Illustrating Three Approaches to GPU Computing: The Mandelbrot Set
This example shows how to adapt your MATLAB® code to compute the Mandelbrot Set using a GPU.
Starting with an existing algorithm, this example shows how to adapt your code using Parallel Computing Toolbox™ to make use of GPU hardware in three ways:
Using the existing algorithm but with GPU data as input
Using
arrayfun
to perform the algorithm on each element independentlyUsing the MATLAB/CUDA interface to run some existing CUDA/C++ code
Setup
The values below specify a highly zoomed part of the Mandelbrot Set in the valley between the main cardioid and the p/q bulb to its left.
A 1000x1000 grid of real parts (X) and imaginary parts (Y) is created between these limits and the Mandelbrot algorithm is iterated 500 times at each grid location.
maxIterations = 500; gridSize = 1000; xlim = [-0.748766713922161, -0.748766707771757]; ylim = [ 0.123640844894862, 0.123640851045266];
The Mandelbrot Set in MATLAB
Below is an implementation of the Mandelbrot Set using standard MATLAB commands running on the CPU. This is based on the code provided in Cleve Moler's Experiments with MATLAB e-book. Time the execution on the CPU using tic
and toc
.
Setup the two-dimensional grid of complex values.
tic; x = linspace(xlim(1),xlim(2),gridSize); y = linspace(ylim(1),ylim(2),gridSize); [xGrid,yGrid] = meshgrid(x,y); z0 = xGrid + 1i*yGrid;
For 500 iterations, calculate the next value of a point on the complex grid z
by squaring the previous value and adding its initial value, z0
. Count the number of iterations for which the magnitude of z
is less than or equal to two. This calculation is vectorized such that every location is updated at once.
cpuCount = ones(size(z0)); z = z0; for n = 0:maxIterations z = z.*z + z0; inside = abs(z)<=2; cpuCount = cpuCount + inside; end cpuCount = log(cpuCount); cpuTime = toc
cpuTime = 4.4007
Plot the natural logarithm of the count.
figure imagesc(x,y,cpuCount); c = colormap([jet;flipud(jet);0 0 0]); axis off title(sprintf("CPU Execution: %1.3f s",cpuTime));
Using gpuArray
When MATLAB encounters data on the GPU, calculations with that data are performed on the GPU. The class gpuArray
provides GPU versions of many functions that you can use to create data arrays, including the linspace
, logspace
, and meshgrid
functions needed here. Similarly, the count
array is initialized directly on the GPU using the function ones
.
Ensure that your desired GPU is available and selected.
gpu = gpuDevice;
disp(gpu.Name + " GPU selected.")
NVIDIA RTX A5000 GPU selected.
Call the naiveGPUMandelbrot
function. The supporting function naiveGPUMandelbrot
applies the Mandelbrot algorithm for each point on the grid on the GPU and is provided at the end of this examples.
[x,y,naiveGPUCount] = naiveGPUMandelbrot(xlim,ylim,gridSize,maxIterations);
Time the execution of the function on the GPU using gputimeit
. For functions that use the GPU, gputimeit
is better than tic
and toc
or timeit
because it ensures that all operations on the GPU finish before recording the elapsed time.
naiveGPUTime = gputimeit(@() naiveGPUMandelbrot(xlim,ylim,gridSize,maxIterations))
naiveGPUTime = 0.2181
Element-wise Operation
Noting that the algorithm is operating equally on every element of the input, we can place the code in a function and call it using arrayfun
. The function processMandelbrotElement
is provided as a supporting function at the end of this example. For gpuArray
inputs, the function used with arrayfun
gets compiled into native GPU code.
An early abort has been introduced into the function processMandelbrotElement
because this function processes only a single element. For most views of the Mandelbrot Set a significant number of elements stop very early and this can save a lot of processing. The for
-loop has also been replaced by a while
-loop because they are usually more efficient. This function makes no mention of the GPU and uses no GPU-specific features.
Using arrayfun
causes MATLAB to make one call to a parallelized GPU operation that performs the whole calculation, instead of many thousands of calls to separate GPU-optimized operations (at least 6 per iteration). The first time you call arrayfun
to run a particular function on the GPU, there is some overhead time to set up the function for GPU execution. Subsequent calls of arrayfun
with the same function can run faster.
Setup the two-dimensional grid of complex values.
xGrid = gpuArray(xGrid); yGrid = gpuArray(yGrid);
Using arrayfun
, apply the Mandelbrot algorithm for each point on the grid.
gpuArrayfunCount = arrayfun(@processMandelbrotElement, ...
xGrid,yGrid,maxIterations);
Time the arrayfun
execution using gputimeit
.
gpuArrayfunTime = gputimeit(@() arrayfun(@processMandelbrotElement, ...
xGrid,yGrid,maxIterations))
gpuArrayfunTime = 0.0308
Working with CUDA
In Experiments in MATLAB performance is improved by converting the basic algorithm to a C-Mex function. If you are willing to do some work in C/C++, then you can use Parallel Computing Toolbox to call pre-written CUDA kernels using MATLAB data. For more details on using CUDA kernels in MATLAB, see Run CUDA or PTX Code on GPU.
