Exploring Commutative Matrix Multiplication Schemes via Flip Graphs

Isaac Wood

REVIEW 3 major objections 5 minor 18 references

Exploring Commutative Matrix Multiplication Schemes via Flip Graphs

T0 review · 3 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read This paper adapts flip-graph search to commutative matrix multiplication, introducing a quotient-tensor flip graph and a construction modeled on a 1986 3x3 algorithm, and shows both recover the best known bounds for every matrix size up…

desk verdict A creditable first application of flip-graph search to commutative matrix multiplication, with a real gap between the quotient model and actual commutative algorithms. read the letter →

arxiv 2506.22113 v1 pith:SOCVMQIN submitted 2025-06-27 cs.SC

Isaac Wood This is my paper · ORCID

classification cs.SC MSC 15A6968Q2568W30

keywords matrixmultiplicationcommutativealgorithmsflipgraphstensorrankbilinearcomplexitysearchheuristicsalgebraicquotientspaces

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Standard searches for fast matrix multiplication assume non-commutative products, but allowing commutativity can reduce the number of multiplications needed. This paper adapts the flip-graph search, which walks between schemes by flips and reductions, to the commutative setting in two ways: a construction modeled on a known 1986 3×3 algorithm that the standard flip graph can search, and a fully commutative flip graph defined on a quotient tensor space where swapping the two matrix factors is identified. Across all sizes up to 5×5, both approaches recover the best known commutative bounds; the fully commutative graph matches them in every case, and the hybrid matches in almost all cases after extra computation. No improvement on existing bounds was found, but the paper argues the tools are ready for larger sizes.

What carries the argument

The argument runs on two search spaces. The first is a tensor construction modeled on a 1986 algorithm for 3×3 matrices, which reorders even- and odd-indexed entries so a commutative multiplication appears as a rank-one term and the standard adaptive flip graph can search it. The second is the commutative tensor space $U = (U_1\otimes U_1\otimes U_2)/S$, quotiented by identifying $u_{11}\otimes u_{12}\otimes u_2$ with $u_{12}\otimes u_{11}\otimes u_2$; this captures commutativity, and flips, reductions, and pluses are redefined on these quotient rank-one terms. The commutative adaptive flip graph is asserted to be connected over $\mathbb{F}_2$ following the non-commutative proof. Searches walk this graph from the standard scheme or from smaller schemes in search of short decompositions of the commutative matrix multiplication tensor.

What would settle it

Take a tiny case such as (2,2,2) or (3,3,3) over $\mathbb{F}_2$ and exhaustively generate every scheme in the commutative flip graph to check whether it is connected and whether every minimal scheme lies in the component of the standard scheme; a disconnected graph or an unreachable minimal scheme would refute the search's completeness. A second check is to take a known commutative algorithm and verify each of its multiplications can be written as a quotient-space rank-one term; any algorithm that cannot be written that way falsifies the model.

Watch

Extended reading notes

Core claim

The central claim is that flip-graph search works for commutative matrix multiplication if commutativity is handled either before the search or inside the graph. The first strategy rewrites the tensor so each commutative multiplication is a rank-one term in a reordered space, letting the standard adaptive flip graph run unchanged. The second quotients the tensor space so $a\otimes b\otimes c$ and $b\otimes a\otimes c$ are the same element, then redefines flip, reduction, and plus on these quotient terms. Over $\mathbb{F}_2$, the quotient-based graph matches the best known commutative bound for every matrix size up to 5×5; the hybrid of both strategies matches it in almost all sizes with extended computation, and the first strategy is fastest but sometimes misses the bound. No improvement on the known bounds is found, and the paper presents the methods as viable automated tools for larger commutative searches.

Load-bearing premise

The whole search assumes that every commutative way to multiply matrices can be captured by the quotient-tensor formalism, and that the graph of schemes it searches is connected; if either assumption fails, the search could silently miss good algorithms.

