https://github.com/cran/pracma
Tip revision: 10ae2bb8daa6ba60ffc49143525900a7978d54b7 authored by HwB on 08 August 2013, 00:00:00 UTC
version 1.5.0
version 1.5.0
Tip revision: 10ae2bb
ridders.Rd
\name{ridders}
\alias{ridders}
\title{
Ridders' Method
}
\description{
Ridders' root finding method is a powerful variant of `regula falsi' (and
`false position'). In reliability and speed, this method is competitive
with Brent-Dekker and similar approaches.
}
\usage{
ridders(f, a, b, maxiter = 100, tol = .Machine$double.eps^0.5)
}
\arguments{
\item{f}{function whose root is to be found.}
\item{a, b}{left and right interval bounds.}
\item{maxiter}{maximum number of iterations (function calls).}
\item{tol}{tolerance, length of the last interval.}
}
\details{
Given a bracketing interval $[x_1, x_2]$, the method first calculates the
midpoint \eqn{x_3 = (x_1 + x_2)/2} and the uses an updating formula
\deqn{x_4 = x_3 + (x_3 - x_1) \frac{sgn(f(x_1) - f(x_2)) f(x_3)}{\sqrt{f(x_3)^2 - f(x_1) f(x_2)}}}
}
\value{
Returns a list with components
\item{root}{root of the function.}
\item{f.root}{value of the function at the found root.}
\item{niter}{number of iterations,or more specifically: number of function calls.}
\item{estim.prec}{the estimated precision, coming from the last brackett.}
}
\note{
See function \code{f12} whose zero at \eqn{\sqrt{e}} is difficult to find
exactly for all root finders.
}
\author{
HwB email: <hwborchers@googlemail.com>
}
\references{
Press, Teukolsky, Vetterling, and Flannery (1992). Numerical Recipes in C.
Cambridge University Press.
}
\seealso{
\code{\link{brent_dekker}}
}
\examples{
## Test functions
f1 <- function(x) # [0, 1.2], 0.399 422 2917
x^2 * (x^2/3 + sqrt(2)*sin(x)) - sqrt(3)/18
f2 <- function(x) 11*x^11 - 1 # [0.4, 1.6], 0.804 133 0975
f3 <- function(x) 35*x^35 - 1 # [-0.5, 1.9], 0.903 407 6632
f4 <- function(x) # [-0.5, 0.7], 0.077 014 24135
2*(x*exp(-9) - exp(-9*x)) + 1
f5 <- function(x) x^2 - (1 - x)^9 # [-1.4, 1], 0.259 204 4937
f6 <- function(x) (x-1)*exp(-9*x) + x^9 # [-0.8, 1.6], 0.536 741 6626
f7 <- function(x) x^2 + sin(x/9) - 1/4 # [-0.5, 1.9], 0.4475417621
f8 <- function(x) 1/8 * (9 - 1/x) # [0.001, 1.201], 0.111 111 1111
f9 <- function(x) tan(x) - x - 0.0463025 # [-0.9, 1.5], 0.500 000 0340
f10 <- function(x) # [0.4, 1], 0.679 808 9215
x^2 + x*sin(sqrt(75)*x) - 0.2
f11 <- function(x) x^9 + 0.0001 # [-1.2, 0], -0.359 381 3664
f12 <- function(x) # [1, 3.4], 1.648 721 27070
log(x) + x^2/(2*exp(1)) - 2 * x/sqrt(exp(1)) + 1
r <- ridders(f1 , 0, 1.2); r$root; r$niter # 18
r <- ridders(f2 , 0.4, 1.6); r$root; r$niter # 14
r <- ridders(f3 ,-0.5, 1.9); r$root; r$niter # 20
r <- ridders(f4 ,-0.5, 0.7); r$root; r$niter # 12
r <- ridders(f5 ,-1.4, 1); r$root; r$niter # 16
r <- ridders(f6 ,-0.8, 1.6); r$root; r$niter # 20
r <- ridders(f7 ,-0.5, 1.9); r$root; r$niter # 16
r <- ridders(f8 ,0.001, 1.201); r$root; r$niter # 18
r <- ridders(f9 ,-0.9, 1.5); r$root; r$niter # 20
r <- ridders(f10,0.4, 1); r$root; r$niter # 14
r <- ridders(f11,-1.2, 0); r$root; r$niter # 12
r <- ridders(f12,1, 3.4); r$root; r$niter # 30, err = 1e-5
\dontrun{
## Use ridders() with Rmpfr
options(digits=16)
library("Rmpfr") # unirootR
prec <- 256
.N <- function(.) mpfr(., precBits = prec)
f12 <- function(x) {
e1 <- exp(.N(1))
log(x) + x^2/(2*e1) - 2*x/sqrt(e1) + 1
}
sqrte <- sqrt(exp(.N(1))) # 1.648721270700128...
f12(sqrte) # 0
unirootR(f12, interval=mpfr(c(1, 3.4), prec), tol=1e-20)
# $root
# 1 'mpfr' number of precision 200 bits
# [1] 1.648721270700128...
ridders(f12, .N(1), .N(3.4), maxiter=200, tol=1e-20)
# $root
# 1 'mpfr' number of precision 200 bits
# [1] 1.648721270700128...
}
}
\keyword{ math }