A CUDA/C++ implementation of the element processing algorithm is provided with this example, pctdemo_processMandelbrotElement.cu
. The part of the CUDA/C++ code that executes the Mandelbrot algorithm for a single location is given below.
__device__ unsigned int doIterations( double const realPart0, double const imagPart0, unsigned int const maxIters ) { // Initialize: z = z0 double realPart = realPart0; double imagPart = imagPart0; unsigned int count = 0; // Loop until escape while ( ( count <= maxIters ) && ((realPart*realPart + imagPart*imagPart) <= 4.0) ) { ++count; // Update: z = z*z + z0; double const oldRealPart = realPart; realPart = realPart*realPart - imagPart*imagPart + realPart0; imagPart = 2.0*oldRealPart*imagPart + imagPart0; } return count; }
Compile this file into a parallel thread execution (PTX) file using mexcuda
.
mexcuda -ptx pctdemo_processMandelbrotElement.cu
Building with 'NVIDIA CUDA Compiler'. MEX completed successfully.
Create a parallel.gpu.CUDAKernel
object by passing the CUDA file and the PTX file to the parallel.gpu.CUDAKernel
function.
cudaFilename = "pctdemo_processMandelbrotElement.cu"; ptxFilename = "pctdemo_processMandelbrotElement.ptx"; kernel = parallel.gpu.CUDAKernel(ptxFilename,cudaFilename);
One GPU thread is required per location in the Mandelbrot Set, with the threads grouped into blocks. The kernel indicates how big a thread-block is. Calculate the number of thread-blocks required, and set the GridSize
property of the kernel (effectively the number of thread blocks that will be launched independently by the GPU) accordingly.
numElements = numel(xGrid); kernel.ThreadBlockSize = [kernel.MaxThreadsPerBlock,1,1]; kernel.GridSize = [ceil(numElements/kernel.MaxThreadsPerBlock),1];
Evaluate the kernel using feval
.
count = zeros(size(xGrid),"gpuArray");
gpuCUDAKernelCount = feval(kernel,count,xGrid,yGrid,maxIterations,numElements);
Time the kernel execution using gputimeit
.
gpuCUDAKernelTime = gputimeit(@() feval(kernel,count,xGrid,yGrid,maxIterations,numElements));
Summary
Plot the results from the different methods and compare the execution times.
method = ["Naive GPU Execution" "GPU Execution Using arrayfun" "CUDAKernel Execution"]; count = cat(3,naiveGPUCount,gpuArrayfunCount,gpuCUDAKernelCount); time = [naiveGPUTime gpuArrayfunTime gpuCUDAKernelTime]; figure colormap(c) tiledlayout("flow") nexttile imagesc(x,y,cpuCount); axis off title(sprintf("CPU Execution: %1.3f s",cpuTime)); for idx = 1:3 nexttile imagesc(x,y,count(:,:,idx)) axis off title(sprintf("%s: %1.3f s \n (%1.1fx faster)", ... method(idx),time(idx),cpuTime/time(idx))) end
This example has shown three ways in which a MATLAB algorithm can be adapted to make use of GPU hardware:
Convert the input data to be on the GPU using
gpuArray
, leaving the algorithm unchanged.Use
arrayfun
on agpuArray
input to perform the algorithm on each element of the input independently.Use a
parallel.gpu.CUDAKernel
to run some existing CUDA/C++ code using MATLAB data.
The code in this example was timed on a Windows® 10, Intel® Xeon® W-2133 @ 3.60 GHz test system with an NVIDIA® RTX A5000 GPU.
Supporting Functions
The supporting function naiveGPUMandelbrot
creates a two-dimensional grid of complex values and counts the number of iterations before the complex value number (x0,y0)
jumps outside a circle of radius 2 on the complex plane. Each iteration involves mapping z = z^2 + z0
where z0 = x0 + i*y0
. The function returns the grid coordinate vectors and the log of the iteration count at escape or maxIterations
if the point did not escape. By initializing data on the GPU and operating on this data, the naiveGPUMandelbrot
function executes the Mandelbrot algorithm on the GPU.
function [x,y,count] = naiveGPUMandelbrot(xlim,ylim,gridSize,maxIterations) x = gpuArray.linspace(xlim(1),xlim(2),gridSize); y = gpuArray.linspace(ylim(1),ylim(2),gridSize); [xGrid,yGrid] = meshgrid(x,y); z0 = complex(xGrid,yGrid); count = ones(size(z0),"gpuArray"); z = z0; for n = 0:maxIterations z = z.*z + z0; inside = abs(z)<=2; count = count + inside; end count = log(count); end
The supporting function processMandelbrotElement
creates a two-dimensional grid of complex values and counts the number of iterations before the complex value number (x0,y0)
jumps outside a circle of radius 2 on the complex plane. Each iteration involves mapping z = z^2 + z0
where z0 = x0 + i*y0
. The function returns the log of the iteration count at escape or maxIterations
if the point did not escape.
function count = processMandelbrotElement(x0,y0,maxIterations) z0 = complex(x0,y0); z = z0; count = 1; while (count <= maxIterations) && (abs(z) <= 2) count = count + 1; z = z*z + z0; end count = log(count); end