solving nonlinear overdetermined equations using newton raphson

can any one suggest me the matlab code for newton raphson to solve overdetermined equations given below

(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

solve X(1),X(2),X(3)

Answer 1

[arushi]

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.