"premature optimization is the root of all evil"  Donald Knuth in paper "Structured Programming With Go To Statements"
Software optimisationEdit
 Profiling code
 benchmarking ^{[1]}
How to ask about optimisation ?Edit
 "If you've actually profiled the code, have specific snippets so that everyone can run the same code to see its performance, and you have this library published somewhere where it can be seen (GitHub, Bitbucket, etc.), then I don't necessarily see a problem with it going on Code Review. Including the results from your profiler of choice with identified bottlenecks would go a long way towards keeping it ontopic.
 If you're just starting the code but have profiled a small amount of code that exhibits the aberrant performance, then asking it on Stack Overflow would be acceptable. Including the results from your profiler of choice with identified bottlenecks would go a long way towards keeping it ontopic.
 If you're asking for people to just help you optimize the code, don't ask it anywhere. Getting a code dump and being asked to find the performance bottleneck is only something I do professionally, not on my free time." (Makoto)^{[2]}
See also : code review^{[3]}
Inner loopEdit
 inner loop in wikipedia ^{[4]}
Optimisation of numerical computationsEdit
DistanceEdit
 Distance approximation
 Koebe 1/4 theorem^{[5]}
SymmetryEdit
Mirror symmetryEdit
Parameter plane and the Mandelbrot set is symmetric with respect to the xaxis in the plane :^{[6]}
colour(x,y) = colour(x,y)
Here is a C code that shows how to divide symmetric image (with axis in the middle of its height) into 3 parts :
 axis of symmetry
 2 symmetric parts : above and below axis
// fill array using mirror symmetry of image
// uses global var : ...
int FillArray(unsigned char data[] )
{
unsigned char Color; // gray from 0 to 255
// draw axis of symmetry
iy = iyAxisOfSymmetry;
for(ix=ixMin;ix<=ixMax;++ix) PlotPoint(ix, iy, GiveColor(ix, iy));
// draw symmetric parts : above and below axis
for(iyAbove = iyAboveMin; iyAbove<=iyAboveMax; ++iyAbove)
for(ix=ixMin; ix<=ixMax; ++ix)
{ // above axis compute color and save it to the array
iy = iyAxisOfSymmetry + iyAbove;
Color = GiveColor(ix, iy);
PlotPoint(ix, iy, Color );
// below the axis only copy Color the same as above without computing it
PlotPoint(ix, iyAxisOfSymmetry  iyAbove , Color );
}
return 0;
}
See also Pascal code ( Delphi) for general case ^{[7]}
Rotational symmetryEdit
Dynamical plane and the Julia set have rotational symmetry. It can be used to speed up computing :
// fill array using symmetry of image
// uses global var : ...
int FillArraySymmetric(unsigned char data[] )
{
unsigned char Color; // gray from 0 to 255
printf("axis of symmetry \n");
iy = iyAxisOfSymmetry;
for(ix=ixMin;ix<=ixMax;++ix) PlotPoint(ix, iy, GiveColor(ix, iy));
// above and below axis
for(iyAbove = iyAboveMin; iyAbove<=iyAboveMax; ++iyAbove)
{printf(" %d from %d\n", iyAbove, iyAboveMax); //info
for(ix=ixMin; ix<=ixMax; ++ix)
{ // above axis compute color and save it to the array
iy = iyAxisOfSymmetry + iyAbove;
Color = GiveColor(ix, iy);
PlotPoint(ix, iy, Color );
// below the axis only copy Color the same as above without computing it
PlotPoint(ixMaxix, iyAxisOfSymmetry  iyAbove , Color );
}
}
return 0;
}
Complex numbersEdit
Computations without explicit definition of complex numbers or without complex type are usually faster.
