# Matrix Multiplication¶

In this tutorial, you will write a very short high-performance FP16 matrix multiplication kernel that achieves performance on parallel with cuBLAS.

• Block-level matrix multiplications.

• Multi-dimensional pointer arithmetics.

• Program re-ordering for improved L2 cache hit rate.

• Automatic performance tuning.

## Motivations¶

Matrix multiplications are a key building block of most modern high-performance computing systems. They are notoriously hard to optimize, hence their implementation is generally done by hardware vendors themselves as part of so-called “kernel libraries” (e.g., cuBLAS). Unfortunately, these libraries are often proprietary and cannot be easily customized to accommodate the needs of modern deep learning workloads (e.g., fused activation functions). In this tutorial, you will learn how to implement efficient matrix multiplications by yourself with Triton, in a way that is easy to customize and extend.

Roughly speaking, the kernel that we will write will implement the following blocked algorithm to multiply a (M, K) by a (K, N) matrix:

```# Do in parallel
for m in range(0, M, BLOCK_SIZE_M):
# Do in parallel
for n in range(0, N, BLOCK_SIZE_N):
acc = zeros((BLOCK_SIZE_M, BLOCK_SIZE_N), dtype=float32)
for k in range(0, K, BLOCK_SIZE_K):
a = A[m : m+BLOCK_SIZE_M, k : k+BLOCK_SIZE_K]
b = B[k : k+BLOCK_SIZE_K, n : n+BLOCK_SIZE_N]
acc += dot(a, b)
C[m : m+BLOCK_SIZE_M, n : n+BLOCK_SIZE_N] = acc
```

where each iteration of the doubly-nested for-loop is performed by a dedicated Triton program instance.

## Compute Kernel¶

The above algorithm is, actually, fairly straightforward to implement in Triton. The main difficulty comes from the computation of the memory locations at which blocks of `A` and `B` must be read in the inner loop. For that, we need multi-dimensional pointer arithmetics.

### Pointer Arithmetics¶

For a row-major 2D tensor `X`, the memory location of `X[i, j]` is given b y `&X[i, j] = X + i*stride_xi + j*stride_xj`. Therefore, blocks of pointers for `A[m : m+BLOCK_SIZE_M, k:k+BLOCK_SIZE_K]` and `B[k : k+BLOCK_SIZE_K, n : n+BLOCK_SIZE_N]` can be defined in pseudo-code as:

```&A[m : m+BLOCK_SIZE_M, k:k+BLOCK_SIZE_K] =  a_ptr + (m : m+BLOCK_SIZE_M)[:, None]*A.stride(0) + (k : k+BLOCK_SIZE_K)[None, :]*A.stride(1);
&B[k : k+BLOCK_SIZE_K, n:n+BLOCK_SIZE_N] =  b_ptr + (k : k+BLOCK_SIZE_K)[:, None]*B.stride(0) + (n : n+BLOCK_SIZE_N)[None, :]*B.stride(1);
```

Which means that pointers for blocks of A and B can be initialized (i.e., `k=0`) in Triton as the following code. Also note that we need an extra modulo to handle the case where `M` is not a multiple of `BLOCK_SIZE_M` or `N` is not a multiple of `BLOCK_SIZE_N`, in which case we can pad the data with some useless values, which will not contribute to the results. For the `K` dimension, we will handle that later using masking load semantics.

```offs_am = (pid_m * BLOCK_SIZE_M + tl.arange(0, BLOCK_SIZE_M)) % M
offs_bn = (pid_n * BLOCK_SIZE_N + tl.arange(0, BLOCK_SIZE_N)) % N
offs_k = tl.arange(0, BLOCK_SIZE_K)
a_ptrs = a_ptr + (offs_am[:, None]*stride_am + offs_k [None, :]*stride_ak)
b_ptrs = b_ptr + (offs_k [:, None]*stride_bk + offs_bn[None, :]*stride_bn)
```

And then updated in the inner loop as follows:

```a_ptrs += BLOCK_SIZE_K * stride_ak;
b_ptrs += BLOCK_SIZE_K * stride_bk;
```

