Halley's Root Finding Mathod

Finding roots of univariate functions using the Halley method.

halley(fun, x0, maxiter = 500, tol = 1e-08, ...)

Arguments

fun

function whose root is to be found.

x0

starting value for the iteration.

maxiter

maximum number of iterations.

tol

absolute tolerance; default eps^(1/2)

...

additional arguments to be passed to the function.

Details

Well known root finding algorithms for real, univariate, continuous functions; the second derivative must be smooth, i.e. continuous. The first and second derivative are computed numerically.

Value

Return a list with components root, f.root, the function value at the found root, iter, the number of iterations done, and the estimated precision estim.prec

See also

newtonRaphson

Examples

halley(sin, 3.0)        # 3.14159265358979 in 3 iterations
#> $root
#> [1] 3.141593
#> 
#> $f.root
#> [1] 1.224647e-16
#> 
#> $iter
#> [1] 3
#> 
#> $estim.prec
#> [1] 1.841549e-11
#> 
halley(function(x) x*exp(x) - 1, 1.0)
#> $root
#> [1] 0.5671433
#> 
#> $f.root
#> [1] 0
#> 
#> $iter
#> [1] 4
#> 
#> $estim.prec
#> [1] 0
#> 
                        # 0.567143290409784 Gauss' omega constant

# Legendre polynomial of degree 5
lp5 <- c(63, 0, -70, 0, 15, 0)/8
f <- function(x) polyval(lp5, x)
halley(f, 1.0)          # 0.906179845938664
#> $root
#> [1] 0.9061798
#> 
#> $f.root
#> [1] -4.440892e-16
#> 
#> $iter
#> [1] 4
#> 
#> $estim.prec
#> [1] 4.218847e-15
#>