CUDA Matrix Trace: Efficient Diagonal Sum on GPU

Introduction

Matrix trace—the sum of diagonal elements—is one of the simplest matrix operations but appears frequently in machine learning (regularization terms, loss functions) and physics (quantum mechanics, tensor contractions). While trivially parallel, efficient trace computation requires attention to memory access patterns. The diagonal elements are strided in memory, so naive implementations suffer from poor cache utilization.

Common Performance Issues

• Strided memory access - diagonal elements are N+1 apart in row-major storage
• Underutilizing GPU for single small matrix - kernel launch overhead dominates
• Not batching multiple trace operations - missing parallelism opportunity
• Numerical precision for large matrices - many small values can lose precision

Optimization Techniques

Implementation Comparison

Before (Naive Implementation)

Single-threaded loop completely wastes GPU parallelism.

__global__ void trace_naive(float* A, int n, float* result) {
    if (threadIdx.x == 0 && blockIdx.x == 0) {
        float sum = 0.0f;
        for (int i = 0; i < n; i++) {
            sum += A[i * n + i];
        }
        *result = sum;
    }
}

After (Optimized Implementation)

Parallel reduction with warp shuffles for efficient diagonal summation.

__global__ void trace_optimized(float* A, int n, float* result) {
    __shared__ float shared[32];
    int tid = threadIdx.x;
    int lane = tid % 32;
    int warp = tid / 32;

    float sum = 0.0f;
    for (int i = tid; i < n; i += blockDim.x) {
        sum += A[i * n + i];
    }

    // Warp reduction
    for (int offset = 16; offset > 0; offset /= 2)
        sum += __shfl_down_sync(0xffffffff, sum, offset);

    if (lane == 0) shared[warp] = sum;
    __syncthreads();

    if (warp == 0) {
        sum = (tid < blockDim.x / 32) ? shared[lane] : 0.0f;
        for (int offset = 16; offset > 0; offset /= 2)
            sum += __shfl_down_sync(0xffffffff, sum, offset);
        if (tid == 0) *result = sum;
    }
}

Performance Results

Frequently Asked Questions

Related Guides

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