Solving 3 simultaneous cubic equations

Question

I have three equations of the form:

$$i_1^3L_1+i_1K+V_1+(i_2+i_3+C)Z_n=0$$ $$i_2^3L_2+i_2K+V_2+(i_1+i_3+C)Z_n=0$$ $$i_3^3L_3+i_3K+V_3+(i_1+i_2+C)Z_n=0$$

where $L_1,L_2,L_3,K,V_1,V_2,V_3,C$ and $Z_n$ are all known constants.

What methods can I use to obtain the values of $i_1,i_2$ and $i_3$ ?

Answer 1

A numerical way to solve this would be to use the Newton-Raphson method. This method can be extended to 3 dimensions as follows:

$$\vec{i_{n+1}}=\vec{i_n}-J^{-1}(i_n)\vec{f}(i_n)$$

Where $J$ is the Jacobian matrix of the system:

$$J= \begin{bmatrix} 3i_1^2L_1+K & Z_n & Z_n \\ Z_n & 3i_2^2L_2+K & Z_n \\ Z_n & Z_n & 3i_3^2L_3+K \\ \end{bmatrix} $$

Choose an initial "guess" $\vec{i_0}$, and repeat this process. Since it's an iterative process, the more times you evaluated it, the closer you get to the solution.

Answer 2

Any interactive method for solve this equations is difficult to calculate each interate manually. I recommend an already established method that has many theorems that guarantee convergence. Newton's method with guaranteed convergence theorems for Kantorovich ( see in ArXiV a expository Article on the Kantorovich's Theorem on Newton's Method ) is easy to implement in many computational software for numerical or algebraic computing.