Simultaneous Solution of Nonlinear Algebraic Equations

Whereas the procedures for solving systems of linear equations are straightforward, those for solving sets of nonlinear equations are not nearly so well formulated. The method that will be used for the solution 

If there exists a present set of guesses, for the solution of (2.3.21), the function / may be written for any other a as a Taylor series expansion about ar as follows  

The partial derivative in equation (2.3.22) is called the Jacobian matrix and is given in terms of the following first partials  

The Jacobian can be evaluated analytically by using analytic first partials for each entry (2.3.23) however, it is usually estimated numerically by 

Ideally, the next set of guesses, would be the solution to equation (2.3.21). If this were the case, (2.3.22) becomes, truncating the nonlinear higher-order terms. 

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