bisection.method {animation} | R Documentation |
This is a visual demonstration of finding the root of an equation f(x) = 0 on an interval using the Bisection Method.
bisection.method(FUN = function(x) x^2 - 4, rg = c(-1, 10), tol = 0.001, interact = FALSE, main, xlab, ylab, ...)
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 |
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 |
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.
A list containing
root |
the root found by the algorithm |
value |
the value of |
iter |
number of
iterations; if it is equal to |
The maximum number of iterations is specified in
ani.options('nmax')
.
Yihui Xie
http://en.wikipedia.org/wiki/Bisection_method
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)