Matrix multiplication in Mojo

Learn how to leverage Mojo’s various functions to write a high-performance matmul.

This notebook describes how to write a matrix multiplication (matmul) algorithm in Mojo. We will start with a pure Python implementation, transition to a naive implementation that is essentially a copy of the Python one, then add types, then continue the optimizations by vectorizing, tiling, and parallelizing the implementation.

First, let’s define matrix multiplication. Given two dense matrices \(A\) and \(B\) of dimensions \(M\times K\) and \(K\times N\) respectively, we want to compute their dot product \(C = A . B\) (also known as matmul). The dot product \(C += A . B\) is defined by

\[C_{i,j} += \sum_{k \in [0 \cdots K)} A_{i,k} B_{k,j}\]

Please take look at our blog post on matmul and why it is important for ML and DL workloads.

The format of this notebook is to start with an implementation which is identical to that of Python (effectively renaming the file extension), then look at how adding types to the implementation helps performance before extending the implementation by leveraging the vectorization and parallelization capabilities available on modern hardware. Throughout the execution, we report the GFlops achieved.

Python Implementation

Let’s first implement matmul in Python directly from the definition.

%%python
def matmul_python(C, A, B):
    for m in range(C.rows):
        for k in range(A.cols):
            for n in range(C.cols):
                C[m, n] += A[m, k] * B[k, n]

Let’s benchmark our implementation using 128 by 128 square matrices and report the achieved GFLops.

Install numpy if it’s not already:

%%python
from importlib.util import find_spec
import shutil
import subprocess

fix = """
-------------------------------------------------------------------------
fix following the steps here:
    https://github.com/modularml/mojo/issues/1085#issuecomment-1771403719
-------------------------------------------------------------------------
"""

def install_if_missing(name: str):
    if find_spec(name):
        return

    print(f"{name} not found, installing...")
    try:
        if shutil.which('python3'): python = "python3"
        elif shutil.which('python'): python = "python"
        else: raise ("python not on path" + fix)
        subprocess.check_call([python, "-m", "pip", "install", name])
    except:
        raise ImportError(f"{name} not found" + fix)

install_if_missing("numpy")

%%python
from timeit import timeit
import numpy as np

class Matrix:
    def __init__(self, value, rows, cols):
        self.value = value
        self.rows = rows
        self.cols = cols

    def __getitem__(self, idxs):
        return self.value[idxs[0]][idxs[1]]

    def __setitem__(self, idxs, value):
        self.value[idxs[0]][idxs[1]] = value

def benchmark_matmul_python(M, N, K):
    A = Matrix(list(np.random.rand(M, K)), M, K)
    B = Matrix(list(np.random.rand(K, N)), K, N)
    C = Matrix(list(np.zeros((M, N))), M, N)
    secs = timeit(lambda: matmul_python(C, A, B), number=2)/2
    gflops = ((2*M*N*K)/secs) / 1e9
    print(gflops, "GFLOP/s")
    return gflops

python_gflops = benchmark_matmul_python(128, 128, 128).to_float64()

0.0018947946413032505 GFLOP/s

Importing the Python implementation to Mojo

Using Mojo is as simple as Python. First, let’s include that modules from the Mojo stdlib that we are going to use:

Import utilities and define Matrix (click to show/hide)

from benchmark import Unit
from sys.intrinsics import strided_load
from math import div_ceil, min
from memory import memset_zero
from memory.unsafe import DTypePointer
from random import rand, random_float64
from sys.info import simdwidthof
from runtime.llcl import Runtime

Then, we can copy and paste our Python code. Mojo is a superset of Python, so the same Python code will run as Mojo code

# This exactly the same Python implementation,
# but is infact Mojo code!
def matmul_untyped(C, A, B):
    for m in range(C.rows):
        for k in range(A.cols):
            for n in range(C.cols):
                C[m, n] += A[m, k] * B[k, n]

