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Newton-Raphson Method for Non-linear System of 3 variables in Matlab

  1. Jun 17, 2014 #1

    wel

    User Avatar
    Gold Member

    I am trying to solve 3 non-linear system of 3 variables using the newton-raphson method in matlab. Here are the 3 non-linear equations:

    \begin{equation} c[\alpha I+ k_f+k_d+k_ns+k_p(1-q)]-I \alpha =0 \end{equation}
    \begin{equation} s[\lambda_b c P_C +\lambda_r (1-q)]- \lambda_b c P_C =0 \end{equation}
    \begin{equation} q[\gamma +c k_p \frac{P_C}{P_Q}]- c k_p \frac{P_C}{P_Q}=0 \end{equation}

    I need to find the values of c,s, and q using the newton-raphson method.

    =>
    This is my matlab code :

    Code (Text):
     format long
        clear;
     
     %values of parameters
        I=1200;
        k_f= 6.7*10.^7;
        k_d= 6.03*10.^8;
        k_n=2.92*10.^9;
        k_p=4.94*10.^9;
        lambda_b= 0.0087;
        lambda_r =835;
        gamma =2.74;
        alpha =1.14437*10.^-3;
        P_C= 3 * 10.^(11);
        P_Q= 2.87 * 10.^(10);    
    tol = 10.^-4;  %tol is a converge tolerance    
    %initial guess or values
      c=1;
      s=0.015;
      q=0.98;
      iter= 0; %iterations
      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
     
    can someone please check my code, there are no errors but did i got accurate values of c,s, and q?. Thanks in advance.
     
    Last edited: Jun 17, 2014
  2. jcsd
  3. Jun 17, 2014 #2

    phyzguy

    User Avatar
    Science Advisor

    I'm not a Matlab expert, and I didn't check your code in detail, but clearly your definition of the Jacobian is wrong.

    You have: J = [dfdc dfds dfds; dgds dgds dgds; dhds dhds dhds];

    You want: J = [dfdc dfds dfdq; dgdc dgds dgdq; dhdc dhds dhdq];
     
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