< Fractals < Computer graphic techniques < 2D 
          "premature optimization is the root of all evil" - Donald Knuth in paper "Structured Programming With Go To Statements"
           "good algorithms beat optimized code" Bruce Dawson ( Fractal extreme )

Software optimisation

  • Profiling code
  • benchmarking [1]

How to ask about optimisation ?

  • "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 on-topic.
  • 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 on-topic.
  • 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 loop

  • inner loop in wikipedia [4]

Optimisation of numerical computations

Distance

Symmetry

Mirror symmetry

Parameter plane and the Mandelbrot set is symmetric with respect to the x-axis 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 symmetry

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(ixMax-ix, iyAxisOfSymmetry - iyAbove , Color ); 
   } 
}  
 return 0;
}

Complex numbers

Computations without explicit definition of complex numbers or without complex type are usually faster.

Parallel computing

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]

Vectorisation

C

c++

R

This image is made with use of vectorised code. Its creation time is very short.

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)

Multicore

Image and src code using OpenMP and symmetry

"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 separate 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(ixMax-ix, iyAxisOfSymmetry - iyAbove , Color ); 
   } 
}  
 return 0;
}

Cell

"The Sony PlayStation 3 is one of the cheapest parallel computers available on the consumer market."[31][32]

GPGPU

Canvas : list of supporting browsers

Effects

Effects on 2 core CPU ( with 1 thread per core )

Methodtime [minutes]relative speed
CPU & no optimisation801
CPU & symmetry392
CPU & symmetry & OpenMP243
GPU? ( to do )

GPU should be:

  • 62 time faster than CPU [35]
  • 760 times faster [36]

Perturbation algorithm with series approximation ( 100 times faster on zoom levels around e100! ) [37]


Using WinXP Intel 2.9 GHz CPU (1 CPU used) with a GTX 480 GPU I get the following using 1000x1000 plot with 1000 max iterations :[38]

typetimedescription
gpu0.07sgpu is a pure CUDA solution on the GPU
gpuarray3.4sgpuarray uses a numpy-like CUDA wrapper in Python on the GPU
numpy43.4snumpy is a pure Numpy (C-based) solution on the CPU
python (serial)1605.6spython is a pure Python solution on the CPU with numpy arrays

References

  1. ↑ The Computer Language Benchmarks Game
  2. ↑ Stack overflow meta : Is it okay to ask code optimization help?
  3. ↑ stack exchanfge : codereview
  4. ↑ Inner loop in wikipedia
  5. ↑ Adaptive super-sampling using distance estimate by Claude Heiland-Allen
  6. ↑ wikipedia : Reflection symmetry
  7. ↑ Mirror symmetry around x axis at fraktal.republika.pl
  8. ↑ embarrassingly parallel problem
  9. ↑ wikipedia : Parallel_computers - Classes_of_parallel_computers
  10. ↑ Mandelbrot using OpenCl and Parallella
  11. ↑ wikipedia : Intel MIC
  12. ↑ parallel mandelbrot set (C Code with Message Passing Interface (MPI) library) by Omar U. Florez
  13. ↑ Guide into OpenMP: Easy multithreading programming for C++ by Joel Yliluoma
  14. ↑ Parallel Mandelbrot with OpenMP by dcfrogle
  15. ↑ claudiusmaximus : exponential mapping and openmp
  16. ↑ Part 1: OpenCL™ – Portable Parallelism By manythreads
  17. ↑ A Mandelbrot Set on the GPU in Matlab by Loren Shure
  18. ↑ progressive-julia-fractal using webgl by Felix Woitzel
  19. ↑ glsl sandbox at heroku.com
  20. ↑ gcc : Vector Extensions
  21. ↑ First code example using gcc vector support by bert hubert
  22. ↑ Mandelbrot calculation using SIMD by EJRH Edmund Horner
  23. ↑ The Computer Language Benchmarks Game
  24. ↑ Intel Advanced Vector Extensions
  25. ↑ ICC and Mandelbrot by Tim Horton
  26. ↑ Vectorization (parallel computing)
  27. ↑ Matlab - Code Vectorization Guide
  28. ↑ mandelbrot set in r by J.D. Long
  29. ↑ /programming/what-is-the-general-approach-to-threading-for-2d-plotting/ Fractal forums discussion : What is the general approach to threading for 2d plotting
  30. ↑ fractalforums discussion : most-powerful-computer-possible-for-a-reasonable-price/?PHPSESSID=7db41da33f499eb25ef85f46866438f2
  31. ↑ Algorithms for Specific Hardware -Mathematical Image Analysis Group Saarland University, Germany
  32. ↑ Jenks Parallel Programming Blog
  33. ↑ Programming high-performance applications on the Cell BE processor, Part 1: An introduction to Linux on the PLAYSTATION 3
  34. ↑ par4all
  35. ↑ MandelCPU vs MandelGPU Written by David Bucciarelli
  36. ↑ mathworks : llustrating-three-approaches-to-gpu-computing-the-mandelbrot-set
  37. ↑ Kalles Fraktaler 2 by Bernard Geiger
  38. ↑ 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
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