### L2 Cache Optimizations¶

As mentioned above, each program instance computes a `[BLOCK_SIZE_M, BLOCK_SIZE_N]` block of `C`. It is important to remember that the order in which these blocks are computed does matter, since it affects the L2 cache hit rate of our program. and unfortunately, a a simple row-major ordering

```pid = triton.program_id(0);
grid_m = (M + BLOCK_SIZE_M - 1) // BLOCK_SIZE_M;
grid_n = (N + BLOCK_SIZE_N - 1) // BLOCK_SIZE_N;
pid_m = pid / grid_n;
pid_n = pid % grid_n;
```

is just not going to cut it.

One possible solution is to launch blocks in an order that promotes data reuse. This can be done by ‘super-grouping’ blocks in groups of `GROUP_M` rows before switching to the next column:

```# Program ID
pid = tl.program_id(axis=0)
# Number of program ids along the M axis
num_pid_m = tl.cdiv(M, BLOCK_SIZE_M)
# Number of programs ids along the N axis
num_pid_n = tl.cdiv(N, BLOCK_SIZE_N)
# Number of programs in group
num_pid_in_group = GROUP_SIZE_M * num_pid_n
# Id of the group this program is in
group_id = pid // num_pid_in_group
# Row-id of the first program in the group
first_pid_m = group_id * GROUP_SIZE_M
# If `num_pid_m` isn't divisible by `GROUP_SIZE_M`, the last group is smaller
group_size_m = min(num_pid_m - first_pid_m, GROUP_SIZE_M)
# *Within groups*, programs are ordered in a column-major order
# Row-id of the program in the *launch grid*
pid_m = first_pid_m + (pid % group_size_m)
# Col-id of the program in the *launch grid*
pid_n = (pid % num_pid_in_group) // group_size_m
```

For example, in the following matmul where each matrix is 9 blocks by 9 blocks, we can see that if we compute the output in row-major ordering, we need to load 90 blocks into SRAM to compute the first 9 output blocks, but if we do it in grouped ordering, we only need to load 54 blocks. In practice, this can improve the performance of our matrix multiplication kernel by more than 10% on some hardware architecture (e.g., 220 to 245 TFLOPS on A100).

