Multivariable Calculus with MATLAB® by Ronald L. Lipsman & Jonathan M. Rosenberg

Author:Ronald L. Lipsman & Jonathan M. Rosenberg
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham

searches for a minimum of the function , starting at . This command produces the output:

This reflects the fact that has a minimum value of when . If instead we wanted to find a local maximum for f, we could simply apply the same algorithm to the function , since f has a local maximum where has a local minimum.

7.1.3 Newton’s Method

Newton’s method is a process for solving nonlinear equations numerically. We might want to do this for reasons that have nothing to do with finding local extrema, but nevertheless the topics of equation-solving and optimization (looking for extrema) are closely linked. As we have already seen, looking for a local extremum of f forces us to try to solve the equation . We can also go in the other direction. If we want to solve an equation , one way to do this is to look at the function . The function f is nonnegative, so the smallest it could ever be is 0, and f takes the value 0 exactly where the equation is satisfied. So the solutions of occur at local minimum points of f.

To explain Newton’s method for solving equations , let us take a simple example. Suppose we want to solve an equation such as . We start by formulating an initial guess as to where a solution might be found. Since and , it looks as if a good starting guess might be . Next, we rewrite the equation in the form . In this case we would take . Near , we have, by the tangent line approximation , the estimate . Setting this equal to zero, we get a linear equation for a (presumably) better approximation to a solution. In other words, solving

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