Question

What are the 3 Newton-Raphson formulas to solve a system of three nonlinear equations in Matlab?  

Answer 1

This is the code for Newton raphson method for three variable

xnew =[c;s;q];
xold = zeros(size(xnew));
while norm(xnew - xold) > tol
    iter= iter + 1;
    xold = xnew;

    % update c, s, and q
    c = xold(1);
    s = xold(2);
    q = xold(3);

    %Defining the functions for c,s and q.
    f = c * (alpha*I + k_f + k_d + k_n * s + k_p*(1-q))-I *alpha;
    g = s * (lambda_b * c* P_C + lambda_r *(1-q))- lambda_b* c * P_C; 
    h = q * ( gamma + c * k_p *(P_C / P_Q))- (c * k_p * (P_C / P_Q));

    %Partial derivatives in terms of c,s and q.
    dfdc = alpha*I + k_f + k_d + k_n * s + k_p*(1-q);
    dfds = k_n *c ;
    dfdq = - k_p *c;

    dgdc = lambda_b * P_C *(s-1);
    dgds = lambda_b * c* P_C + lambda_r *(1-q);
    dgdq = - lambda_r * s;

    dhdc = k_p *(P_C / P_Q)*(q-1);
    dhds = 0;
    dhdq = gamma + c * k_p *(P_C / P_Q);

    %Jacobian matrix 
    J = [dfdc dfds dfdq; dgdc dgds dgdq; dhdc dhds dhdq];    
    % Applying the Newton-Raphson method
    xnew = xold - J\[f;g;h];
    disp(sprintf('iter=%6.15f,  c=%6.15f,  s=%6.15f, q=%6.15f', iter,xnew)); 
end

The output of the program will be

iter=1.000000000000000,  c=0.000000000389029,  s=0.015000000287216, q=0.979999999955780
iter=2.000000000000000,  c=0.000000001356998,  s=0.158028331191731, q=0.923765241962920
iter=3.000000000000000,  c=0.000000001181617,  s=0.104156261426515, q=0.952886937302707
iter=4.000000000000000,  c=0.000000001216663,  s=0.085849634576983, q=0.958360887077671
iter=5.000000000000000,  c=0.000000001224388,  s=0.084367460093642, q=0.958463129596494
iter=6.000000000000000,  c=0.000000001224423,  s=0.084367992582976, q=0.958463625488450