\name{bisect}
\alias{bisect}
\alias{regulaFalsi}
\title{
Rootfinding Through Bisection
}
\description{
Finding roots of univariate functions in bounded intervals.
}
\usage{
bisect(f, a, b, maxiter = 100, tol = NA)
regulaFalsi(f, a, b, maxiter = 100, tol = .Machine$double.eps^0.5)
}
\arguments{
\item{f}{Function or its name as a string.}
\item{a, b}{interval end points.}
\item{maxiter}{maximum number of iterations; default 100.}
\item{tol}{absolute tolerance; default \code{eps^(1/2)}}
}
\details{
``Bisection'' is a well known root finding algorithms for real, univariate,
continuous functions. Bisection works in any case if the function has
opposite signs at the endpoints of the interval.
\code{bisect} stops when floating point precision is reached, attaching
a tolerance is no longer needed. This version is trimmed for exactness,
not speed. Special care is taken when 0.0 is a root of the function.
Argument 'tol' is deprecated and not used anymore.
``Regula falsi'' combines bisection and secant methods.
The so-called `Illinois' improvement is used.
}
\value{
Return a list with components \code{root}, \code{f.root},
the function value at the found root, \code{iter}, the number of iterations
done, and \code{root}, and the estimated accuracy \code{estim.prec}
}
\references{
Quarteroni, A., R. Sacco, and F. Saleri (2007). Numerical Mathematics.
Second Edition, Springer-Verlag, Berlin Heidelberg.
}
\seealso{
\code{\link{ridders}}
}
\examples{
bisect(sin, 3.0, 4.0)
# $root $f.root $iter $estim.prec
# 3.1415926536 1.2246467991e-16 52 4.4408920985e-16
bisect(sin, -1.0, 1.0)
# $root $f.root $iter $estim.prec
# 0 0 2 0
# Legendre polynomial of degree 5
lp5 <- c(63, 0, -70, 0, 15, 0)/8
f <- function(x) polyval(lp5, x)
bisect(f, 0.6, 1) # 0.9061798453 correct to 15 decimals
regulaFalsi(f, 0.6, 1) # 0.9061798459 correct to 10 decimals
}
\keyword{ math }