Resources

Video Presentation

Watch the visual explanation of berkeDet, meryemSign, and meryemPer:

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Benchmark & Verification Scripts

Everything in the paper is executable. Fixed seeds; correctness asserted in-script; runs abort on any mismatch. All are single-file Python. The first group needs only a CPU (Google Colab's free tier is enough); the Version 4 streaming scripts additionally use a GPU where indicated.

• verify.py — determinants vs LU (orders 1–10), structural lex-order and sign checks
• colab_sign_benchmark.py — isolated sign generation: meryemSign vs inversion count, merge parity, and Narayana + $O(1)$ flip
• colab_benchmark.py — serial end-to-end Leibniz determinant, four methods
• colab_parallel_benchmark.py — vectorized tree vs chains: permutations, signs, and the full exact determinant

Version 4 — streaming execution. Rank-locality windows, the block-size overflow law, exact big-integer reduction, and the fused GPU kernel. Every determinant these produce is certified in-script against independent exact fraction-free Bareiss elimination.

• phaseA_stream.py — CPU streaming core and full verification battery (exhaustive line-and-sign checks, the block-size law table, prepare/execute pipeline, LU comparison). No GPU required.
• colab_t4_phaseB.py — streaming on a free Tesla T4: the exact $14 \times 14$ determinant (87.2 billion terms) in 194 MB, plus the dual-stream prepare/execute measurement (the honest 0.99× finding). GPU runtime.
• runpod_summit_n18_v2.py — fused register-resident CUDA kernel: $n = 16$, 20,922,789,888,000 terms exact in 361 s at 57.9 G lines/s on one NVIDIA B300, with checkpointing, rank-partitioning across devices, and measured horizons for $n = 17$–$19$. Large-GPU runtime.

Preprint Repositories

Quick Start (Python)

Copy this code, paste it into any Python environment (such as online-python.com), and run it. Full code with detailed comments is on the Algorithm page.

def meryemSign(n):
    b = [1]
    for e in range(1, n):
        c = b[:]
        for d in range(e):
            if d % 2 == 0:
                f = [-x for x in c]
            else:
                f = c[:]
            b.extend(f)
    return b

def meryemPer(n):
    a = [[1]]
    for b in range(2, n + 1):
        c = list(range(1, b + 1))
        d = []
        for f in c:
            g = [e for e in c if e != f]
            for h in a:
                j = [g[i - 1] for i in h]
                d.append([f] + j)
        a = d
    return a

def berkeDet(matrix):
    n = len(matrix)
    signs = meryemSign(n)
    perms = meryemPer(n)
    det = 0
    for sign, perm in zip(signs, perms):
        product = 1
        for i in range(n):
            product *= matrix[i][perm[i] - 1]
        det += sign * product
    return det

# Try it!
A = [[6, 1, 1],
     [4, -2, 5],
     [2, 8, 7]]

print("Determinant:", berkeDet(A))  # Output: -306

Citation

If you use berkeDet in your work, please cite:

Gülmen, B., Gülmen, M., & Gülmen, Ö. (2026).
berkeDet: Data-Parallel Lexicographic Permutation Generation and
Streaming Exact Determinant Evaluation via Block Replication and
Index Mapping (Version 4).
Zenodo. https://doi.org/10.5281/zenodo.21709734

Version History

• Version 4 (July 2026) — Generator-first rework for a parallel-computing readership. New: streaming execution with the block-size overflow law and exact big-integer reduction; prepare/execute pipeline with the honest single-GPU 0.99× finding; fused register-resident CUDA kernel; Tesla T4 results (n=14 exact in 194 MB) and NVIDIA B300 results (n=16: 20.9 trillion terms exact in 361 s at 57.9 G/s, Bareiss-certified); measured horizons for n=17/18/19; Bareiss added as independent exact baseline. Archived: DOI 10.5281/zenodo.21709734.
• Version 3 (July 2026, DOI 10.5281/zenodo.21326563) — Complete rewrite. Isolated sign-generation benchmarks (~100× / 350–400× / 15–18×); formal parallel-complexity analysis (factorial tree, work–span model: span $O(n)$ vs chains' $\Omega(n!)$); vectorized NumPy implementations with benchmarks (6× over C itertools; $S_{12}$ signs in 0.19 s; vectorized determinant 15× over SJT); verification extended to order 10; the serial n=10 crossover vs SJT explained and resolved.
• Version 2 (April 2026) — Strengthened meryemPer analysis with explicit comparison to naive (suffix sort) and optimized (Narayana Pandita) lexicographic generators. Added sort-free column to comparison table. Updated title.
• Version 1 (February 2026) — Initial release.