Scientific Computing

This module demonstrates the secant method for solving a nonlinear
equation *f*(*x*) = 0*x*_{0} and
*x*_{1}, the secant method produces a new approximate
solution given by *x*_{1} −
*f*(*x*_{1}) (*x*_{1} −
*x*_{0}) ⁄ (*f*(*x*_{1}) −
*f*(*x*_{0})),*x* axis. The new approximate solution
replaces one of the old ones, and the process is repeated until
convergence, which is usually quite 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 two starting points *x*_{0} and
*x*_{1} or accept default values. The successive steps of
the secant 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 the secant method, the approximating secant line
at the current points is drawn, the next approximate solution is taken
to be the intersection of the secant line with the *x* axis, and
the process is then repeated. If the starting guesses are close enough
to the true solution, then the secant method converges to it, typically
with a superlinear convergence rate.

**Reference:** Michael T. Heath, *Scientific Computing,
An Introductory Survey*, 2nd edition, McGraw-Hill, New York,
2002. See Section 5.5.4, especially Algorithm 5.3 and Example 5.12.

**Developers:** Jeffrey Naisbitt and Michael Heath