animation (version 0.2-0)

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

Description

In mathematics, 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. This function gives a visual demonstration of this process of finding the root of an equation f(x) = 0.

Usage

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

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?

control

control parameters for the animation; see ani.control

...

other arguments passed to ani.control

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 control$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

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()
bisection.method(saveANI = TRUE, width = 600, height = 500)
ani.stop()