Parallel computingEdit
In Mandelbrot and Julia sets each point/pixel is calculated independently, so it can be easly divided into an smaller tasks and carried out simultaneously.^{[8]}
Types of parallel computing : Wikipedia description^{[9]}
 Multicore (computing)
 Parallella and OpenCl^{[10]}
 Intel MIC^{[11]}
 MPI ^{[12]}
 OpenMP^{[13]}^{[14]}^{[15]}
 Generalpurpose computing on graphics processing units (GPGPU):
 OpenCl^{[16]}
 CUDA^{[17]}
 WebGl ^{[18]}^{[19]}
VectorisationEdit
CEdit
 gcc vector extension ^{[20]}^{[21]}^{[22]}^{[23]}
 Intel® Advanced Vector Extensions^{[24]}^{[25]}
REdit
Vectorisation ^{[26]}^{[27]}
Here is R code of above image with comments: ^{[28]}
## based on the R code by Jarek Tuszynski ## http://commons.wikimedia.org/wiki/File:Mandelbrot_Creation_Animation_%28800x600%29.gif iMax=20; # number of frames and iterations ## world coordinate ( float ) cxmin = 2.2; cxmax = 1.0; cymax = 1.2; cymin = 1.2; ## screen coordinate ( integer) dx=800; dy=600 ; # define grid size ## sequences of values = rows and columns cxseq = seq(cxmin, cxmax, length.out=dx); cyseq = seq(cymin, cymax, length.out=dy); ## sequences of values = rows * columns cxrseq = rep(cxseq, each = dy); cyrseq = rep(cyseq, dx); C = complex( real=cxrseq, imag = cyrseq); # complex vector C = matrix(C,dy,dx) # convert from vector to matrix # Now C is a matrix of c points coordinates (cx,cy) # allocate memory for all the frames F = array(0, dim = c(dy,dx,iMax)) # dy*dx*iMax array filled with zeros Z = 0 # initialize Z to zero for (i in 1:iMax) # perform iMax iterations { Z = Z^2+C # iteration; uses n point to compute n+1 point F[,,i] = exp(abs(Z)) # an omitted index is used to represent an entire column or row # store magnitude of the complex number, scale it using an exponential function to emphasize fine detail, } # color and save F array to the file library(caTools) # load library with write.gif function # mapped to color palette jetColors . Dark red is a very low number, dark blue is a very high number jetColors = colorRampPalette(c("#00007F", "blue", "#007FFF", "cyan", "#7FFF7F", "yellow", "#FF7F00", "red", "#7F0000")) write.gif(F, "Mandelbrot.gif", col=jetColors, delay=100, transparent=0)
MulticoreEdit
"I always create as many worker threads as I have cores. More than that and your system spends too much time task switching"  Duncan C^{[29]}^{[30]}
Using OpenMP :
// fill array using symmetry of image
// uses global var : ...
int FillArraySymmetric(unsigned char data[] )
{
unsigned char Color; // gray from 0 to 255
printf("axis of symmetry \n");
iy = iyAxisOfSymmetry;
#pragma omp parallel for schedule(dynamic) private(ix,Color) shared(ixMin,ixMax, iyAxisOfSymmetry)
for(ix=ixMin;ix<=ixMax;++ix) PlotPoint(ix, iy, GiveColor(ix, iy));
/*
The use of ‘shared(variable, variable2) specifies that these variables should be shared among all the threads.
The use of ‘private(variable, variable2)’ specifies that these variables should have a seperate instance in each thread.
*/
#pragma omp parallel for schedule(dynamic) private(iyAbove,ix,iy,Color) shared(iyAboveMin, iyAboveMax,ixMin,ixMax, iyAxisOfSymmetry)
// above and below axis
for(iyAbove = iyAboveMin; iyAbove<=iyAboveMax; ++iyAbove)
{printf(" %d from %d\n", iyAbove, iyAboveMax); //info
for(ix=ixMin; ix<=ixMax; ++ix)
{ // above axis compute color and save it to the array
iy = iyAxisOfSymmetry + iyAbove;
Color = GiveColor(ix, iy);
PlotPoint(ix, iy, Color );
// below the axis only copy Color the same as above without computing it
PlotPoint(ixMaxix, iyAxisOfSymmetry  iyAbove , Color );
}
}
return 0;
}
CellEdit
"The Sony PlayStation 3 is one of the cheapest parallel computers available on the consumer market."^{[31]}^{[32]}
GPGPUEdit
 CUDA

 Yellow Dog Linux for CUDA^{[33]}
 par4all^{[34]}

 OpenCl
 GLSL  Tools for GLSL editing
 Fragmentarium : playground for working with GPU (GLSL) based pixel graphics built in C++, OpenGL/GLSL, and Qt 4.
 shader editor using kickjs
 GLSL Sandbox by Ricardo Cabello Miguel using three.js
 jsfiddle and example
 WebGl : GLSL thru html. It is a 3D graphics API for JavaScript based on the OpenGL ES 2.0 API, that developers can use to create fully 3D web apps.
 shadertoy by Inigo Quilez, Pol Jeremias : a browser based Fragment Shader editor for 2D effects with it.