We can then benchmark the implementation. As before we use a 128 by 128 matrix

fn matrix_getitem(self: object, i: object) raises -> object:
    return self.value[i]


fn matrix_setitem(self: object, i: object, value: object) raises -> object:
    self.value[i] = value
    return None


fn matrix_append(self: object, value: object) raises -> object:
    self.value.append(value)
    return None


fn matrix_init(rows: Int, cols: Int) raises -> object:
    let value = object([])
    return object(
        Attr("value", value), Attr("__getitem__", matrix_getitem), Attr("__setitem__", matrix_setitem),
        Attr("rows", rows), Attr("cols", cols), Attr("append", matrix_append),
    )

def benchmark_matmul_untyped(M: Int, N: Int, K: Int, python_gflops: Float64):
    C = matrix_init(M, N)
    A = matrix_init(M, K)
    B = matrix_init(K, N)
    for i in range(M):
        c_row = object([])
        b_row = object([])
        a_row = object([])
        for j in range(N):
            c_row.append(0.0)
            b_row.append(random_float64(-5, 5))
            a_row.append(random_float64(-5, 5))
        C.append(c_row)
        B.append(b_row)
        A.append(a_row)

    @parameter
    fn test_fn():
        try:
            _ = matmul_untyped(C, A, B)
        except:
            pass

    let secs = benchmark.run[test_fn](max_runtime_secs=0.5).mean()
    _ = (A, B, C)
    let gflops = ((2*M*N*K)/secs) / 1e9
    let speedup : Float64 = gflops / python_gflops
    print(gflops, "GFLOP/s, a", speedup, "x speedup over Python")

benchmark_matmul_untyped(128, 128, 128, python_gflops)

0.0092210218211624933 GFLOP/s, a 4.866501952327785 x speedup over Python

Note the huge speedup with no effort that we have gotten.

Adding types to the Python implementation

The above program, while achieving better performance than Python, is still not the best we can get from Mojo. If we tell Mojo the types of the inputs, it can optimize much of the code away and reduce dispatching costs (unlike Python, which only uses types for type checking, Mojo exploits type info for performance optimizations as well).

To do that, let’s first define a Matrix struct. The Matrix struct contains a data pointer along with size fields. While the Matrix struct can be parametrized on any data type, here we set the data type to be Float32 for conciseness.

alias type = DType.float32

struct Matrix:
    var data: DTypePointer[type]
    var rows: Int
    var cols: Int

    # Initialize zeroeing all values
    fn __init__(inout self, rows: Int, cols: Int):
        self.data = DTypePointer[type].alloc(rows * cols)
        memset_zero(self.data, rows * cols)
        self.rows = rows
        self.cols = cols

    # Initialize taking a pointer, don't set any elements
    fn __init__(inout self, rows: Int, cols: Int, data: DTypePointer[DType.float32]):
        self.data = data
        self.rows = rows
        self.cols = cols

    ## Initialize with random values
    @staticmethod
    fn rand(rows: Int, cols: Int) -> Self:
        let data = DTypePointer[type].alloc(rows * cols)
        rand(data, rows * cols)
        return Self(rows, cols, data)

    fn __getitem__(self, y: Int, x: Int) -> Float32:
        return self.load[1](y, x)

    fn __setitem__(self, y: Int, x: Int, val: Float32):
        return self.store[1](y, x, val)

    fn load[nelts: Int](self, y: Int, x: Int) -> SIMD[DType.float32, nelts]:
        return self.data.simd_load[nelts](y * self.cols + x)

    fn store[nelts: Int](self, y: Int, x: Int, val: SIMD[DType.float32, nelts]):
        return self.data.simd_store[nelts](y * self.cols + x, val)

Note that we implement getitem and setitem in terms of load and store. For the naive implementation of matmul it does not make a difference, but we will utilize this later in a more optimized vectorized version of matmul.

With the above Matrix type we can effectively copy and paste the Python implementation and just add type annotations:

# Note that C, A, and B have types.
fn matmul_naive(C: Matrix, A: Matrix, B: Matrix):
    for m in range(C.rows):
        for k in range(A.cols):
            for n in range(C.cols):
                C[m, n] += A[m, k] * B[k, n]

We are going to benchmark the implementations as we improve, so let’s write a helper function that will do that for us:

alias M = 1024
alias N = 1024
alias K = 1024

@always_inline
fn bench[
    func: fn (Matrix, Matrix, Matrix) -> None](base_gflops: Float64):
    var C = Matrix(M, N)
    var A = Matrix.rand(M, K)
    var B = Matrix.rand(K, N)

    @always_inline
    @parameter
    fn test_fn():
        _ = func(C, A, B)

    let secs = benchmark.run[test_fn](max_runtime_secs=1).mean()
    # Prevent the matrices from being freed before the benchmark run
    A.data.free()
    B.data.free()
    C.data.free()
    let gflops = ((2 * M * N * K) / secs) / 1e9
    let speedup: Float64 = gflops / base_gflops
    # print(gflops, "GFLOP/s", speedup, " speedup")
    print(gflops, "GFLOP/s, a", speedup, "x speedup over Python")

Benchmarking shows significant speedups. We increase the size of the matrix to 512 by 512, since Mojo is much faster than Python.

bench[matmul_naive](python_gflops)

3.6469022982798753 GFLOP/s, a 1924.695277674796 x speedup over Python

Adding type annotations gives a huge improvement compared to the original untyped version.

