Abstract
Using open root-finding methods we can find a zero of nonlinear function using a single initial guess. The most famous of these is Newton's method, which uses the function's derivative to where the tangent line to the root estimate crosses zero. Newton's method is rapidly convergent for many functions, though it does require knowledge of the derivative. The inexact Newton and secant methods are presented for cases where the derivative is not known. The secant method, however, is not self-starting and needs to use another method for the first iteration. Finally, we demonstrate how Newton's method can be used for systems by extending the notion of a derivative to define a Jacobian matrix.
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