Editorial extensions

If this is right

The quotient-based flip graph matches the best known commutative bound for every matrix size up to 5×5, so it can serve as an automated upper-bound certifier in that range.
Seeding the commutative search with the first construction's schemes matches the bound in almost all sizes while taking about 15 hours instead of the full quotient search's 50, so the two strategies complement each other.
The first construction alone runs about 30 times faster than the quotient graph but misses several bounds, so speed and completeness trade off.
A large run up to (7,7,7) matched the known bounds except for some sizes of the form (a,7,b), locating a concrete weakness of the current pipeline.
No tested size beat the existing bound, so the contribution is a search pipeline that reproduces known results rather than a new record.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

A natural next step is to apply the quotient-space flip graph to other bilinear problems whose tensors are symmetric under swapping two factors; the formalism is not specific to matrix multiplication.
The asserted connectivity of the commutative flip graph is load-bearing but only said to follow from the non-commutative proof; an exhaustive connectivity check for very small sizes would be a cheap way to de-risk the whole search.
If the misses at sizes of the form (a,7,b) persist under longer runs, that points to starting-scheme quality or connectivity rather than raw compute as the bottleneck, suggesting better seeds over more processors.
The speed gap between the first construction and the quotient graph hints that the quotient search could be made much faster with implementation effort, which would make searches beyond 7 practical.

Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Isaac Wood's paper adapts the flip-graph search method to the commutative setting and reports that it recovers Rosowski's bounds for all matrix sizes up to 5x5. That claim is supported by the table, and the paper is honest that no new bounds were found.

The genuinely new pieces are the quotient tensor U=(U1⊗U1⊗U2)/S, which identifies a⊗b with b⊗a in the first two factors, and a Marakov-inspired tensor built by splitting entries into two sets before applying the standard flip graph. The hybrid method is a natural combination. The comparison across three methods with timings is a useful data point for anyone doing this kind of search.

My main reservation is conceptual. The paper defines the rank of M' in the quotient space and calls the resulting decompositions 'commutative schemes.' But a rank-one term in U is any α⊗β⊗γ with α,β ∈ U1, so α and β can be arbitrary linear combinations of A and B entries. In a standard commutative bilinear algorithm, each multiplication is a product of a linear form in A with a linear form in B. The quotient does not enforce that separation. It is therefore not obvious that rk'(M') equals the minimum number of multiplications in a commutative algorithm. The paper never addresses this. Since the authors only match Rosowski and find nothing better, no false claim is made, but the 'strong potential' claim rests on an unproven equivalence.

Second, Theorem 3.5 asserts the commutative flip graph is connected, with the proof dismissed as 'exactly following' the non-commutative case. That transfer is not routine in the quotient: rank-one terms have no unique representative, so the common-factor argument in the original proof does not obviously carry over. If connectivity fails, the method only explores a component. The empirical success up to 5x5 suggests the component is rich, but that does not establish completeness.

Third, Definition 3.1 has a concrete indexing bug: the basis sets as written put the odd-j a's and even-i b's in the wrong factors for the tensor's sum. Fixable, but as printed the Marakov tensor is ill-defined. The paper also ships no code or data, gives no random seeds or detailed parameters, and the two hybrid misses (3,3,4) and (3,5,4) are later reported as matching after extended runs, which the table doesn't reflect.

Who is this for? People working on automated search for matrix multiplication schemes, specifically the commutative sub-question. It's a niche but real audience. I would send it to a referee. The gaps are addressable and the core search heuristic is plausible. A referee should ask for a precise statement of what the quotient rank means, a real proof of connectivity, and the experimental code.

Referee Report

3 major / 5 minor

Summary. This paper proposes three flip-graph based heuristics for finding low-rank commutative matrix multiplication schemes: (i) a 'Marakov-like' tensor that encodes the parity-based partition from Makarov's 3×3 algorithm, searched with the standard noncommutative flip graph; (ii) a commutative flip graph defined on the quotient space U=(U1⊗U1⊗U2)/S, in which the first two factors are identified; and (iii) a hybrid that seeds the commutative search with low-rank Marakov-like schemes. The main experimental claim is that across all l,m,n up to 5, the commutative search matches Rosowski's upper bounds, the Marakov search is fastest but occasionally misses, and the hybrid is nearly as accurate and about three times faster; no improvements over Rosowski are found. The paper also states Theorem 3.5 that the commutative adaptive flip graph is connected, with the proof deferred to the noncommutative case.

