animation (version 1.1-4)

bisection.method: Demonstration of the Bisection Method for Root-finding on an Interval...

Description

Demonstration of the Bisection Method for Root-finding on an Interval A visual demonstration of finding the root of an equation $f(x) = 0$ on an interval using the Bisection Method.

Usage

bisection.method(FUN=function(x) x^2 - 4, rg=c(-1, 10), tol=0.001,
    interact=FALSE, main, xlab, ylab, ...)

Arguments

FUN

the function in the equation to solve (univariate)

rg

a vector containing the end-points of the interval to be searched for the root; in a c(a, b) form

tol

the desired accuracy (convergence tolerance)

interact

logical; whether choose the end-points by cliking on the curve (for two times) directly?

xlab,ylab,main

axis and main titles to be used in the plot

...

other arguments passed to curve

Value

• A list containing
• root
  the root found by the algorithm
• value
  the value of FUN(root)
• iter
  number of iterations; if it is equal to ani.options('nmax'), it's quite likely that the root is not reliable because the maximum number of iterations has been reached

Details

Suppose we want to solve the equation $f(x) = 0$. Given two points a and b such that $f(a)$ and $f(b)$ have opposite signs, we know by the intermediate value theorem that $f$ must have at least one root in the interval $[a, b]$ as long as $f$ is continuous on this interval. The bisection method divides the interval in two by computing $c = (a + b) / 2$. There are now two possibilities: either $f(a)$ and $f(c)$ have opposite signs, or $f(c)$ and $f(b)$ have opposite signs. The bisection algorithm is then applied recursively to the sub-interval where the sign change occurs. During the process of searching, the mid-point of subintervals are annotated in the graph by both texts and blue straight lines, and the end-points are denoted in dashed red lines. The root of each iteration is also plotted in the right margin of the graph.

References

http://en.wikipedia.org/wiki/Bisection_method http://animation.yihui.name/compstat:bisection_method

See Also

deriv, uniroot

Examples

# default example 
xx = bisection.method() 
xx$root  # solution

# a cubic curve 
f = function(x) x^3 - 7 * x - 10 
xx = bisection.method(f, c(-3, 5)) 

# interaction: use your mouse to select the end-points 
bisection.method(f, c(-3, 5), interact = TRUE) 

# HTML animation pages 
ani.start(nmax = 50, ani.height = 400, ani.width = 600, interval = 1,
title = "The Bisection Method for Root-finding on an Interval",
description = "The bisection method is a root-finding algorithm
which works by repeatedly dividing an interval in half and then
selecting the subinterval in which a root exists.")
par(mar = c(4, 4, 1, 2))
bisection.method(main = "")
ani.stop()