animation (version 2.0-5)

bisection.method: Demonstration of the Bisection Method for root-finding on an interval

Description

This is 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, curve

Examples

oopt = ani.options(nmax = ifelse(interactive(), 30, 
    2))

## 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 two end-points
if (interactive()) bisection.method(f, c(-3, 5), interact = TRUE)

## HTML animation pages
saveHTML({
    par(mar = c(4, 4, 1, 2))
    bisection.method(main = "")
}, img.name = "bisection.method", htmlfile = "bisection.method.html", 
    ani.height = 400, ani.width = 600, interval = 1, title = "The Bisection Method for Root-finding on an Interval", 
    description = c("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."))

ani.options(oopt)