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This example looks at how we can benchmark the solving of a linear system on the GPU. The MATLAB® code to solve for `x` in `A*x = b` is very simple. Most frequently, we use matrix left division, also known as `mldivide` or the backslash operator (\), to calculate `x` (that is, `x = A\b`).

Related examples:

Benchmarking A\b using distributed arrays.

The code shown in this example can be found in this function:

```
function results = paralleldemo_gpu_backslash(maxMemory)
```

It is important to choose the appropriate matrix size for the computations. We can do this by specifying the amount of system memory in GB available to the CPU and the GPU. The default value is based only on the amount of memory available on the GPU, and you can specify a value that is appropriate for your system.

if nargin == 0 g = gpuDevice; maxMemory = 0.4*g.FreeMemory/1024^3; end

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We want to benchmark matrix left division (\), and not the cost of transferring data between the CPU and GPU, the time it takes to create a matrix, or other parameters. We therefore separate the data generation from the solving of the linear system, and measure only the time it takes to do the latter.

function [A, b] = getData(n, clz) fprintf('Creating a matrix of size %d-by-%d.\n', n, n); A = rand(n, n, clz) + 100*eye(n, n, clz); b = rand(n, 1, clz); end function time = timeSolve(A, b, waitFcn) tic; x = A\b; %#ok<NASGU> We don't need the value of x. waitFcn(); % Wait for operation to complete. time = toc; end

As with a great number of other parallel algorithms, the performance of solving a linear system in parallel depends greatly on the matrix size. As seen in other examples, such as Benchmarking A\b, we compare the performance of the algorithm for different matrix sizes.

% Declare the matrix sizes to be a multiple of 1024. maxSizeSingle = floor(sqrt(maxMemory*1024^3/4)); maxSizeDouble = floor(sqrt(maxMemory*1024^3/8)); step = 1024; if maxSizeDouble/step >= 10 step = step*floor(maxSizeDouble/(5*step)); end sizeSingle = 1024:step:maxSizeSingle; sizeDouble = 1024:step:maxSizeDouble;

**Comparing Performance: Gigaflops**

We use the number of floating point operations per second as our measure of performance because that allows us to compare the performance of the algorithm for different matrix sizes.

Given a matrix size, the benchmarking function creates the matrix `A` and the right-hand side `b` once, and then solves `A\b` a few times to get an accurate measure of the time it takes. We use the floating point operations count of the HPC Challenge, so that for an n-by-n matrix, we count the floating point operations as `2/3*n^3 + 3/2*n^2`.

The function is passed in a handle to a 'wait' function. On the CPU, this function does nothing. On the GPU, this function waits for all pending operations to complete. Waiting in this way ensures accurate timing.

function gflops = benchFcn(A, b, waitFcn) numReps = 3; time = inf; % We solve the linear system a few times and calculate the Gigaflops % based on the best time. for itr = 1:numReps tcurr = timeSolve(A, b, waitFcn); time = min(tcurr, time); end % Measure the overhead introduced by calling the wait function. tover = inf; for itr = 1:numReps tic; waitFcn(); tcurr = toc; tover = min(tcurr, tover); end % Remove the overhead from the measured time. Don't allow the time to % become negative. time = max(time - tover, 0); n = size(A, 1); flop = 2/3*n^3 + 3/2*n^2; gflops = flop/time/1e9; end % The CPU doesn't need to wait: this function handle is a placeholder. function waitForCpu() end % On the GPU, to ensure accurate timing, we need to wait for the device % to finish all pending operations. function waitForGpu(theDevice) wait(theDevice); end

Having done all the setup, it is straightforward to execute the benchmarks. However, the computations can take a long time to complete, so we print some intermediate status information as we complete the benchmarking for each matrix size. We also encapsulate the loop over all the matrix sizes in a function, to benchmark both single- and double-precision computations.

function [gflopsCPU, gflopsGPU] = executeBenchmarks(clz, sizes) fprintf(['Starting benchmarks with %d different %s-precision ' ... 'matrices of sizes\nranging from %d-by-%d to %d-by-%d.\n'], ... length(sizes), clz, sizes(1), sizes(1), sizes(end), ... sizes(end)); gflopsGPU = zeros(size(sizes)); gflopsCPU = zeros(size(sizes)); gd = gpuDevice; for i = 1:length(sizes) n = sizes(i); [A, b] = getData(n, clz); gflopsCPU(i) = benchFcn(A, b, @waitForCpu); fprintf('Gigaflops on CPU: %f\n', gflopsCPU(i)); A = gpuArray(A); b = gpuArray(b); gflopsGPU(i) = benchFcn(A, b, @() waitForGpu(gd)); fprintf('Gigaflops on GPU: %f\n', gflopsGPU(i)); end end

We then execute the benchmarks in single and double precision.

