bisection.method {animation} | R Documentation |
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.
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 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 |
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 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 |
Yihui Xie
http://en.wikipedia.org/wiki/Bisection_method
http://animation.yihui.name/compstat:bisection_method
# default example xx = bisection.method() xx$root # solution ## Not run: # 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() ## End(Not run)