Newtons Method for Simultaneous Nonlinear Equations

If the mathematical model involves two (or more) simultaneous nonlinear equations in two (or more) unknowns, the Newton-Raphson method can be extended to solve these equations simultaneously. In what follows, we will first develop the Newton-Raphson method for two equations and then expand the algorithm to a system of k equations. 

The model for two unknowns will have the general form 

The superscript (1) will be used to designate the iteration number of the estimate. 

Setting the left sides of Eqs. (1.58) to zero and truncating the second-order and higher derivatives of the Taylor series, we obtain the following equations  

61) are a set of simultaneous linear algebraic equations, where the unknowns are 6 

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