## Final Result¶

```import torch

import triton
import triton.language as tl

# `triton.jit`'ed functions can be auto-tuned by using the `triton.autotune` decorator, which consumes:
#   - A list of `triton.Config` objects that define different configurations of
#       meta-parameters (e.g., `BLOCK_SIZE_M`) and compilation options (e.g., `num_warps`) to try
#   - An auto-tuning *key* whose change in values will trigger evaluation of all the
#       provided configs
@triton.autotune(
configs=[
triton.Config({'BLOCK_SIZE_M': 128, 'BLOCK_SIZE_N': 256, 'BLOCK_SIZE_K': 64, 'GROUP_SIZE_M': 8}, num_stages=3,
num_warps=8),
triton.Config({'BLOCK_SIZE_M': 64, 'BLOCK_SIZE_N': 256, 'BLOCK_SIZE_K': 32, 'GROUP_SIZE_M': 8}, num_stages=4,
num_warps=4),
triton.Config({'BLOCK_SIZE_M': 128, 'BLOCK_SIZE_N': 128, 'BLOCK_SIZE_K': 32, 'GROUP_SIZE_M': 8}, num_stages=4,
num_warps=4),
triton.Config({'BLOCK_SIZE_M': 128, 'BLOCK_SIZE_N': 64, 'BLOCK_SIZE_K': 32, 'GROUP_SIZE_M': 8}, num_stages=4,
num_warps=4),
triton.Config({'BLOCK_SIZE_M': 64, 'BLOCK_SIZE_N': 128, 'BLOCK_SIZE_K': 32, 'GROUP_SIZE_M': 8}, num_stages=4,
num_warps=4),
triton.Config({'BLOCK_SIZE_M': 128, 'BLOCK_SIZE_N': 32, 'BLOCK_SIZE_K': 32, 'GROUP_SIZE_M': 8}, num_stages=4,
num_warps=4),
triton.Config({'BLOCK_SIZE_M': 64, 'BLOCK_SIZE_N': 32, 'BLOCK_SIZE_K': 32, 'GROUP_SIZE_M': 8}, num_stages=5,
num_warps=2),
triton.Config({'BLOCK_SIZE_M': 32, 'BLOCK_SIZE_N': 64, 'BLOCK_SIZE_K': 32, 'GROUP_SIZE_M': 8}, num_stages=5,
num_warps=2),
],
key=['M', 'N', 'K'],
)
@triton.jit
def matmul_kernel(
# Pointers to matrices
a_ptr, b_ptr, c_ptr,
# Matrix dimensions
M, N, K,
# The stride variables represent how much to increase the ptr by when moving by 1
# element in a particular dimension. E.g. `stride_am` is how much to increase `a_ptr`
# by to get the element one row down (A has M rows).
stride_am, stride_ak,  #
stride_bk, stride_bn,  #
stride_cm, stride_cn,
# Meta-parameters
BLOCK_SIZE_M: tl.constexpr, BLOCK_SIZE_N: tl.constexpr, BLOCK_SIZE_K: tl.constexpr,  #
GROUP_SIZE_M: tl.constexpr,  #
ACTIVATION: tl.constexpr  #
):
"""Kernel for computing the matmul C = A x B.
A has shape (M, K), B has shape (K, N) and C has shape (M, N)
"""
# -----------------------------------------------------------
# Map program ids `pid` to the block of C it should compute.
# This is done in a grouped ordering to promote L2 data reuse.
# See above `L2 Cache Optimizations` section for details.
pid = tl.program_id(axis=0)
num_pid_m = tl.cdiv(M, BLOCK_SIZE_M)
num_pid_n = tl.cdiv(N, BLOCK_SIZE_N)
num_pid_in_group = GROUP_SIZE_M * num_pid_n
group_id = pid // num_pid_in_group
first_pid_m = group_id * GROUP_SIZE_M
group_size_m = min(num_pid_m - first_pid_m, GROUP_SIZE_M)
pid_m = first_pid_m + (pid % group_size_m)
pid_n = (pid % num_pid_in_group) // group_size_m

# ----------------------------------------------------------
# Create pointers for the first blocks of A and B.
# We will advance this pointer as we move in the K direction
# and accumulate
# `a_ptrs` is a block of [BLOCK_SIZE_M, BLOCK_SIZE_K] pointers
# `b_ptrs` is a block of [BLOCK_SIZE_K, BLOCK_SIZE_N] pointers
# See above `Pointer Arithmetics` section for details
offs_am = (pid_m * BLOCK_SIZE_M + tl.arange(0, BLOCK_SIZE_M)) % M
offs_bn = (pid_n * BLOCK_SIZE_N + tl.arange(0, BLOCK_SIZE_N)) % N
offs_k = tl.arange(0, BLOCK_SIZE_K)
a_ptrs = a_ptr + (offs_am[:, None] * stride_am + offs_k[None, :] * stride_ak)
b_ptrs = b_ptr + (offs_k[:, None] * stride_bk + offs_bn[None, :] * stride_bn)

# -----------------------------------------------------------
# Iterate to compute a block of the C matrix.
# We accumulate into a `[BLOCK_SIZE_M, BLOCK_SIZE_N]` block
# of fp32 values for higher accuracy.
# `accumulator` will be converted back to fp16 after the loop.
accumulator = tl.zeros((BLOCK_SIZE_M, BLOCK_SIZE_N), dtype=tl.float32)
for k in range(0, tl.cdiv(K, BLOCK_SIZE_K)):
# Load the next block of A and B, generate a mask by checking the K dimension.
# If it is out of bounds, set it to 0.
a = tl.load(a_ptrs, mask=offs_k[None, :] < K - k * BLOCK_SIZE_K, other=0.0)
b = tl.load(b_ptrs, mask=offs_k[:, None] < K - k * BLOCK_SIZE_K, other=0.0)
# We accumulate along the K dimension.
accumulator += tl.dot(a, b)
# Advance the ptrs to the next K block.
a_ptrs += BLOCK_SIZE_K * stride_ak
b_ptrs += BLOCK_SIZE_K * stride_bk
# You can fuse arbitrary activation functions here
# while the accumulator is still in FP32!
if ACTIVATION == "leaky_relu":
accumulator = leaky_relu(accumulator)
c = accumulator.to(tl.float16)

# -----------------------------------------------------------
# Write back the block of the output matrix C with masks.
offs_cm = pid_m * BLOCK_SIZE_M + tl.arange(0, BLOCK_SIZE_M)
offs_cn = pid_n * BLOCK_SIZE_N + tl.arange(0, BLOCK_SIZE_N)
c_ptrs = c_ptr + stride_cm * offs_cm[:, None] + stride_cn * offs_cn[None, :]
c_mask = (offs_cm[:, None] < M) & (offs_cn[None, :] < N)

# We can fuse `leaky_relu` by providing it as an `ACTIVATION` meta-parameter in `_matmul`.
@triton.jit
def leaky_relu(x):
x = x + 1
return tl.where(x >= 0, x, 0.01 * x)
```

