CUDA Band Matrix Solve: Efficient Banded System Solver

// Treating banded matrix as dense - extremely wasteful!
void band_solve_naive(float* d_A, float* d_b, float* d_x, int n, int bandwidth) {
    // Allocate full n x n matrix (wasteful!)
    // Use dense LU solve (O(n³) instead of O(n·k²))
    cusolverDnSgetrf(solver, n, n, d_A, n, d_work, d_pivot, d_info);
    cusolverDnSgetrs(solver, CUBLAS_OP_N, n, 1, d_A, n, d_pivot, d_b, n, d_info);
}

// Tridiagonal solve (bandwidth = 1)
void tridiagonal_solve(cusolverSpHandle_t solver, int n,
                       float* d_lower, float* d_diag, float* d_upper, float* d_b) {
    // cuSOLVER tridiagonal solver - O(n) complexity
    cusolverSpSgtsv2(solver, n, 1, d_lower, d_diag, d_upper, d_b, ...);
}

// Batched tridiagonal (many independent systems)
void tridiagonal_solve_batched(cusolverSpHandle_t solver, int n, int batch,
                                float* d_lower, float* d_diag, float* d_upper, float* d_b) {
    cusolverSpSgtsv2StridedBatch(solver, n, d_lower, d_diag, d_upper, d_b, batch, n);
}

// General band solve via band LU
void band_solve(float* d_AB, float* d_b, int n, int kl, int ku) {
    // d_AB in band storage format: (2*kl + ku + 1) x n
    // Band LU factorization
    int ldab = 2 * kl + ku + 1;

    // Use cuSOLVER or custom band LU
    band_lu_factor(d_AB, n, kl, ku);
    band_lu_solve(d_AB, d_b, n, kl, ku);
}

__global__ void thomas_algorithm(float* a, float* b, float* c, float* d, float* x, int n) {
    // Forward elimination
    for (int i = 1; i < n; i++) {
        float w = a[i] / b[i-1];
        b[i] -= w * c[i-1];
        d[i] -= w * d[i-1];
    }
    // Back substitution
    x[n-1] = d[n-1] / b[n-1];
    for (int i = n-2; i >= 0; i--) {
        x[i] = (d[i] - c[i] * x[i+1]) / b[i];
    }
}