Muller's Method

Muller's root finding method, similar to the secant method, using a parabola through three points for approximating the curve.

muller(f, p0, p1, p2 = NULL, maxiter = 100, tol = 1e-10)

Arguments

f

function whose root is to be found; function needs to be defined on the complex plain.

p0, p1, p2

three starting points, should enclose the assumed root.

tol

relative tolerance, change in successive iterates.

maxiter

maximum number of iterations.

Details

Generalizes the secant method by using parabolic interpolation between three points. This technique can be used for any root-finding problem, but is particularly useful for approximating the roots of polynomials, and for finding zeros of analytic functions in the complex plane.

Value

List of root, fval, niter, and reltol.

Note

Muller's method is considered to be (a bit) more robust than Newton's.

References

Pseudo- and C code available from the `Numerical Recipes'; pseudocode in the book `Numerical Analysis' by Burden and Faires (2011).

See also

secant, newtonRaphson, newtonsys

Examples

muller(function(x) x^10 - 0.5, 0, 1)  # root: 0.9330329915368074
#> $root
#> [1] 0.933033
#> 
#> $fval
#> [1] -5.551115e-17
#> 
#> $niter
#> [1] 9
#> 
#> $reltol
#> [1] 5.828663e-11
#> 

f <- function(x) x^4 - 3*x^3 + x^2 + x + 1
p0 <- 0.5; p1 <- -0.5; p2 <- 0.0
muller(f, p0, p1, p2)
#> $root
#> [1] -0.3390928-0.4466301i
#> 
#> $fval
#> [1] 2.220446e-16-5.551115e-17i
#> 
#> $niter
#> [1] 10
#> 
#> $reltol
#> [1] 4.568803e-16
#> 
## $root
## [1] -0.3390928-0.4466301i
## ...

##  Roots of complex functions:
fz <- function(z) sin(z)^2 + sqrt(z) - log(z)
muller(fz, 1, 1i, 1+1i)
#> $root
#> [1] 0.2555197+0.8948303i
#> 
#> $fval
#> [1] -3.608225e-16+2.220446e-16i
#> 
#> $niter
#> [1] 8
#> 
#> $reltol
#> [1] 3.656538e-13
#> 
## $root
## [1] 0.2555197+0.8948303i
## $fval
## [1] -4.440892e-16+0i
## $niter
## [1] 8
## $reltol
## [1] 3.656219e-13