Newton's Method
This module demonstrates Newton's method for solving a nonlinear
equation f(x) = 0 in one dimension. Given
an approximate solution x, Newton's method produces a new
approximate solution given by x − f(x)
⁄ f′(x), based on local linearization
about the current point (the tangent line in one dimension). 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 a desired function f(x).
The user can also select a starting point x 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 values of x and
f(x) are indicated by bullets on the
plot and are also shown numerically in the table below. At each
iteration of Newton's method, the approximating tangent line at the
current point is drawn, the next approximate solution is taken to be
the intersection of the tangent line with the x axis, and the
process is then repeated. If the starting guess is close enough to the
solution, then Newton's method converges to it, typically with a
quadratic convergence rate.
Example 1 shows Newton's method quickly finding the solution of the sum
of a polynomial and a trigonometric function. Example 2 shows a case
in which Newton's method fails because it is started too far away from
the solution. With the default starting value of
x0 = 1, the method is trapped in an
infinite loop alternating between x = 1 and
x = −1.
Reference: Michael T. Heath, Scientific Computing,
An Introductory Survey, 2nd edition McGraw-Hill, New York,
2002. See Section 5.5.3, especially Algorithm 5.2 and Example 5.10.
Developers: Jeffrey Naisbitt and Michael Heath
Department of Computer Science