Vectorizing the inner most loop

We can do better than the above implementation by utilizing the vector instructions. Rather than assuming a vector width, we query the simd width of the specified dtype using simd_width. This makes our code portable as we transition to other hardware. Leverage SIMD instructions is as easy as:

# Mojo has SIMD vector types, we can vectorize the Matmul code as follows.
# nelts = number of float32 elements that can fit in SIMD register
alias nelts = simdwidthof[DType.float32]()
fn matmul_vectorized_0(C: Matrix, A: Matrix, B: Matrix):
    for m in range(C.rows):
        for k in range(A.cols):
            for nv in range(0, C.cols, nelts):
                C.store[nelts](m,nv, C.load[nelts](m,nv) + A[m,k] * B.load[nelts](k,nv))

            # Handle remaining elements with scalars.
            for n in range(nelts*(C.cols//nelts), C.cols):
                C[m,n] += A[m,k] * B[k,n]

We can benchmark the above implementation. Note that many compilers can detect naive loops and perform optimizations on them. Mojo, however, allows you to be explicit and precisely control what optimizations are applied.

bench[matmul_vectorized_0](python_gflops)

12.535805674163893 GFLOP/s, a 6615.9178419154141 x speedup over Python

Vectorization is a common optimization, and Mojo provides a higher-order function that performs vectorization for you. The vectorize function takes a vector width and a function which is parametric on the vector width and is going to be evaluated in a vectorized manner.

# Simplify the code by using the builtin vectorize function
from algorithm import vectorize
fn matmul_vectorized_1(C: Matrix, A: Matrix, B: Matrix):
    for m in range(C.rows):
        for k in range(A.cols):
            @parameter
            fn dot[nelts : Int](n : Int):
                C.store[nelts](m,n, C.load[nelts](m,n) + A[m,k] * B.load[nelts](k,n))
            vectorize[nelts, dot](C.cols)

There is only a slight difference in terms of performance between the two implementations:

bench[matmul_vectorized_1](python_gflops)

12.541864071124047 GFLOP/s, a 6619.1152316631433 x speedup over Python

Parallelizing Matmul

With Mojo we can easily run code in parallel with the parallelize function.

Let’s modify our matmul implementation and make it multi-threaded (for simplicity, we only parallelize on the M dimension).

In parallelize below we’re overpartitioning by distributing the work more evenly among processors. This ensures they all have something to work on even if some tasks finish before others, or some processors are stragglers. Intel and Apple now have separate performance and efficiency cores and this mitigates the problems that can cause.

# Parallelize the code by using the builtin parallelize function
from algorithm import parallelize
fn matmul_parallelized(C: Matrix, A: Matrix, B: Matrix):
    @parameter
    fn calc_row(m: Int):
        for k in range(A.cols):
            @parameter
            fn dot[nelts : Int](n : Int):
                C.store[nelts](m,n, C.load[nelts](m,n) + A[m,k] * B.load[nelts](k,n))
            vectorize[nelts, dot](C.cols)
    parallelize[calc_row](C.rows, C.rows)

We can benchmark the parallel matmul implementation.

bench[matmul_parallelized](python_gflops)

107.98687292023116 GFLOP/s, a 56991.333290850547 x speedup over Python

Tiling Matmul

Tiling is an optimization performed for matmul to increase cache locality. The idea is to keep sub-matrices resident in the cache and increase the reuse. The tile function itself can be written in Mojo as:

from algorithm import Static2DTileUnitFunc as Tile2DFunc

# Perform 2D tiling on the iteration space defined by end_x and end_y.
fn tile[tiled_fn: Tile2DFunc, tile_x: Int, tile_y: Int](end_x: Int, end_y: Int):
    # Note: this assumes that ends are multiples of the tiles.
    for y in range(0, end_y, tile_y):
        for x in range(0, end_x, tile_x):
            tiled_fn[tile_x, tile_y](x, y)

The above will perform 2 dimensional tiling over a 2D iteration space defined to be between \(([0, end_x], [0, end_y])\). Once we define it above, we can use it within our matmul kernel. For simplicity we choose 4 as the tile height and since we also want to vectorize we use 4 * nelts as the tile width (since we vectorize on the columns).

