Compiling a Function: Mandelbrot Set

Computing the Mandelbrot set is a classic example of the need to Compile a Mathematica function (see Ruskeepää, p. 451). You just need to get the syntax right for a successful Compile. Below, the first argument is the parameter, c, and its type, Complex. The entire Module is the second argument, the expression to be compiled. Definitely try this at home!

Clear@mandelbrot;mandelbrot=Compile[{{c,_Complex}},Module[{z=0+0I,n=0},While [Abs@z<2&&n<50,z=z^2+c;n++];n  ] ]

CompiledFunction[Argument count: 1
Argument types: {_Complex}

The CompiledFunction can be applied to individual numbers. The criteria for inclusion in the Mandelbrot set is, after a limited number of iterations of z = z^2 + c (in this case 50), is its absolute value less than a given number (in this case 2). The absolute value of a complex number is its Euclidean distance from the origin. If the absolute value of a number exceeds 2 before 50 iterations, e.g. 4, it is excluded.

mandelbrot[0.2+I]

4

mandelbrot[0.2+0.2I]

50

Let's do some heavy-duty computing. As a precaution we'll increase the RAM allocated to Mathematica.

Needs@"JLink`";
ReinstallJava[JVMArguments->"-Xmx6000m"]

LinkObject[[]

Start with a couple of individual plots. I use ImageSize->Full so when you do this yourself a large image will automatically be generated. On my Dell Optiplex 900 these take 2.68 seconds each.

Timing[DensityPlot[mandelbrot[x+I y],{x,-2,1},{y,-1.5,1.5},ColorFunction->Hue,PlotPoints->200,Mesh->False,ImageSize->Full,FrameTicks->{{-2,-1,0,1},{-1,0,1}}]]//Column

 Timing[DensityPlot[mandelbrot[x+I y],{x,-2,1},{y,-1.5,1.5},ColorFunction->"Rainbow",PlotPoints->200,Mesh->False,ImageSize->Full,FrameTicks->{{-2,-1,0,1},{-1,0,1}}]]//Column

 I'm kind of curious about all the built-in ColorFunctions. Let's plot Mandelbrot sets using each of them. Here they are. I reverse the order since I like some of the ones at the end best.

colorFunctions=ColorData@"Gradients"

{AlpineColors,Aquamarine,ArmyColors,AtlanticColors,AuroraColors,AvocadoColors,BeachColors,BlueGreenYellow,BrassTones,BrightBands,BrownCyanTones,CandyColors,CherryTones,CMYKColors,CoffeeTones,DarkBands,DarkRainbow,DarkTerrain,DeepSeaColors,FallColors,FruitPunchColors,FuchsiaTones,GrayTones,GrayYellowTones,GreenBrownTerrain,GreenPinkTones,IslandColors,LakeColors,LightTemperatureMap,LightTerrain,MintColors,NeonColors,Pastel,PearlColors,PigeonTones,PlumColors,Rainbow,RedBlueTones,RedGreenSplit,RoseColors,RustTones,SandyTerrain,SiennaTones,SolarColors,SouthwestColors,StarryNightColors,SunsetColors,TemperatureMap,ThermometerColors,ValentineTones,WatermelonColors}

Length@colorFunctions

51

I'll make a Grid of them that will correspond to the Grid of the plots themselves so we can easily look up the ColorFunction of one we like by finding its name in the corresponding place in the Grid. Although each row will have 3 plots and 3 divides 51 evenly, by habit I use the syntax for Partition that would not drop an overhang of a ragged List (the empty List {} in the position of the padding argument).

Reverse@colorFunctions//Partition[#,3,3,1,{}]&//Grid

The iterator, predefined, in this Table iterates over all the ColorFunctions. Note you do not need to use the cumbersome {i, Length@colorFunctions} requiring the iterator to be a number.

Table[DensityPlot[mandelbrot[x+I y],{x,-2,1},{y,-1.5,1.5},ColorFunction->predefined,PlotPoints->200,Mesh->False,FrameTicks->{{-2,-1,0,1},{-1,0,1}},ImageSize->Medium],{predefined,Reverse@colorFunctions}]//Partition[#,3,3,1,{}]&//Grid

Recommended:

Memory Management Tools

While Mathematica is designed to manage memory for you, under certain circumstances it can get bogged down, mainly because it keeps a record of all your inputs and outputs with In and Out. So if you're using functions that output a lot of computation, or working with large files, you may notice Mathematica slowing down.

There are a number of ways that you can manage memory in Mathematica. Here is a summary (see also How to Find Memory Used in Computations).

Command	Effect
?Global`*	Shows all Symbols in a non-accessible table
Names@”Global`*”	Returns a List of all Symbols that you can access
Clear@symbol	Clears the value of symbol but leaves its name in memory
Clear@”Global`*”	Clears the values of all Symbols but leaves their names in memory
Remove@symbol	Removes the name symbol and its value from memory
Remove@”Global`*”	Removes all Symbols and their values from memory

If you're going to go as far as removing all Global Symbols, consider starting a new session by entering Quit[] in your Notebook or Quit Kernel → Local under the Evaluation menu.

Beginners hesitate to Quit the kernel, but there's little downside. Even if you haven't saved your Notebooks, the kernel is a separate entity and you can save them.

To automate resuming after quitting the kernel or in general, use Initialization Cells. You can set Initialization in the menu under Cell → Cell Properties or by right-clicking on the cell and selecting Initialization Cell. A little downward tick mark appears in the upper right corner of the cell.

Then when you re-start the kernel by selecting any cell, selecting Evaluation → Evaluate Initialization Cells, or re-open the Notebook, all the Initialization cells are automatically re-Evaluated. In this way you lose very little time by quitting the kernel and re-starting.

Memory-Management Commands to Use Occasionally

Memory currently used by the kernel:

In[157]:= MemoryInUse[]

Out[157]= 135450976

Memory currently used by the front end (all of your open Notebooks):

In[158]:= MemoryInUse@$FrontEnd

Out[158]= 543264768

The maximum memory used by the kernel during your current Mathematica session:

In[159]:= MaxMemoryUsed[]

Out[159]= 137155304

Clear a cell that consumed lots of memory in your session:

Unprotect[Out]; Out[537] =.;
Protect@Out;