Newtons method for a single equation

We now use this truncated Taylor series to develop an iterative technique for solving a nonlinear algebraic equation /(x) = 0 known as Newton s method. To start Newton s method, we make an initial guess xt of the solution that we hope is close to the true value Xs where /(xs) = 0. We use a Taylor series to approximate /(x) in the vicinity ofxl l. 

In this case, as long as the first derivative is nonzero ntx , we obtain a reasonable approximation of the solution, x l, from the rule 

Successive application of this rule yields Newton s method for solving a single nonlinear algebraic equation. 

Performance of Newton s method fora single equation 

We demonstrate the performance of Newton s method for various cubic polynomials, starting with one that possesses only a single real root, 

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