Ridders' Root Finding Method
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.
ridders(fun, a, b, maxiter = 500, tol = 1e-12, ...)
fun |
function whose root is to be found. |
a, b |
left and right interval bounds. |
maxiter |
maximum number of iterations (function calls). |
tol |
tolerance, length of the last interval. |
... |
additional parameters passed on to the function. |
Given a bracketing interval $[x_1, x_2]$, the method first calculates the midpoint x_3 = (x_1 + x_2)/2 and the uses an updating formula
x_4 = x_3 + (x_3 - x_1) \frac{sgn(f(x_1) - f(x_2)) f(x_3)}{√{f(x_3)^2 - f(x_1) f(x_2)}}
Returns a list with components
root |
root of the function. |
f.root |
value of the function at the found root. |
niter |
number of iterations,or more specifically: number of function calls. |
estim.prec |
the estimated precision, coming from the last brackett. |
See function f12 whose zero at √{e} is difficult to find
exactly for all root finders.
HwB email: <hwborchers@googlemail.com>
Press, Teukolsky, Vetterling, and Flannery (1992). Numerical Recipes in C. Cambridge University Press.
## 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
## Not run:
## 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...
## End(Not run)Please choose more modern alternatives, such as Google Chrome or Mozilla Firefox.