MATLAB: Results of Newton-Raphson method to find the root

matricesnewton's methodroot

Hi,
 I need to solve the following equation by using Newton-raphson method :
a.    f(lambda)=-z'*inv(M)*inv(W)*Q*inv(W)*inv(M)*z,
        W=inv(M)+lambda*Q
 f'(lambda)=z'*inv(M)*inv(W)*(Q^2*inv(W)+inv(W)*Q^2)*inv(W)*inv(M)*z
 Q and W are complex square matrices ,also z and M are metrices .
I wrote a code to find the root by Newton's method ,but the value of the root is complex ,but it must be real.I am not sure a bout the derivative of f(x).
 I have another form to the function f(x) ,but I don't know if it's suitable to be solved by Newton's method in matlab,the other form is:
                m       gamma_k*|x_k|^2
 b.  f(lambda)=sum   -----------------------
                k=1   (1+lambda(gamma_k))^2
                m       (gamma_k)^2*|x_k|^2
f'(lambda)= 2*  sum   --------------------------
                k=1       (1+lambda(gamma_k))^3
 where    gamma_k is sub indices  , gamma is eigenvalue
           m is total number of eigenvalues of  T=(M)^(1/3)*Q*(M)^(1/2)
           x=U'*M^(-1/2)*z   ,U is eigen vectors of T.
Is this form in (b) of equation suitable to be solved by Newton Raphson method.
 The code for form (a):
      delta=1e-12;
     epsilon=1e-12;
      max1=500;
    lambda=-1/(2*gamma);
    for k=1:max1
       zeta=cos(theta);
       I=eye(n,n);
       Q=zeta*I-p*p';
      W=inv(M)+lambda*Q;
      y1=2*z'*inv(M)*inv(W)*Q.^2*inv(W)*inv(M)*z;
      y=-z'*inv(M)*inv(W)*Q*inv(W)*inv(M)*z;
        p1=lambda-y/y1;
      err=abs(p1-lambda);
      lambda=p1
        if (err<delta)
          break
      end
      k
      err
  end

Best Answer

Are you able to find the eigenvalues and eigenvectors of T? If so, then (b) should be straight-forward to code (though perhaps tedious.)
plot the real and imaginary parts of solutions on the same graph, but in different colors or symbols. Use "hold on" and plot each complex point as it is generated. Newton's method will move back and forth on x values, so it is probably best not to attempt to connect the solutions with lines until after all the solutions have been generated, after which you would sort based on the x coordinate.
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