coefficients to find

I need to solve the system such that the values of x I find are close to x_original. The system is badly scaled and I need to use a pre conditioner.

Does anyone knows how to make a good pre conditioner ? or how to solve this problem.

I have done this till far

Pre_Conditioner = [1;1;1;1;1;1;1;1;1;1].^(-1);
A = bsxfun(@times,A,Pre_Conditioner);
b = Pre_Conditioner.*b;

% Solve the equation Ax = b using Least Squares Method

%  Numerical Method
x_2 = inv(A.'*A)*A.'*b; % Least Square Estimation
x_2 = abs(x_2);

x_original = [800000000;200000000;10000;10000]; % coefficients to find

   
% Error calculation
for i =1:1:size(x_original,1)
    error_percent(i,1) = (abs((x_original(i)-x_2(i)))/x_original(i))*100
end

Does anybody know...?

Answer 1

[Rishabh Mishra]

Hi,

I would suggest you use following preconditioner column vectors

Pre_Conditioner = [1e-9;1e-9;1e-9;1e-9;1e-9;1e-9;1e-9;1e-9;1e-9;1e-9].^(-1); 

To simply put, the above statement can be simplified as:

Pre_Conditioner = 1e9 

Additionally, I would advise you to use any 4 rows (as number of unknown variables = 4) from the ‘A’ matrix to evaluate the solution of the system of linear equations.