How to solve rachford - race function using newton method

I am beginner in MatLab. I don't understand how to define this function. I need to find value of fRg. I have massives with values of ci and Ki.

Answer 1

[Torsten]

Multiply the equation by product_{i=1}^{n_c} (1+f_Rg*(K_i-1)). This gives you a polynomial of degree n_c-1 in f_Rg which can be solved using the "roots" function.

Or if you are lazy, just use:

K = [3 5 -7 9 -10 2];
c = [0.4 0.6 1.2 -0.7 1.1 0.8];
syms f_Rg
fun = sum(c.*(K-1)./(1+f_Rg*(K-1)));
f_Rg_num = double(solve(fun==0,f_Rg))

f_Rg_num = 
   0.1072 + 0.0000i
  -0.1606 - 0.0949i
  -0.1606 + 0.0949i
  -0.4091 + 0.0000i
  -0.7971 + 0.0000i

But don't use Newton's method in this case because the equation has nc-1 solutions, and it were pure luck if Newton's method converged to the "correct" one.

[Ruslan]

Thank you for the answer! I will try it today. Can you please explain, what is the second part of answer -0.0949i for example.

[Torsten]

Your equation rewritten is a polynomial of degree n_c-1 in the variable f_Rg. A polynomial can have real and complex roots. A value for the "i" component in the solution for f_Rg_num that is different from 0 indicates complex roots. You will have to sort out the "correct" solution for your case from the n_c-1 solutions returned.

[Ruslan]

Oh. Now I understand. Thank you. Actual answer is in range from 0 to 1, so it will be easy to sort. Thank you for your help!