MATLAB: How can i solve an equation in matlab using Newton-Raphson method

monotonicallynewton raphsonnumeric equation solving

I want to solve monotonically increasing function g(lambda)and has a unique zero,by Newton-Raphson method in matlab.The equation is : g(lambda)= -zm'*inv(M)*inv(inv(M)+lambda.*Q)*Q*inv(inv(M)+lambda.*Q)*inv(M)*zm; the variable is lambda

Best Answer

A specific Newton-Raphson routine is not provided in MATLAB (at least not that I have seen.) You can find MATLAB coding for it in a number of places.

I have to ask, though, whether any efficiency gains you might hypothetically gain using this instead of using fzero() will be worth the time you spend programming and debugging this? (I am not convinced that NR would be any faster than fzero()'s adaptive method, but that is a different point.) Has using NR been imposed on you? Or did fzero() fail?

Related Solutions

MATLAB: CODE

[Edited to define the anonymous function as a separate step]

      function lambda = drive_zr17t9001
        format short e;
        Params = load('saved_data.mat');
        theta = pi/2;
        zeta = cos(theta);
        I = eye(Params.n,Params.n);
        Q = zeta*I-Params.p*Params.p';

%T is a matrix(5,5)

        Mroot = Params.M.^(1/2);  %optimization

        T = Mroot*Q*Mroot;

%find the eigen values

        E =real( eig(T));

%find the negative eigen values -- %find the smallest negative eigen value

        gamma = min(E);

%now solve for lambda

        bounds = sort([0, -1/gamma]);  %in case gamma is positive
        M_inv = inv(Params.M);  %optimization
        zm = Params.zm;
        Lz = @(lambda) zr17t9001(lambda,M_inv,Q,zm);
lambda = fzero(Lz,bounds);  %do the searching
     % end   
     function r = zr17t9001(lambda, M_inv, Q, zm)
        Winv = inv(M_inv+lambda.*Q);
        r = -zm'*M_inv*Winv*Q*Winv*M_inv*zm;
      %end

MATLAB: CODE ASSISTANCE

Remove the final "end" statement from drive_zr17t9 and the code would likely start working.

This is a side effect of "poofing variables into existence" through a load() statement.

Matt's approach is a better one over the long term.

[Addition: code with Matt's suggested changes]

[Code modified to reflect the fact p is in the .mat file after all]

[Code modified: Minv -> M_inv]

    function lambda = drive_zr17t9
      format short e;
      Params = load('saved_data.mat');
      theta = pi/2;
      zeta = cos(theta);
      I = eye(Params.n,Params.n);
      Q = zeta*I-Params.p*Params.p';
      %T is a matrix(5,5)
      Mroot = Params.M.^(1/2);  %optimization

      T = Mroot*Q*Mroot;
      %find the eigen values
      E = eig(T);
      %find the negative eigen values -- this section still needs to be fixed
      ind = find(E<0);
      G = E(ind);
      %find the smallest negative eigen value
      gamma = min(abs(G));  %this is almost certainly wrong!
      %now solve for lambda
      bounds = sort([0, -1/gamma]);  %in case gamma is positive
      M_inv = inv(Params.M);  %optimization
      lambda = fzero(@(lambda) zr17t9(lambda,  M_inv, Q, Params.zm), bounds);  %do the searching
    end
    function r = zr17t9(lambda, M_inv, Q, zm)
      Winv = inv(M_inv+lambda.*Q);
      r = -zm'*M_inv*Winv*Q*Winv*M_inv*zm;
    end

Best Answer

Related Solutions

MATLAB: CODE

MATLAB: CODE ASSISTANCE

Related Question