Significance. If the central claims held, this would be the first successful automated search for commutative matrix multiplication schemes and would provide a practical toolkit for larger sizes. The paper's positive points are the explicit construction of a quotient tensor model, the empirical comparison across 42 size triples, and the honest reporting that no new bounds were found. However, the current value is limited by two unresolved load-bearing issues: the Marakov-like tensor (Definition 3.1) is not well-defined as written, and the connectivity of the commutative flip graph (Theorem 3.5) is asserted without the non-routine transfer of the proof. Because the paper makes no new rank bounds, its significance depends on the reliability and reproducibility of the search methods.

major comments (3)

[§3.1, Definition 3.1] Definition 3.1 as stated does not define a tensor in the declared vector spaces: V'1,l,m is declared to be F^{l×⌊m/2⌋} but its generator set {aij : j even} ∪ {bij : i even} has cardinality l⌈m/2⌉ + m⌈n/2⌉ (e.g., for l=m=n=3, six elements in a three-dimensional space), and similarly V'2,m,n. Moreover, in the displayed sum the odd-j terms place aij in the first factor although aij is not among the listed generators of V'1,l,m, and the even-j terms place aij in the second factor although it is not among the generators of V'2,m,n. The Marakov-like tensor Vl,m,n is therefore not a well-defined element of V'1⊗V'2⊗V'3, and the entire M column of Table 1 rests on an ill-defined object. Please correct the parity assignments, the dimensions, or explain the intended quotient or identification.
[§3.2, Theorem 3.5] The proof of Theorem 3.5 consists of one sentence saying it 'exactly follows' the noncommutative case. This is not a routine transfer: the noncommutative proof in [2] works with a fixed tensor product V1⊗V2⊗V3 and rank-one terms u⊗v⊗w with unique representatives, and its key step is factoring a common vector a out of two terms. In the quotient U=(U1⊗U1⊗U2)/S, a rank-one class α⊗β⊗γ+S does not have a unique representative (since α⊗β and β⊗α are identified), so the common-factor operation is not well-defined on classes. If the graph is not connected, then the search explores only one component and the fact that Table 1 matches Rosowski's bounds does not demonstrate coverage of all commutative schemes. Please provide a complete proof, a counterexample, or a clearly stated weakening of the claim to a heuristic statement.
[§4, Table 1 and experimental setup] The paper's central empirical claim cannot be reproduced from the information given. There is no released code or data, and the text does not specify the search parameters that are known to matter for adaptive flip graph methods: the plus-operation frequency (mentioned only qualitatively in Remark 3.6), the compute budget and stopping rule, the random seed, the neighbor and selection strategy, and the exact field arithmetic (stated as F2 but without confirmation for all runs). In addition, the table and the text disagree on the hybrid M-C for (3,3,4) and (3,5,4): the table lists 28 and 46 while Rosowski's bounds are 27 and 45, and the text says extended computation matched Rosowski. Please clarify whether the table reports initial runs or final runs, and add enough experimental detail (or a code/data repository) to make the numerical claims checkable.

minor comments (5)

[Throughout] The name 'Marakov' appears throughout the text, while reference [8] is 'Makarov'; please standardize the spelling.
[Definition 2.3] The rank decomposition is written as sum_{p=0}^r t_p; the lower limit should presumably be p=1.
[§3.1] The tensor Vl,m,n in Definition 3.1 collides notationally with the flip-graph vertex set Vl,m,n from Definition 2.9; choose a distinct symbol such as Wl,m,n.
[§4, Table 1] The claim that differing entries are shown in bold is not visible in the table as typeset; list the differing entries explicitly in the caption or text.
[Remark 2.12] The remark mentions Hensel lifting but gives no details or reference for this step; if it is used in the experiments, it should be specified.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the searches optimize independently defined tensors and are benchmarked externally against Rosowski's bounds, with no fitted parameter or load-bearing self-citation.

full rationale

The paper's claimed derivations are not circular. The two search targets are defined independently of the benchmark: Definition 3.1 constructs a Marakov-like tensor directly from the matrix entries, and Definition 3.2 defines the commutative rank via an explicit quotient tensor space; neither definition refers to Rosowski's bounds. The reported ranks come from randomized flip-graph searches over these tensors and are compared to Theorem 2.16/Corollary 2.17 only as an external benchmark; no parameter is fitted to reproduce the target ranks, and the hybrid method's starting points are schemes produced by the same search process rather than imported from the benchmark. The sole self-citation, reference [6], appears in an introductory list of prior flip-graph work and is not load-bearing for any result. The main gap is Theorem 3.5, whose proof is asserted to 'exactly follow' the non-commutative case; this is a missing proof, not a circularity, because the theorem provides a connectivity precondition rather than the source of the matching ranks. No equation reduces to its own input, and the reported ranks are not constructed to equal Rosowski's bounds.

