# Learning from Examples » Matrix Multiplication (cudaFlow)

Following up on Matrix Multiplication, this page studies how to accelerate a matrix multiplication workload on a GPU using tf::cudaFlow.

## Define a Matrix Multiplication Kernel

GPU can perform a lot of parallel computations more than CPUs. It is especially useful for data-intensive computing such as matrix multiplication. With GPU, we express the parallel patterns at a fine-grained level. The kernel, written in CUDA, is described as follows:

    // CUDA kernel to perform matrix multiplication
    __global__ void matmul(int *A, int *B, int *C, int M, int K, int N) {
      int row = blockIdx.y * blockDim.y + threadIdx.y;
      int col = blockIdx.x * blockDim.x + threadIdx.x;
      int sum = 0;
      if(col < N && row < M) {
        for(int i = 0; i < K; i++) {
          sum += a[row * K + i] * b[i * N + col];
        }
        c[row * N + col] = sum;
      }
    }

Each CUDA thread corresponds to an element of `C` and compute its result. Instead of storing each matrix in a 2D array, we use 1D layout to ease the data transfer between CPU and GPU. In a row-major layout, an element `(x, y)` in the 2D matrix can be addressed at `x * width + y` in the transformed 1D layout.

## Define a cudaFlow for Matrix Multiplication

The next step is to allocate memory for `A`, `B`, and `C` at a GPU. We create three tasks each calling `cudaMalloc` to allocate space for one matrix. Then, we create a cudaFlow to offload matrix multiplication to a GPU. The entire code is described as follows:

    void matrix_multiplication(int* A, int* B, int* C, int M, int K, int N) {

      tf::Taskflow taskflow;
      tf::Executor executor;

      // allocate the host and gpu storage for A
      tf::Task allocate_a = taskflow.emplace([&](){
        cudaMalloc(&da, M*K*sizeof(int));
      }).name("allocate_a");

      // allocate the host and gpu storage for B
      tf::Task allocate_b = taskflow.emplace([&](){
        cudaMalloc(&db, K*N*sizeof(int));
      }).name("allocate_b");

      // allocate the host and gpu storage for C
      tf::Task allocate_c = taskflow.emplace([&](){
        cudaMalloc(&dc, M*N*sizeof(int));
      }).name("allocate_c");

      // create a cudaFlow task to run the matrix multiplication
      tf::Task cudaFlow = taskflow.emplace([&](){

        tf::cudaFlow cf;

        // copy data to da, db, and dc
        tf::cudaTask copy_da = cf.copy(da, A, M*K).name("H2D_A");
        tf::cudaTask copy_db = cf.copy(db, B, K*N).name("H2D_B");
        tf::cudaTask copy_hc = cf.copy(C, dc, M*N).name("D2H_C");

        dim3 grid  ((K+16-1)/16, (M+16-1)/16);
        dim3 block (16, 16);

        tf::cudaTask kmatmul = cf.kernel(grid, block, 0, matmul, da, db, dc, M, K, N)
                                 .name("matmul");

        kmatmul.succeed(copy_da, copy_db)
               .precede(copy_hc);

        // launch the cudaFlow
        tf::cudaStream stream;
        cf.run(stream);
        stream.synchronize();

      }).name("cudaFlow");

      // free the gpu storage
      auto free = taskflow.emplace([&](){
        cudaFree(da);
        cudaFree(db);
        cudaFree(dc);
      }).name("free");

      // create dependency
      cudaFlow.succeed(allocate_a, allocate_b, allocate_c)
              .precede(free);

      // dump the graph without unfolding the cudaFlow
      taskflow.dump(std::cout);

      // run the taskflow
      executor.run(taskflow).wait();

      // dump the entire execution graph including unfolded cudaFlow
      taskflow.dump(std::cout);
    }

Within the cudaFlow, we create two host-to-device (H2D) tasks that copy data from `A` and `B` to `da` and `db`, one device-to-host (D2H) task that copies the result from `dc` to `C`, and one kernel task that launches `matmul` on the GPU (by default, GPU 0). H2D tasks precede the kernel and the kernel precedes the D2H task. These GPU operations form a GPU task graph managed by a cudaFlow. The first dump of the taskflow gives the following graph:

A cudaFlow encapsulates a GPU task dependency graph similar to a tf::Subflow (see Subflow Tasking). In order to visualize it, we need to execute the graph first and then dump the taskflow.

## Benchmarking

We run three versions of matrix multiplication, sequential CPU, parallel CPUs, and one GPU, on a machine of 12 Intel i7-8700 CPUs at 3.20 GHz and a Nvidia RTX 2080 GPU using various matrix sizes of `A`, `B`, and `C`.

A | B | C | CPU Sequential | CPU Parallel | GPU Parallel
--- | --- | --- | --- | --- | ---
10x10 | 10x10 | 10x10 | 0.142 ms | 0.414 ms | 82 ms
100x100 | 100x100 | 100x100 | 1.641 ms | 0.733 ms | 83 ms
1000x1000 | 1000x1000 | 1000x1000 | 1532 ms | 504 ms | 85 ms
2000x2000 | 2000x2000 | 2000x2000 | 25688 ms | 4387 ms | 133 ms
3000x3000 | 3000x3000 | 3000x3000 | 104838 ms | 16170 ms | 214 ms
4000x4000 | 4000x4000 | 4000x4000 | 250133 ms | 39646 ms | 427 ms

As the matrix size increases, the speed-up of GPU over CPUs becomes prominent. For example, at `4000x4000`, the GPU runtime is 585.8 times faster than the sequential CPU runtime and is 92.8 times faster than the parallel CPU solutions.

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