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% Generated by roxygen2: do not edit by hand
% Please edit documentation in R/bisection.method.R
\name{bisection.method}
\alias{bisection.method}
\title{Demonstration of the Bisection Method for root-finding on an interval}
\usage{
bisection.method(
FUN = function(x) x^2 - 4,
rg = c(-1, 10),
tol = 0.001,
interact = FALSE,
main,
xlab,
ylab,
...
)
}
\arguments{
\item{FUN}{the function in the equation to solve (univariate)}
\item{rg}{a vector containing the end-points of the interval to be searched
for the root; in a \code{c(a, b)} form}
\item{tol}{the desired accuracy (convergence tolerance)}
\item{interact}{logical; whether choose the end-points by cliking on the
curve (for two times) directly?}
\item{xlab, ylab, main}{axis and main titles to be used in the plot}
\item{\dots}{other arguments passed to \code{\link{curve}}}
}
\value{
A list containing \item{root }{the root found by the algorithm}
\item{value }{the value of \code{FUN(root)}} \item{iter}{number of
iterations; if it is equal to \code{ani.options('nmax')}, it's quite likely
that the root is not reliable because the maximum number of iterations has
been reached}
}
\description{
This is a visual demonstration of finding the root of an equation \eqn{f(x) =
0} on an interval using the Bisection Method.
}
\details{
Suppose we want to solve the equation \eqn{f(x) = 0}. Given two points a and
b such that \eqn{f(a)} and \eqn{f(b)} have opposite signs, we know by the
intermediate value theorem that \eqn{f} must have at least one root in the
interval \eqn{[a, b]} as long as \eqn{f} is continuous on this interval. The
bisection method divides the interval in two by computing \eqn{c = (a + b) /
2}. There are now two possibilities: either \eqn{f(a)} and \eqn{f(c)} have
opposite signs, or \eqn{f(c)} and \eqn{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.
}
\note{
The maximum number of iterations is specified in
\code{ani.options('nmax')}.
}
\references{
Examples at \url{https://yihui.org/animation/example/bisection-method/}
For more information about Bisection method, please see
\url{https://en.wikipedia.org/wiki/Bisection_method}
}
\seealso{
\code{\link{deriv}}, \code{\link{uniroot}}, \code{\link{curve}}
}
\author{
Yihui Xie
}
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