Faster matrix exponential

Compute the exponential of a 555 × 555 random matrix faster than scipy.linalg.expm, to a relative tolerance of 1e-5.

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1	# Faster matrix exponential
2
3	Compute the exponential $e^{A}$ of a dense random matrix faster than SciPy's `expm`, to five significant digits. The matrix exponential solves linear systems of differential equations and appears in control theory, Markov chains and quantum dynamics.
4
5	## Task
6
7	Input: $A\in\mathbb{R}^{n\times n}$ with independent standard normal entries; in the benchmark $n=555$. Output:
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9	$$e^{A}=\sum_{k=0}^{\infty}\frac{A^k}{k!}.$$
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11	An output is accepted when it matches `scipy.linalg.expm(A)` entrywise under NumPy's `allclose` with relative tolerance $10^{-5}$ and absolute tolerance $10^{-8}$. The reference solver is `scipy.linalg.expm`, which implements the scaling-and-squaring algorithm with Padé approximants of Al-Mohy and Higham (2009).
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13	Score: the reference's run time divided by the candidate's, averaged over the test instances; it is 1 if any output fails the check, and speed-ups below 1 count as 1. Higher is better.
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15	## Background
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17	Moler and Van Loan's classic survey, "Nineteen dubious ways to compute the exponential of a matrix", was updated in 2003; SciPy uses scaling and squaring with Padé approximants, as refined by Al-Mohy and Higham (2009), who choose the scaling and the Padé degree from the data. Its cost is a handful of matrix products, a linear solve and the squarings, so speed-ups must come from fewer products, cheaper ones, or a different method that still meets the tolerance.
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19	## Data and evaluation
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21	- Instances: 100 test and 100 development instances from the task's generator at $n=555$, in the [AlgoTune dataset](https://huggingface.co/datasets/oripress/AlgoTune); the reference needs about 107 ms per instance on the authors' machine.
22	- Evaluation code: the AlgoTune harness (`./algotune.sh evaluate` in the [repository](https://github.com/oripress/AlgoTune)), which runs the task's checker on every output.
23	- Hardware: speed-ups depend on the machine, so reference and candidate are timed in the same environment. The paper keeps the fastest of 10 timed runs per instance, each after a warm-up, on an AMD EPYC 9454 CPU with 14 GB of memory; agents had 8 cores, and compilation is not timed.
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25	## What is known
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27	- Best on the [AlgoTune leaderboard](https://algotune.io) as of 2026-10-20: 2.13×, by the AlgoTuner agent driven by Gemini 3.1 Pro Preview, ahead of Gemini 3 Pro Preview (2.12×) and Claude Opus 4.5 (1.01×).
28	- Only two models' runs exceed 1.1×; the next best is 1.01×.
29	- Press et al. (2025) describe most agent speed-ups as surface-level: better library calls, better library usage or low-level rewrites, for example with Numba.
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31	## What would count as progress
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33	- A solver beating 2.13× on all 100 test instances under the AlgoTune harness, with its code; anyone can rerun the checker and the timing.
34	- An error analysis showing that the faster method stays within the tolerance for matrices with larger norms than the test inputs.
35
36	## References
37
38	1. O. Press, B. Amos, H. Zhao, et al. AlgoTune: Can language models speed up general-purpose numerical programs? NeurIPS 2025. [arXiv:2507.15887](https://arxiv.org/abs/2507.15887)
39	2. A. H. Al-Mohy, N. J. Higham. A new scaling and squaring algorithm for the matrix exponential. SIAM Journal on Matrix Analysis and Applications 31, 970–989 (2009). [doi:10.1137/09074721X](https://doi.org/10.1137/09074721X)
40	3. C. Moler, C. Van Loan. Nineteen dubious ways to compute the exponential of a matrix, twenty-five years later. SIAM Review 45, 3–49 (2003). [doi:10.1137/S00361445024180](https://doi.org/10.1137/S00361445024180)
41
42	## Source
43
44	Imported from [AlgoTune task matrix_exponential](https://github.com/oripress/AlgoTune/tree/main/AlgoTuneTasks/matrix_exponential), curated by Ori Press and the AlgoTune contributors. Open according to the source as of 2026-10-20. The statement above is written in our own words; see the source for the original wording, history and discussion.