 Chrome Experiments
 WebGL playground  new is not working !!
 examples :
Canvas : list of supporting browsers
EffectsEdit
Effects on 2 core CPU ( with 1 thread per core )
Method  time [minutes]  relative speed 

CPU & no optimisation  80  1 
CPU & symmetry  39  2 
CPU & symmetry & OpenMP  24  3 
GPU  ? ( to do ) 
GPU should 62 time faster than CPU ^{[35]}
Perturbation algorithm with series approximation ( 100 times faster on zoom levels around e100! ) ^{[36]}
Using WinXP Intel 2.9GHz CPU (1 CPU used) with a GTX 480 GPU I get the following using 1000x1000 plot with 1000 max iterations : ^{[37]}
type  time  description 

gpu  0.07s  gpu is a pure CUDA solution on the GPU 
gpuarray  3.4s  gpuarray uses a numpylike CUDA wrapper in Python on the GPU 
numpy  43.4s  numpy is a pure Numpy (Cbased) solution on the CPU 
python (serial)  1605.6s  python is a pure Python solution on the CPU with numpy arrays 
ReferencesEdit
 ↑ The Computer Language Benchmarks Game
 ↑ Stack overflow meta : Is it okay to ask code optimization help?
 ↑ stack exchanfge : codereview
 ↑ Inner loop in wikipedia
 ↑ Adaptive supersampling using distance estimate by Claude HeilandAllen
 ↑ wikipedia : Reflection symmetry
 ↑ Mirror symmetry around x axis at fraktal.republika.pl
 ↑ embarrassingly parallel problem
 ↑ wikipedia : Parallel_computers  Classes_of_parallel_computers
 ↑ Mandelbrot using OpenCl and Parallella
 ↑ wikipedia : Intel MIC
 ↑ parallel mandelbrot set (C Code with Message Passing Interface (MPI) library) by Omar U. Florez
 ↑ Guide into OpenMP: Easy multithreading programming for C++ by Joel Yliluoma
 ↑ Parallel Mandelbrot with OpenMP by dcfrogle
 ↑ claudiusmaximus : exponential mapping and openmp
 ↑ Part 1: OpenCL™ – Portable Parallelism By manythreads
 ↑ A Mandelbrot Set on the GPU in Matlab by Loren Shure
 ↑ progressivejuliafractal using webgl by Felix Woitzel
 ↑ glsl sandbox at heroku.com
 ↑ gcc : Vector Extensions
 ↑ First code example using gcc vector support by bert hubert
 ↑ Mandelbrot calculation using SIMD by EJRH Edmund Horner
 ↑ The Computer Language Benchmarks Game
 ↑ Intel Advanced Vector Extensions
 ↑ ICC and Mandelbrot by Tim Horton
 ↑ Vectorization (parallel computing)
 ↑ Matlab  Code Vectorization Guide
 ↑ mandelbrot set in r by J.D. Long
 ↑ /programming/whatisthegeneralapproachtothreadingfor2dplotting/ Fractal forums discussion : What is the general approach to threading for 2d plotting
 ↑ fractalforums discussion : mostpowerfulcomputerpossibleforareasonableprice/?PHPSESSID=7db41da33f499eb25ef85f46866438f2
 ↑ Algorithms for Specific Hardware Mathematical Image Analysis Group Saarland University, Germany
 ↑ Jenks Parallel Programming Blog
 ↑ Programming highperformance applications on the Cell BE processor, Part 1: An introduction to Linux on the PLAYSTATION 3
 ↑ [par4all]
 ↑ MandelCPU vs MandelGPU Written by David Bucciarelli
 ↑ Kalles Fraktaler 2 by Bernard Geiger
 ↑ [http://wiki.tiker.net/PyCuda/Examples/Mandelbrot  Mandelbrot calculate using GPU, Serial numpy and faster numpy Use to show the speed difference between CPU and GPU calculations by Ian Ozsvald ianozsvald.com July 2010]