Assumptions & free parameters 2 free parameters · 3 assumptions · 2 invented entities

The paper contributes heuristic search methods rather than a parameter-free derivation. The central recovery claim rests on an assumed connection between the quotient tensor model and actual commutative algorithms, on the asserted connectivity of the commutative flip graph, and on Rosowski's bounds as the external benchmark. No numerical constants are fitted to data beyond the tuning choices listed above, and no physical entity is introduced beyond the two defined tensor constructions.

free parameters (2)

Plus-operation frequency in the commutative flip graph
Remark 3.6 says the frequency was increased relative to the non-commutative search because reductions were harder to find; no exact value, schedule, or ablation is reported, and this choice affects the search trajectory and results.
Compute budget and stopping time per size
The table reports one search outcome per size; no iteration counts are given, and for (3,3,4) and (3,5,4) the hybrid method received extended computation until it matched Rosowski's bound, so the reported recovery depends on unstated budget choices.

assumptions (3)

domain assumption The commutative adaptive flip graph over F2 is connected.
Theorem 3.5 states this with proof deferred to the corresponding non-commutative proof; connectivity is load-bearing for the search's ability to reach low-rank schemes.
domain assumption Rank in the quotient tensor space equals the number of commutative multiplications in a Strassen-like commutative algorithm.
Definition 3.2 defines rank in U, and Remark 2.6's correspondence is assumed to extend to the quotient; no formal proof is given that every commutative scheme is represented.
domain assumption Rosowski's bounds are the best known commutative bounds for all tested sizes.
The table uses Ros. as the reference for best known and improvement; the paper cites Rosowski and adds a transpose-based corollary, but lower bounds are not proven.

invented entities (2)

Marakov-like tensor Vl,m,n
purpose: Encodes commutative matrix multiplication in a form suitable for the existing non-commutative flip graph.
Introduced in Definition 3.1 and chosen after experimentation; as printed the definition has inconsistent basis assignments, and no external verification is provided.
Commutative quotient tensor space U = (U1 tensor U1 tensor U2)/S
purpose: Formalizes commutativity of the two multiplication factors so that flips can swap them.
Introduced in Definition 3.2. The adequacy of this quotient to capture all commutative algorithms is assumed; matching Rosowski's bounds is the only evidence given.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Exploring Commutative Matrix Multiplication Schemes via Flip Graphs." pith.science (2026). https://pith.science/paper/SOCVMQIN

@misc{pith2026250622113,
  author       = {Pith},
  title        = {Pith review of: Exploring Commutative Matrix Multiplication Schemes via Flip Graphs},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SOCVMQIN}},
  note         = {Machine review of arXiv:2506.22113}
}

read the original abstract

We explore new approaches for finding matrix multiplication algorithms in the commutative setting by adapting the flip graph technique: a method previously shown to be effective for discovering fast algorithms in the non-commutative case. While an earlier attempt to apply flip graphs to commutative algorithms saw limited success, we overcome both theoretical and practical obstacles using two strategies: one inspired by Marakov's algorithm to multiply 3x3 matrices, in which we construct a commutative tensor and approximate its rank using the standard flip graph; and a second that introduces a fully commutative variant of the flip graph defined via a quotient tensor space. We also present a hybrid method that combines the strengths of both. Across all matrix sizes up to 5x5, these methods recover the best known bounds on the number of multiplications and allow for a comparison of their efficiency and efficacy. Although no new improvements are found, our results demonstrate strong potential for these techniques at larger scales.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

18 extracted references · 9 canonical work pages

[2]

