CODEWITHSUNDEEP
matlabmathnewtons-method
Kevin K
Kevin K

Reputation: 106

Writing a system of general equilibrium function

I am trying to create a system of equation with theoretically that represent the functions seen below (we know Kt, alpha, and beta but not Kt+q) - any help is greatly appreciated:

Picture of the system of the equations

How should I represent this and pass it into my newton raphson code (any input here is also appreciated):

% newton raphson 
function I = NR(f,x0) 
    x=x0; %starting point 
    fx=f(x);  
    J = CDJac(f,x); 
    I = x - fx/J;     
end 

% the jacobian below is validated;

function [DCD] = CDJac(f,xbar)%the jacobian 
    jk=length(xbar); %find the dimension of x
    hstar=eps^(1/3); %choose value of h based upon Heer and Maussner machine eps
    e=zeros(1,jk); %1 x j vector of zeros; j coresspond to the derivative 
    %with respect to the jth varibale. If j=1, I am taking the derivative of
    %this multivraite function with respect to x1. Creates a bunch of zeros. AS
    %we go through and evlaute everything. We replace that zeros with a one. 
    for j=1:length(xbar) %if j is 1:10. xbar is the vector of 
        %10 different points. you have 10 differetn x s. 
        e(j)=1; %replace the jth entry to 1 in the zero vector. (1,0). In a 
        %of loop, j become 2 after it is done with 1. We then take the second
        %element of it and change it to a 1- (0,1). 
        fxbarph=f([xbar+e.*hstar]); %function evaluated at point xbar plus h
        fxbarmh=f([xbar-e.*hstar]); %function evaluated at point xbar minus h
        DCD(:,j)=(fxbarph-fxbarmh)./(2*hstar);
        e=zeros(1,jk); %create the ej row vector of zeros. For instance, when j
        %goes to 2, you need to have 0s everywhere except the second column. 
    end
end

Upvotes: 1

Views: 98

Answers (1)

duffymo
duffymo

Reputation: 308968

I think you need to perform the following steps:

  1. Express each function as f(k) = 0: subtract the RHS from both sides of each equation.
  2. Calculate the Jacobean for the linear increments: J*dk = f
  3. Make an initial guess for the unknown k vector
  4. Solve for unknown increment vector dk
  5. Update the vector k: k(new) = k(old) + dk
  6. Iterate to convergence

Upvotes: 1

Related Questions