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Matrix-vector multiplication (GEMV) is ubiquitous in linear algebra, neural networks, and scientific computing. Unlike GEMM which is compute-bound, GEMV is typically memory-bound. Optimization focuses on maximizing memory bandwidth utilization through coalescing and parallel reduction.
Assign one warp per row, use shuffle for fast reduction.
__global__ void sgemv_warp(float* A, float* x, float* y, int M, int N) {
int row = blockIdx.x;
int lane = threadIdx.x;
float sum = 0.0f;
for (int j = lane; j < N; j += 32) {
sum += A[row * N + j] * x[j];
}
// Warp reduction
for (int offset = 16; offset > 0; offset /= 2) {
sum += __shfl_down_sync(0xffffffff, sum, offset);
}
if (lane == 0) y[row] = sum;
}One thread per row - serialized inner loop.
__global__ void sgemv_naive(float* A, float* x, float* y, int M, int N) {
int row = blockIdx.x * blockDim.x + threadIdx.x;
if (row < M) {
float sum = 0.0f;
for (int j = 0; j < N; j++) {
sum += A[row * N + j] * x[j];
}
y[row] = sum;
}
}Caches x vector in shared memory, uses block reduction.
__global__ void sgemv_opt(float* A, float* x, float* y, int M, int N) {
__shared__ float xs[256];
int row = blockIdx.x;
int tid = threadIdx.x;
int blockSize = blockDim.x;
float sum = 0.0f;
for (int tile = 0; tile < N; tile += blockSize) {
// Collaborative load of x into shared memory
if (tile + tid < N) xs[tid] = x[tile + tid];
__syncthreads();
int end = min(blockSize, N - tile);
for (int j = 0; j < end; j++) {
sum += A[row * N + tile + j] * xs[j];
}
__syncthreads();
}
// Block reduction
__shared__ float sdata[256];
sdata[tid] = sum;
__syncthreads();
for (int s = blockSize/2; s > 0; s >>= 1) {
if (tid < s) sdata[tid] += sdata[tid + s];
__syncthreads();
}
if (tid == 0) y[row] = sdata[0];
}| Metric | Naive | Optimized | Improvement |
|---|---|---|---|
| Throughput (4096x4096) | 45 GB/s | 380 GB/s | 8.4x faster |
| vs cuBLAS | 12% | 89% | 7.4x closer |
Yes for production. cuBLAS sgemv is highly optimized. Custom kernels only make sense for fused operations or unusual matrix shapes.
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