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 x] as follows:_{1 }, x_{2 },… , x_{n }where To solve the system of non-linear equations the function Solving the above system of linear equations, we have: It means: 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:
= [ x] as a starting point_{1 }, x_{2 },… , x_{n }.
<e then let root = X, else X ; go to step 2._{0} = X
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