The Newton-Raphson method which is employed for solving a single non-linear equation can be extended to solve a system of non-linear equations. Using multi-dimensional Taylor series, a system of non-linear equations can be written near an arbitrary starting point Xi = [ x1 , x2 ,… , xn ] as follows:
To solve the system of non-linear equations the function F(X) should equal zero at the position Xi+1. It means:
Solving the above system of linear equations, we have:
The stability of the Newton-Raphson method is very sensitive to the starting point. A good knowledge about the behavior of every function of the system of non-linear equations is very important for choosing a suitable starting point as near as possible to the accurate position of the root.
The algorithm of the Newton-Raphson method is as follows:
Step 1: Choose X0 = [ x1 , x2 ,… , xn ] as a starting point.
Step 2: Let
Step 3: if ||x-x0||<e then let root = X, else X0 = X ; go to step 2.
Step 4: End.
e: Acceptable approximated error.
(Source Code in C++)