Hi Archa,
Given your system of equations, the goal is to find (X = [X(1), X(2), X(3)]) that minimizes the residuals of the equations. Here's how you can implement this approach in MATLAB:
function X = solveOverdeterminedSystem()
% Initial guess for X
X = [0, 0, 0];
% Tolerance and maximum number of iterations
tol = 1e-6;
maxIter = 100;
for iter = 1:maxIter
% Evaluate the function vector
F = [
(X(1)-500)^2 + (X(2)-900)^2 + (X(3)-20200)^2 - 500^2;
(X(1)-600)^2 + (X(2)-1000)^2 + (X(3)-20100)^2 - 1000^2;
(X(1)-1000)^2 + (X(2)-700)^2 + (X(3)-23000)^2 - 1200^2;
(X(1)-100)^2 + (X(2)-800)^2 + (X(3)-22000)^2 - 1300^2
];
% Evaluate the Jacobian matrix
J = [
2*(X(1)-500), 2*(X(2)-900), 2*(X(3)-20200);
2*(X(1)-600), 2*(X(2)-1000), 2*(X(3)-20100);
2*(X(1)-1000), 2*(X(2)-700), 2*(X(3)-23000);
2*(X(1)-100), 2*(X(2)-800), 2*(X(3)-22000)
];
% Compute the change in X using the normal equations approach
deltaX = -pinv(J' * J) * J' * F;
% Update X
X = X + deltaX';
% Check for convergence
if norm(deltaX) < tol
fprintf('Converged to [%f, %f, %f] after %d iterations.\n', X(1), X(2), X(3), iter);
return;
end
end
error('Did not converge within the maximum number of iterations');
end
Hope this helps.