Adaptive flip graph algorithm for ma- trix multiplication

Yamato Arai, Yuma Ichikawa, and Koji Hukushima. Adaptive flip graph algorithm for ma- trix multiplication. In Proceedings of the 2024 International Symposium on Symbolic and Algebraic Computation, ISSAC ’24, page 292–298, New York, NY, USA, 2024. Association for Computing Machinery. doi:10.1145/3666000.3669701

arXiv 2024
[1]

More Asymmetry Yields Faster Matrix Multiplication , pages 2005–

Josh Alman, Ran Duan, Virginia Vassilevska Williams, Yinzhan Xu, Zixuan Xu, and Renfei Zhou. More Asymmetry Yields Faster Matrix Multiplication , pages 2005–

work page 2005
[3]

A non-commutative algorithm for multiplying 4x4 matrices using 48 non-complex multiplications, 2025

Jean-Guillaume Dumas, Cl´ ement Pernet, and Alexandre Sedoglavic. A non-commutative algorithm for multiplying 4x4 matrices using 48 non-complex multiplications, 2025. URL: https://arxiv.org/abs/2506.13242, arXiv:2506.13242

arXiv 2025
[4]

Marijn J. H. Heule, Manuel Kauers, and Martina Seidl. Local search for fast matrix multipli- cation. In Mikol´ aˇ s Janota and Inˆ es Lynce, editors,Theory and Applications of Satisfiability Testing – SAT 2019 , pages 155–163, Cham, 2019. Springer International Publishing

work page 2019
[5]

Flip graphs for matrix multiplication

Manuel Kauers and Jakob Moosbauer. Flip graphs for matrix multiplication. In Proceedings of the 2023 International Symposium on Symbolic and Algebraic Computation , ISSAC ’23, page 381–388, New York, NY, USA, 2023. Association for Computing Machinery. doi:10. 1145/3597066.3597120

arXiv 2023
[6]

Consequences of the moosbauer-poole algorithms

Manuel Kauers and Isaac Wood. Consequences of the moosbauer-poole algorithms. arXiv preprint arXiv:2505.05896, 2025

arXiv 2025
[7]

Laderman

John D. Laderman. A noncommutative algorithm for multiplying 3 ×3 matrices using 23 multiplications. Bulletin of the American Mathematical Society , 82:126–128, 1976. doi:10. 1090/S0002-9904-1976-13988-2

work page 1976
[8]

O.M. Makarov. An algorithm for multiplying 3 ×3 matrices. USSR Compu- tational Mathematics and Mathematical Physics , 26(1):179–180, 1986. URL: https://www.sciencedirect.com/science/article/pii/004155538690203X, doi:https: //doi.org/10.1016/0041-5553(86)90203-X

arXiv 1986

Show all 18 references

[9]

Search Techniques for Matrix Algorithms

Jakob Moosbauer. Search Techniques for Matrix Algorithms . PhD thesis, Johannes Kepler University, 2023. URL: https://epub.jku.at/obvulihs/content/titleinfo/9217131

2023
[10]

Flip graphs with symmetry and new matrix multipli- cation schemes, 2025

Jakob Moosbauer and Michael Poole. Flip graphs with symmetry and new matrix multipli- cation schemes, 2025. URL: https://arxiv.org/abs/2502.04514, arXiv:2502.04514

2025 arXiv
[11]

Alexander Novikov, Ngˆ an V˜ u, Marvin Eisenberger, Emilien Dupont, Po-Sen Huang, Adam Zsolt Wagner, Sergey Shirobokov, Borislav Kozlovskii, Francisco J. R. Ruiz, Ab- bas Mehrabian, M. Pawan Kumar, Abigail See, Swarat Chaudhuri, George Holland, Alex Davies, Sebastian Nowozin, ...

