Scientific Computing

This module demonstrates Newton's method for solving a nonlinear
equation *f*(*x*) = 0*x*, Newton's method produces a new
approximate solution given by *x* − *f*(*x*)
⁄ *f*′(*x*),

The user selects a problem either by choosing a preset example or
typing in a desired function *f*(*x*)*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*)*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
*x*_{0} = 1*x* = 1*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