[cpu, gpu] = executeBenchmarks('single', sizeSingle); results.sizeSingle = sizeSingle; results.gflopsSingleCPU = cpu; results.gflopsSingleGPU = gpu; [cpu, gpu] = executeBenchmarks('double', sizeDouble); results.sizeDouble = sizeDouble; results.gflopsDoubleCPU = cpu; results.gflopsDoubleGPU = gpu;

Starting benchmarks with 7 different single-precision matrices of sizes ranging from 1024-by-1024 to 19456-by-19456. Creating a matrix of size 1024-by-1024. Gigaflops on CPU: 43.118208 Gigaflops on GPU: 68.428152 Creating a matrix of size 4096-by-4096. Gigaflops on CPU: 87.590552 Gigaflops on GPU: 445.238075 Creating a matrix of size 7168-by-7168. Gigaflops on CPU: 109.641635 Gigaflops on GPU: 703.972169 Creating a matrix of size 10240-by-10240. Gigaflops on CPU: 119.250824 Gigaflops on GPU: 814.339460 Creating a matrix of size 13312-by-13312. Gigaflops on CPU: 125.913442 Gigaflops on GPU: 929.046408 Creating a matrix of size 16384-by-16384. Gigaflops on CPU: 126.280301 Gigaflops on GPU: 1002.235783 Creating a matrix of size 19456-by-19456. Gigaflops on CPU: 133.515248 Gigaflops on GPU: 1020.936350 Starting benchmarks with 5 different double-precision matrices of sizes ranging from 1024-by-1024 to 13312-by-13312. Creating a matrix of size 1024-by-1024. Gigaflops on CPU: 26.660253 Gigaflops on GPU: 49.709032 Creating a matrix of size 4096-by-4096. Gigaflops on CPU: 49.673652 Gigaflops on GPU: 282.736875 Creating a matrix of size 7168-by-7168. Gigaflops on CPU: 58.392763 Gigaflops on GPU: 448.664332 Creating a matrix of size 10240-by-10240. Gigaflops on CPU: 62.992162 Gigaflops on GPU: 534.594961 Creating a matrix of size 13312-by-13312. Gigaflops on CPU: 65.609136 Gigaflops on GPU: 592.661072

We can now plot the results, and compare the performance on the CPU and the GPU, both for single and double precision.

First, we look at the performance of the backslash operator in single precision.

fig = figure; ax = axes('parent', fig); plot(ax, results.sizeSingle, results.gflopsSingleGPU, '-x', ... results.sizeSingle, results.gflopsSingleCPU, '-o') grid on; legend('GPU', 'CPU', 'Location', 'NorthWest'); title(ax, 'Single-precision performance') ylabel(ax, 'Gigaflops'); xlabel(ax, 'Matrix size'); drawnow;

Now, we look at the performance of the backslash operator in double precision.

fig = figure; ax = axes('parent', fig); plot(ax, results.sizeDouble, results.gflopsDoubleGPU, '-x', ... results.sizeDouble, results.gflopsDoubleCPU, '-o') legend('GPU', 'CPU', 'Location', 'NorthWest'); grid on; title(ax, 'Double-precision performance') ylabel(ax, 'Gigaflops'); xlabel(ax, 'Matrix size'); drawnow;

Finally, we look at the speedup of the backslash operator when comparing the GPU to the CPU.

speedupDouble = results.gflopsDoubleGPU./results.gflopsDoubleCPU; speedupSingle = results.gflopsSingleGPU./results.gflopsSingleCPU; fig = figure; ax = axes('parent', fig); plot(ax, results.sizeSingle, speedupSingle, '-v', ... results.sizeDouble, speedupDouble, '-*') grid on; legend('Single-precision', 'Double-precision', 'Location', 'SouthEast'); title(ax, 'Speedup of computations on GPU compared to CPU'); ylabel(ax, 'Speedup'); xlabel(ax, 'Matrix size'); drawnow;

```
end
```

ans = sizeSingle: [1024 4096 7168 10240 13312 16384 19456] gflopsSingleCPU: [1x7 double] gflopsSingleGPU: [1x7 double] sizeDouble: [1024 4096 7168 10240 13312] gflopsDoubleCPU: [26.6603 49.6737 58.3928 62.9922 65.6091] gflopsDoubleGPU: [49.7090 282.7369 448.6643 534.5950 592.6611]

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