2025 arXiv
[12]

Fast commutative matrix algorithms

Andreas Rosowski. Fast commutative matrix algorithms. Journal of Symbolic Compu- tation, 114:302–321, 2023. URL: https://www.sciencedirect.com/science/article/pii/ S0747717122000499, doi:https://doi.org/10.1016/j.jsc.2022.05.002

2023 doi
[13]

A. V. Smirnov. The bilinear complexity and practical algorithms for matrix multiplication. Computational Mathematics and Mathematical Physics , 53(12):1781–1795, 2013. doi:10. 1134/S0965542513120129

2013
[14]

Gaussian elimination is not optimal

Volker Strassen. Gaussian elimination is not optimal. Numerische Mathematik , 13(4):354– 356, 1969. doi:10.1007/BF02165411

1969 doi
[15]

A. Waksman. On winograd’s algorithm for inner products. IEEE Transactions on Computers, C-19(4):360–361, 1970. doi:10.1109/T-C.1970.222926

1970
[16]

Winograd

S. Winograd. A new algorithm for inner product. IEEE Transactions on Computers , C- 17(7):693–694, 1968. doi:10.1109/TC.1968.227420

1968
[17]

Nazrul Islam, and ´Eric Schost

Charles ´Eric Drevet, Md. Nazrul Islam, and ´Eric Schost. Optimization techniques for small matrix multiplication. Theoretical Computer Science, 412(22):2219–2236, 2011. URL: https://www.sciencedirect.com/science/article/pii/S0304397510007036, doi:https:// doi.org/10.1016/j.tc...

2011 doi
[2039]

URL: https://epubs.siam.org/doi/abs/10.1137/1.9781611978322.63, arXiv:https: //epubs.siam.org/doi/pdf/10.1137/1.9781611978322.63, doi:10.1137/1.9781611978322. 63

doi

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

[1] [2]

Adaptive flip graph algorithm for ma- trix multiplication

Yamato Arai, Yuma Ichikawa, and Koji Hukushima. Adaptive flip graph algorithm for ma- trix multiplication. In Proceedings of the 2024 International Symposium on Symbolic and Algebraic Computation, ISSAC ’24, page 292–298, New York, NY, USA, 2024. Association for Computing Machinery. doi:10.1145/3666000.3669701

arXiv 2024

[2] [1]

More Asymmetry Yields Faster Matrix Multiplication , pages 2005–

Josh Alman, Ran Duan, Virginia Vassilevska Williams, Yinzhan Xu, Zixuan Xu, and Renfei Zhou. More Asymmetry Yields Faster Matrix Multiplication , pages 2005–

work page 2005

[3] [3]

A non-commutative algorithm for multiplying 4x4 matrices using 48 non-complex multiplications, 2025

Jean-Guillaume Dumas, Cl´ ement Pernet, and Alexandre Sedoglavic. A non-commutative algorithm for multiplying 4x4 matrices using 48 non-complex multiplications, 2025. URL: https://arxiv.org/abs/2506.13242, arXiv:2506.13242

arXiv 2025

[4] [4]

Marijn J. H. Heule, Manuel Kauers, and Martina Seidl. Local search for fast matrix multipli- cation. In Mikol´ aˇ s Janota and Inˆ es Lynce, editors,Theory and Applications of Satisfiability Testing – SAT 2019 , pages 155–163, Cham, 2019. Springer International Publishing

work page 2019

[5] [5]

Flip graphs for matrix multiplication

Manuel Kauers and Jakob Moosbauer. Flip graphs for matrix multiplication. In Proceedings of the 2023 International Symposium on Symbolic and Algebraic Computation , ISSAC ’23, page 381–388, New York, NY, USA, 2023. Association for Computing Machinery. doi:10. 1145/3597066.3597120

arXiv 2023

[6] [6]

Consequences of the moosbauer-poole algorithms

Manuel Kauers and Isaac Wood. Consequences of the moosbauer-poole algorithms. arXiv preprint arXiv:2505.05896, 2025

arXiv 2025

[7] [7]

Laderman

John D. Laderman. A noncommutative algorithm for multiplying 3 ×3 matrices using 23 multiplications. Bulletin of the American Mathematical Society , 82:126–128, 1976. doi:10. 1090/S0002-9904-1976-13988-2

work page 1976

[8] [8]

O.M. Makarov. An algorithm for multiplying 3 ×3 matrices. USSR Compu- tational Mathematics and Mathematical Physics , 26(1):179–180, 1986. URL: https://www.sciencedirect.com/science/article/pii/004155538690203X, doi:https: //doi.org/10.1016/0041-5553(86)90203-X

arXiv 1986