First: Your script parameter_function_matrix.m is a bit confusing, with your exceedingly long expressions. It would be easier to read and debug if you defined some variables that you could use to build your function expressions. Example: For the first element of F you could write:
f = zeros(3,1);
a = 7.36;
b = 24.2;
c = 6.83;
d = 30.4;
f(1) = a-(b+x(2)*(c-a))/x(3)-(a-(d-a*x(2))/x(3))*exp((b+c*x(2)-d)/x(1)/50);
Convergence of the Newton-Raphson method may depend strongly on the initial values, but starting not too far from the known solution it converges in most cases. In the script below, I start at a random values normally distributed around the correct solution, using a standard deviation equal to the solution value. It converges in about half the cases, although sometimes to another solution. A narrower distribution will converge more often.
xsol = [0.033670275;0.251;1168]; % Sought solution
x0 = xsol.*(1+randn(3,1)); % Try starting a bit away from the solution
% Newton-Raphson:
x = x0;
for i = 1:50
[f,J] = patro(x);
dx = J\f(:);
x = x - dx;
if norm(f(:))< 1e-5
break;
end
end
if i >= 50
disp('No convergence')
else
fprintf('%12s %12s\n','x','g');
for i = 1:3
fprintf('%12g %12g\n',x(i),f(i));
end
end
I have made some minimal edits to your (exhausting) function code, and attached it as Matlab function patro.m.
