# Newton's Method

This module demonstrates Newton's method for minimizing a nonlinear function in one dimension. Given an appoximate solution value, a new approximate solution value is given by the minimum of a quadratic polynomial interpolating the objective function and its first and second derivatives at the given point. The process is repeated until convergence, which is usually very rapid. The method is equivalent to applying Newton's method to compute a zero of the derivative of the objective function.

The user selects a problem either by choosing a preset example or typing in a desired objective function f(x) and initial guess x. The steps of Newton's method are then carried out sequentially by repeatedly clicking on NEXT or on the currently highlighted step. The current point x and corresponding function value f(x) are indicated by bullets on the plot and are also shown numerically in the table below. At each iteration, a quadratic polynomial p(x) is fit to the current function and derivative values, the next approximate solution is taken to be the minimum of the quadratic function, and the process is then repeated. If the starting guess is close enough to the true minimum, then Newton's method converges to it, typically with a quadratic convergence rate.

Reference: Michael T. Heath, Scientific Computing, An Introductory Survey, 2nd edition, McGraw-Hill, New York, 2002. See Section 6.4.3, especially Algorithm 6.2 and Example 6.10.

Developers: Jeffrey Naisbitt and Michael Heath