### Newton's Method of Nonlinear Minimization

Newton's method [162],[166, p. 143] finds the minimum of a nonlinear (scalar) function of several variables by locally approximating the function by a quadratic surface, and then stepping to the bottom of that ``bowl'', which generally requires a matrix inversion. Newton's method therefore requires the function to be ``close to quadratic'', and its effectiveness is directly tied to the accuracy of that assumption. For smooth functions, Newton's method gives very rapid*quadratic convergence*in the last stages of iteration. Quadratic convergence implies, for example, that the number of significant digits in the minimizer approximately doubles each iteration.

Newton's method may be derived as follows: Suppose we wish to minimize the real, positive function with respect to . The vector Taylor expansion [546] of about gives

Applying Eq.(7.13) iteratively, we obtain Newton's method:

where is given as an initial condition. When the is any

*quadratic form*in , then , and Newton's method produces in one iteration; that is, for every .

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