mandelbrot: Mandelbrot iteration with a given number of steps

Description

This function is provided as a basis for visualizing the Mandelbrot set with phasePortrait. While usual visualizations color the points outside the Mandelbrot set dependent on the velocity of divergence, this function produces the information required for coloring the Mandelbrot set itself. For numbers that can be identified as not being elements of the Mandelbrot set, we obtain a NaN+NaNi value; for all other numbers, the function gives back the value after a user-defined number of iterations. The function has been implemented in C++; it runs fairly fast.

Usage

mandelbrot(z, itDepth = 500L)

Value

Either NaN+NaNi or the complex number obtained after

itDepth iterations

Arguments

z: Complex number; the point in the complex plane to which the output of the function is mapped
itDepth: An integer which defines the depth of the iteration, i.e. the maximum number of iteration (default: itDepth = 500)

Details

The Mandelbrot set comprises all complex numbers z for which the sequence a[n+1] = a[n]^2 + z starting with a[0] = 0 remains bounded for all n > 0. This condition is certainly not true, if, at any time, abs(a[]) >= 2. The function mandelbrot performs the iteration for n = 0, ..., itDepth - 1 and permanently checks for abs(a[n+1]) >= 2. If this is the case, it stops the iteration and returns NaN+NaNi. In all other cases, it returns a[itDepth].

Examples

Run this code

# This code shows the famous Mandelbrot figure in total, just in the
# opposite way as usual: the Mandelbrot set itself is colored, while the
# points outside are uniformly black.
# Adjust xlim and ylim to zoom in wherever you like.
# \donttest{
phasePortrait(mandelbrot,
  xlim = c(-2.3, 0.7),
  ylim = c(-1.2, 1.2),
  hsvNaN = c(0, 0, 0),
  nCores = 1)          # Max. two cores on CRAN, not a limit for your use
  # \dontshow{
  # R CMD check: make sure any open connections are closed afterward
  foreach::registerDoSEQ()
  doParallel::stopImplicitCluster()
  # }
# }