# Use the above tile function to perform tiled matmul.
fn matmul_tiled_parallelized(C: Matrix, A: Matrix, B: Matrix):
    @parameter
    fn calc_row(m: Int):
        @parameter
        fn calc_tile[tile_x: Int, tile_y: Int](x: Int, y: Int):
            for k in range(y, y + tile_y):
                @parameter
                fn dot[nelts : Int,](n : Int):
                    C.store[nelts](m,n + x, C.load[nelts](m,n+x) + A[m,k] * B.load[nelts](k,n+x))
                vectorize[nelts, dot](tile_x)

        # We hardcode the tile factor to be 4.
        alias tile_size = 4
        tile[calc_tile, nelts * tile_size, tile_size](A.cols, C.cols)

    parallelize[calc_row](C.rows, C.rows)

Again, we can benchmark the tiled parallel matmul implementation:

bench[matmul_tiled_parallelized](python_gflops)

136.92271321866085 GFLOP/s, a 72262.560930869338 x speedup over Python

One source of overhead in the above implementation is the fact that the we are not unrolling the loops introduced by vectorize of the dot function. We can do that via the vectorize_unroll higher-order function in Mojo:

# Unroll the vectorized loop by a constant factor.
from algorithm import vectorize_unroll
fn matmul_tiled_unrolled_parallelized(C: Matrix, A: Matrix, B: Matrix):
    @parameter
    fn calc_row(m: Int):
        @parameter
        fn calc_tile[tile_x: Int, tile_y: Int](x: Int, y: Int):
            for k in range(y, y + tile_y):
                @parameter
                fn dot[nelts : Int,](n : Int):
                    C.store[nelts](m,n+x, C.load[nelts](m,n+x) + A[m,k] * B.load[nelts](k,n+x))

                # Vectorize by nelts and unroll by tile_x/nelts
                # Here unroll factor is 4
                vectorize_unroll[nelts, tile_x//nelts, dot](tile_x)

        alias tile_size = 4
        tile[calc_tile, nelts*tile_size, tile_size](A.cols, C.cols)

    parallelize[calc_row](C.rows, C.rows)

Again, we can benchmark the new tiled parallel matmul implementation with unrolled and vectorized inner loop:

bench[matmul_tiled_unrolled_parallelized](python_gflops)

173.70727621843133 GFLOP/s, a 91676.043636557093 x speedup over Python

Searching for the tile_factor

from autotune import autotune, search
from time import now
from memory.unsafe import Pointer

alias matmul_fn_sig_type = fn(C: Matrix, A: Matrix, B: Matrix, /) -> None

The choice of the tile factor can greatly impact the performance of the full matmul, but the optimal tile factor is highly hardware-dependent, and is influenced by the cache configuration and other hard-to-model effects. We want to write portable code without having to know everything about the hardware, so we can ask Mojo to automatically select the best tile factor using autotuning.

# Autotune the tile size used in the matmul.
@adaptive
fn matmul_autotune_impl(C: Matrix, A: Matrix, B: Matrix, /):
    @parameter
    fn calc_row(m: Int):
        @parameter
        fn calc_tile[tile_x: Int, tile_y: Int](x: Int, y: Int):
            for k in range(y, y + tile_y):
                @parameter
                fn dot[nelts : Int](n : Int):
                    C.store[nelts](m,n+x, C.load[nelts](m,n+x) + A[m,k] * B.load[nelts](k,n+x))
                vectorize_unroll[nelts, tile_x // nelts, dot](tile_x)

        # Instead of hardcoding to tile_size = 4, search for the fastest
        # tile size by evaluating this function as tile size varies.
        alias tile_size = autotune(1, 2, 4, 8, 16, 32)
        tile[calc_tile, nelts * tile_size, tile_size](A.cols, C.cols)

    parallelize[calc_row](C.rows, C.rows)

This will generate multiple candidates for the matmul function. To teach Mojo how to find the best tile factor, we provide an evaluator function Mojo can use to assess each candidate.

fn matmul_evaluator(funcs: Pointer[matmul_fn_sig_type], size: Int) -> Int:
    print("matmul_evaluator, number of candidates: ", size)

    let eval_begin: Int = now()

