CUDA Matrix Determinant: GPU-Accelerated Computation

Introduction

Matrix determinant computation is fundamental to linear algebra, used in solving linear systems, computing matrix inverses, and evaluating system stability. On GPU, determinants are typically computed via LU decomposition—factoring A = LU where L is lower triangular and U is upper triangular. The determinant then equals the product of diagonal elements of U (with sign adjustment for pivoting). This approach leverages highly optimized GPU primitives and scales well to large matrices.

Common Performance Issues

• Numerical instability without pivoting - small pivots cause catastrophic errors
• Memory overhead for large matrices - LU decomposition requires O(N²) storage
• Single-matrix bottleneck - GPU underutilized for individual small matrices
• Overflow/underflow for large matrices - product of many values exceeds float range
• Not leveraging batched operations - missing parallelism for multiple matrices

Optimization Techniques

Implementation Comparison

Before (Naive Implementation)

Cofactor expansion has factorial complexity - completely impractical for real use.

// WARNING: O(N!) complexity - never use for N > 10
__device__ float naive_det(float* A, int n) {
    if (n == 1) return A[0];
    if (n == 2) return A[0]*A[3] - A[1]*A[2];
    float det = 0.0f;
    for (int j = 0; j < n; j++) {
        det += (j % 2 == 0 ? 1 : -1) * A[j] * cofactor(A, n, 0, j);
    }
    return det;
}

After (Optimized Implementation)

Batched LU decomposition processes thousands of small matrices in parallel.

#include <cublas_v2.h>

void batched_determinant(float** d_matrices, int n, int batch_size, float* d_dets) {
    cublasHandle_t handle;
    cublasCreate(&handle);
    int* d_pivot, *d_info;
    cudaMalloc(&d_pivot, n * batch_size * sizeof(int));
    cudaMalloc(&d_info, batch_size * sizeof(int));

    // Batched LU factorization
    cublasSgetrfBatched(handle, n, d_matrices, n, d_pivot, d_info, batch_size);

    // Compute determinants from LU diagonals
    compute_det_kernel<<<(batch_size+255)/256, 256>>>(d_matrices, d_pivot, n, batch_size, d_dets);
}

__global__ void compute_det_kernel(float** mats, int* pivs, int n, int bs, float* dets) {
    int b = blockIdx.x * blockDim.x + threadIdx.x;
    if (b >= bs) return;
    float* LU = mats[b];
    int* piv = pivs + b * n;
    float det = 1.0f;
    int swaps = 0;
    for (int i = 0; i < n; i++) {
        det *= LU[i * n + i];
        if (piv[i] != i + 1) swaps++;
    }
    dets[b] = (swaps % 2 == 0) ? det : -det;
}

Performance Results

Frequently Asked Questions

Related Guides

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