Interactive Educational Modules in
Scientific Computing

Newton's Method

This module demonstrates Newton's method for solving a system of nonlinear equations f (x, y) = 0 in two dimensions. Given an approximate solution, Newton's method produces a new approximate solution based on local linearization about the current point using the Jacobian matrix, which results in a linear system to be solved for the step to the new approximate solution. This process is repeated until convergence, which is usually very rapid.

The user selects a problem either by choosing a preset example or typing in desired functions f1(x, y) and f2(x, y). The user can also select a starting point (x, y) or accept a default value. The successive 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, y) is indicated by a bullet on the plot, and all values are also shown numerically in the table below. At each iteration of Newton's method, the Newton step s to the next approximate solution is given by the solution to the approximating linear system J s = − f, where J is the Jacobian matrix of f, and the process is then repeated. If the starting guess is close enough to the true solution, 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 5.6.2, especially Algorithm 5.4 and Example 5.15.

Developers: Jeffrey Naisbitt and Michael Heath