    # This size is picked at random, in real code we could use a real size
    # distribution here.
    let M = 512
    let N = 512
    let K = 512
    print("Optimizing for size:", M, "x", N, "x", K)

    var best_idx: Int = -1
    var best_time: Int = -1

    alias eval_iterations = 10
    alias eval_samples = 10

    var C = Matrix(M, N)
    var A = Matrix(M, K)
    var B = Matrix(K, N)
    let Cptr = Pointer[Matrix].address_of(C).address
    let Aptr = Pointer[Matrix].address_of(A).address
    let Bptr = Pointer[Matrix].address_of(B).address

    # Find the function that's the fastest on the size we're optimizing for
    for f_idx in range(size):
        let func = funcs.load(f_idx)

        @always_inline
        @parameter
        fn wrapper():
            func(C, A, B)
        let cur_time = benchmark.run[wrapper](max_runtime_secs=0.5).mean(Unit.ns).to_int()

        if best_idx < 0:
            best_idx = f_idx
            best_time = cur_time
        if best_time > cur_time:
            best_idx = f_idx
            best_time = cur_time

    let eval_end: Int = now()
    # Prevent matrices from being destroyed before we finished benchmarking them.
    A.data.free()
    B.data.free()
    C.data.free()
    print("Time spent in matmul_evaluator, ms:", (eval_end - eval_begin) // 1000000)
    print("Best candidate idx:", best_idx)
    return best_idx

Finally, we need to define an entry function that would simply call the best candidate.

fn matmul_autotune(C: Matrix, A: Matrix, B: Matrix):
    alias best_impl: matmul_fn_sig_type
    search[
        matmul_fn_sig_type,
        VariadicList(matmul_autotune_impl.__adaptive_set),
        matmul_evaluator -> best_impl
    ]()
    # Run the best candidate
    return best_impl(C, A, B)

Let’s benchmark our new implementation:

bench[matmul_autotune](python_gflops)

matmul_evaluator, number of candidates:  6
Optimizing for size: 512 x 512 x 512
Time spent in matmul_evaluator, ms: 8098
Best candidate idx: 5
201.48323262066052 GFLOP/s, a 106335.12900483991 x speedup over Python

Tile and accumulate in registers and reorder loop

Perform 2D tiling on the iteration space defined by end_x and end_y, parallelizing over y.

fn tile_parallel[
    tiled_fn: Tile2DFunc, tile_x: Int, tile_y: Int
](end_x: Int, end_y: Int):
    # Note: this assumes that ends are multiples of the tiles.
    @parameter
    fn row(yo: Int):
        let y = tile_y * yo
        for x in range(0, end_x, tile_x):
            tiled_fn[tile_x, tile_y](x, y)

    parallelize[row](end_y // tile_y, M)

Use stack allocation for tiles to accumulate values efficiently, avoiding repeated reads and writes to memory. Also reorder the loops and do not fully unroll the loop over the reduction dimension.

from memory import stack_allocation

fn accumulate_registers(C: Matrix, A: Matrix, B: Matrix):
    alias tile_k = 8
    alias tile_k_unroll = 8
    alias tile_i = 32
    alias tile_j = nelts * 4

    @parameter
    fn calc_tile[tile_j: Int, tile_i: Int](jo: Int, io: Int):
        # Allocate the tile of accumulators on the stack.
        var accumulators = Matrix(
            tile_i, tile_j, stack_allocation[tile_i * tile_j, DType.float32]()
        )

        for ko in range(0, A.cols, tile_k * tile_k_unroll):
            for _ in range(tile_i):
                for i in range(tile_k):
                    @unroll
                    for k in range(tile_k_unroll):
                        @parameter
                        fn calc_tile_cols[nelts: Int](j: Int):
                            accumulators.store[nelts](
                                i,
                                j,
                                accumulators.load[nelts](i, j)
                                + A[io + i, ko + k]
                                * B.load[nelts](ko + k, jo + j),
                            )

                        vectorize_unroll[
                            nelts, tile_j // nelts, calc_tile_cols
                        ](tile_j)

        # Copy the local tile to the output
        for i in range(tile_i):
            for j in range(tile_j):
                C[io + i, jo + j] = accumulators[i, j]

    tile_parallel[calc_tile, tile_j, tile_i](C.cols, C.rows)

bench[accumulate_registers](python_gflops)

584.69820167517651 GFLOP/s, a 308581.30423728545 x speedup over Python