# Learning from Examples » Matrix Multiplication

We study the classic problem, 2D matrix multiplication. We will start with a short introduction about the problem and then discuss how to solve it parallel CPUs.

## Problem Formulation

We are multiplying two matrices, `A` (`MxK`) and `B` (`KxN`). The numbers of columns of `A` must match the number of rows of `B`. The output matrix `C` has the shape of (MxN) where `M` is the rows of `A` and `N` the columns of `B`. The following example multiplies a `3x3` matrix with a `3x2` matrix to derive a `3x2` matrix.

As a general view, for each element of `C` we iterate a complete row of `A` and a complete column of `B`, multiplying each element and summing them.

We can implement matrix multiplication using three nested loops.

    for(int m=0; m<M; m++) {
      for(int n=0; n<N; n++) {
        C[m][n] = 0;
        for(int k=0; k<K; k++) {
          C[m][n] += A[m][k] * B[k][n];
        }
      }
    }

## Parallel Patterns

At a fine-grained level, computing each element of `C` is independent of each other. Similarly, computing each row of `C` or each column of `C` is also independent of one another. With task parallelism, we prefer coarse-grained model to have each task perform rather large computation to amortize the overhead of creating and scheduling tasks. In this case, we avoid intensive tasks each working on only a single element. by creating a task per row of `C` to multiply a row of `A` by every column of `B`.

    // C = A * B
    // A is a MxK matrix, B is a KxN matrix, and C is a MxN matrix
    void matrix_multiplication(int** A, int** B, int** C, int M, int K, int N) {
      tf::Taskflow taskflow;
      tf::Executor executor;
      for(int m=0; m<M; ++m) {
        taskflow.emplace([m, &] () {
          for(int n=0; n<N; n++) {
            for(int k=0; k<K; k++) {
              C[m][n] += A[m][k] * B[k][n];  // inner product
            }
          }
        });
      }
      executor.run(taskflow).wait();
    }

Instead of creating tasks one-by-one over a loop, you can leverage Taskflow::for_each_index to create a parallel-for task. A parallel-for task spawns a subflow to perform parallel iterations over the given range.

    // perform parallel iterations on the range [0, M) with the step size of 1
    tf::Task task = taskflow.for_each_index(0, M, 1, [&] (int m) {
      for(int n=0; n<N; n++) {
        for(int k=0; k<K; k++) {
          C[m][n] += A[m][k] * B[k][n];
        }
      }
    });

Please visit Parallel Iterations for more details.

## Benchmarking

Based on the discussion above, we compare the runtime of computing various matrix sizes of `A`, `B`, and `C` between a sequential CPU and parallel CPUs on a machine of 12 Intel i7-8700 CPUs at 3.2 GHz.

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

The speed-up of parallel execution becomes clean as we increase the problem size. For example, at `4000x4000`, the parallel runtime is 6.3 times faster than the sequential runtime.

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