We can now create a convenience wrapper function that only takes two input tensors, and (1) checks any shape constraint; (2) allocates the output; (3) launches the above kernel.

```def matmul(a, b, activation=""):
# Check constraints.
assert a.shape == b.shape, "Incompatible dimensions"
assert a.is_contiguous(), "Matrix A must be contiguous"
assert b.is_contiguous(), "Matrix B must be contiguous"
M, K = a.shape
K, N = b.shape
# Allocates output.
c = torch.empty((M, N), device=a.device, dtype=a.dtype)
# 1D launch kernel where each block gets its own program.
grid = lambda META: (triton.cdiv(M, META['BLOCK_SIZE_M']) * triton.cdiv(N, META['BLOCK_SIZE_N']), )
matmul_kernel[grid](
a, b, c,  #
M, N, K,  #
a.stride(0), a.stride(1),  #
b.stride(0), b.stride(1),  #
c.stride(0), c.stride(1),  #
ACTIVATION=activation  #
)
return c
```

## Unit Test¶

We can test our custom matrix multiplication operation against a native torch implementation (i.e., cuBLAS).

```torch.manual_seed(0)
a = torch.randn((512, 512), device='cuda', dtype=torch.float16)
b = torch.randn((512, 512), device='cuda', dtype=torch.float16)
triton_output = matmul(a, b)
torch_output = torch.matmul(a, b)
print(f"triton_output={triton_output}")
print(f"torch_output={torch_output}")
if torch.allclose(triton_output, torch_output, atol=1e-2, rtol=0):
print("✅ Triton and Torch match")
else:
print("❌ Triton and Torch differ")
```
```triton_output=tensor([[-10.9531,  -4.7109,  15.6953,  ..., -28.4062,   4.3320, -26.4219],
[ 26.8438,  10.0469,  -5.4297,  ..., -11.2969,  -8.5312,  30.7500],
[-13.2578,  15.8516,  18.0781,  ..., -21.7656,  -8.6406,  10.2031],
...,
[ 40.2812,  18.6094, -25.6094,  ...,  -2.7598,  -3.2441,  41.0000],
[ -6.1211, -16.8281,   4.4844,  ..., -21.0312,  24.7031,  15.0234],
[-17.0938, -19.0000,  -0.3831,  ...,  21.5469, -30.2344, -13.2188]],
device='cuda:0', dtype=torch.float16)
torch_output=tensor([[-10.9531,  -4.7109,  15.6953,  ..., -28.4062,   4.3320, -26.4219],
[ 26.8438,  10.0469,  -5.4297,  ..., -11.2969,  -8.5312,  30.7500],
[-13.2578,  15.8516,  18.0781,  ..., -21.7656,  -8.6406,  10.2031],
...,
[ 40.2812,  18.6094, -25.6094,  ...,  -2.7598,  -3.2441,  41.0000],
[ -6.1211, -16.8281,   4.4844,  ..., -21.0312,  24.7031,  15.0234],
[-17.0938, -19.0000,  -0.3831,  ...,  21.5469, -30.2344, -13.2188]],
device='cuda:0', dtype=torch.float16)
✅ Triton and Torch match
```

## Benchmark¶

### Square Matrix Performance¶

We can now compare the performance of our kernel against that of cuBLAS. Here we focus on square matrices, but feel free to arrange this script as you wish to benchmark any other matrix shape.

```@triton.testing.perf_report(
triton.testing.Benchmark(
x_names=['M', 'N', 'K'],  # Argument names to use as an x-axis for the plot
x_vals=[128 * i for i in range(2, 33)],  # Different possible values for `x_name`
line_arg='provider',  # Argument name whose value corresponds to a different line in the plot
# Possible values for `line_arg`
line_vals=['cublas', 'triton'],
# Label name for the lines
line_names=["cuBLAS", "Triton"],
# Line styles
styles=[('green', '-'), ('blue', '-')],
ylabel="TFLOPS",  # Label name for the y-axis
plot_name="matmul-performance",  # Name for the plot, used also as a file name for saving the plot.
args={},
))
def benchmark(M, N, K, provider):
a = torch.randn((M, K), device='cuda', dtype=torch.float16)
b = torch.randn((K, N), device='cuda', dtype=torch.float16)
quantiles = [0.5, 0.2, 0.8]
if provider == 'cublas':
ms, min_ms, max_ms = triton.testing.do_bench(lambda: torch.matmul(a, b), quantiles=quantiles)
if provider == 'triton':
ms, min_ms, max_ms = triton.testing.do_bench(lambda: matmul(a, b), quantiles=quantiles)
perf = lambda ms: 2 * M * N * K * 1e-12 / (ms * 1e-3)
return perf(ms), perf(max_ms), perf(min_ms)

benchmark.run(show_plots=True, print_data=True)
``` ```matmul-performance:
M       N       K      cuBLAS      Triton
0    256.0   256.0   256.0    4.096000    4.096000
1    384.0   384.0   384.0   12.288000   12.288000
2    512.0   512.0   512.0   26.214401   23.831273
3    640.0   640.0   640.0   42.666665   39.384616
4    768.0   768.0   768.0   63.195428   58.982401
5    896.0   896.0   896.0   78.051553   82.642822
6   1024.0  1024.0  1024.0  110.376426  104.857603
7   1152.0  1152.0  1152.0  135.726544  129.825388
8   1280.0  1280.0  1280.0  157.538463  163.840004
9   1408.0  1408.0  1408.0  155.765024  132.970149
10  1536.0  1536.0  1536.0  181.484314  157.286398
11  1664.0  1664.0  1664.0  179.978245  179.978245
12  1792.0  1792.0  1792.0  172.914215  212.064605
13  1920.0  1920.0  1920.0  203.294114  168.585369
14  2048.0  2048.0  2048.0  226.719125  192.841562
15  2176.0  2176.0  2176.0  211.827867  205.343354
16  2304.0  2304.0  2304.0  225.357284  229.691080
17  2432.0  2432.0  2432.0  202.118452  203.583068
18  2560.0  2560.0  2560.0  225.986210  221.405396
19  2688.0  2688.0  2688.0  197.567993  197.567993
20  2816.0  2816.0  2816.0  208.680416  210.696652
21  2944.0  2944.0  2944.0  221.493479  221.493479
22  3072.0  3072.0  3072.0  206.653671  207.410628
23  3200.0  3200.0  3200.0  214.046818  219.931269
24  3328.0  3328.0  3328.0  205.103410  207.467716
25  3456.0  3456.0  3456.0  216.143621  217.308808
26  3584.0  3584.0  3584.0  219.305830  211.565625
27  3712.0  3712.0  3712.0  210.310194  217.168134
28  3840.0  3840.0  3840.0  207.101127  210.651436
29  3968.0  3968.0  3968.0  208.945088  216.354501
30  4096.0  4096.0  4096.0  219.310012  215.784121
```

Total running time of the script: ( 0 minutes 40.946 seconds)

Gallery generated by Sphinx-Gallery