A more accurate rational non-commutative algorithm for multiplying 4x4 matrices using 48 multiplications

Alexandre Sedoglavic (CRIStAL); CASC); Cl\'ement Pernet (UGA; Jean-Guillaume Dumas (UGA; LJK; LJK)

REVIEW 3 major objections 4 minor 22 references

A more accurate rational non-commutative algorithm for multiplying 4x4 matrices using 48 multiplications

T0 review · 3 major / 4 minor · reviewed 2026-08-02 · deepseek-v4-flash

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

Pith's one-line read A new 4×4 matrix multiplication scheme uses 48 multiplications and achieves an error-bound exponent of about 2.386, the sharpest among compared sub-cubic algorithms.

desk verdict Plausible new best accuracy exponent for sub-cubic 4×4×4:48, but the central L,R,P identity is unverified and the abstract contradicts the body on both headline numbers. read the letter →

arxiv 2603.18699 v2 pith:VSA7BFIV submitted 2026-03-19 cs.DS cs.SC

Jean-Guillaume Dumas (UGA , LJK , CASC) , Clément Pernet (UGA , LJK) , Alexandre Sedoglavic (CRIStAL) This is my paper · ORCID

classification cs.DScs.SC

keywords 4×4matrixmultiplication48multiplicationsnon-commutativealgorithmnumericalaccuracygrowthfactorerror-boundexponentstraight-lineprogramsub-cubiccomplexity

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

The paper presents a rational, non-commutative algorithm for multiplying 4×4 matrices with 48 multiplications, valid over any ring containing an inverse of 2. Its central claim is improved numerical accuracy: the (∞,2)-growth factor is 27.314, giving an error-bound exponent log_4(27.314) ≈ 2.386, lower than Strassen's, Winograd's, and earlier 48-multiplication variants. The authors also give a concrete straight-line program with a leading complexity constant of about 12.09 n^2.792, plus an alternative-basis version with constant 8 n^2.792. If the decomposition is exact, this becomes the most accurate currently known sub-cubic matrix multiplication scheme in the 2-norm input setting.

What carries the argument

The LRP representation: a triple of matrices L, R, P such that vec(AB) = P·((L·vec A) ⊙ (R·vec B)), where ⊙ is the Hadamard product and the 48 intermediate products are the non-commutative multiplications. Accuracy is governed by the (p,q)-growth factor γ_{p,q} = ||(Σ_i ||L_i,*||_{q*} ||R_i,*||_{q*} |P_{j,i}|)_j||_p; its logarithm in base k gives the recursive error exponent. The new variant was selected using a smoothed relaxation γ_2 ≥ γ_{2,2}, which the paper argues better predicts max-norm behavior in practice.

What would settle it

Symbolically verify the identity for arbitrary 4×4 matrices, or evaluate both sides at random integer matrices over, say, the rationals or a finite field; any mismatch disproves the algorithm. Separately recompute γ∞,2 from the published L, R, P and confirm that log_4(27.314) ≈ 2.386.

Watch

Extended reading notes

Core claim

We exhibit an LRP decomposition of the 4×4×4 matrix-multiplication tensor using 48 rank-one terms, over Z[1/2], whose (∞,2)-growth factor is γ∞,2 = 27.314. This lowers the recursive error-bound exponent from roughly 2.628 (previous rational 4×4×4:48) and 2.577 (best 2×2×2:7 variant) to about 2.386. A straight-line realization uses 355 additions/binary shifts plus 48 multiplications, with leading term (387/32)n^{2+log_4 3}; an alternative-basis variant achieves leading constant 8. Experiments on normally distributed floats show max-norm error close to the conventional cubic algorithm and better than Strassen, Winograd, and the prior 48-multiplication scheme.

Load-bearing premise

The printed L, R, P tables are asserted to exactly satisfy P·((L·vec A) ⊙ (R·vec B)) = vec(AB) for every 4×4 pair A, B, but the paper gives no algebraic proof of this identity in the text.

Editorial extensions

If this is right

Recursive use of this 4×4 block gives an error bound growing like n^2.386, the best among the compared sub-cubic algorithms when inputs are measured in 2-norm.
The concrete straight-line program lets implementers reproduce the algorithm without re-deriving the decomposition.
The claimed accuracy carries over to any ring containing 1/2, including exact rational arithmetic and finite fields of odd characteristic.
The alternative-basis variant reaches a leading complexity constant of 8 n^2.792 while keeping similar accuracy, improving the practical constant over the main variant.
The experiments indicate max-norm output error close to that of the classical cubic algorithm, despite using only 48 scalar products per 4×4 block.

Reading between the lines

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

The 'most accurate sub-cubic algorithm' claim is norm-dependent: for max-norm on the inputs, Strassen's algorithm still holds the best bound; the improvement here applies to the 2-norm input regime.
The abstract quotes the exponent as about 2.335 while the body and Table 2 give 2.386 for the same quantity; the manuscript does not resolve this discrepancy, so comparisons should verify the intended value.
The smoothed relaxation γ_2 used for search is not the true γ_{2,2}; similar relaxations could be applied to other tensor decompositions, and the gap between relaxation and exact growth factor is an open handle for further accuracy tuning.
Because the coefficients are rational with powers of 1/2, the algorithm is likely to be usable in exact and modular arithmetic, though coefficient growth during recursion would need separate testing.

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

You should know two things about this paper. First, it does appear to set a new best for the (∞,2) accuracy exponent among sub-cubic matrix multiplication algorithms, with log_4 γ∞,2 ≈ 2.386 and a concrete straight-line program. Second, the paper never verifies the central L, R, P decomposition, and the abstract disagrees with the body on both headline numbers. That combination makes me treat the results as plausible but not yet established.

What's genuinely new: the specific L, R, P triple in Table 1. It's not in the authors' earlier papers [8] or [9], and it gives γ∞,2 = 27.314, down from 38.163, and γ2,2 = 109.26, down from 139.906. The operation counts in Section 3 are internally consistent: 104 additions for L, 88 + 16 shifts for R, 129 + 18 shifts for P, total 355, leading to the stated constant 387/32 n^{2+log4 3}. The γ values in Table 2 are consistent with Eq. (4). The paper also points to PLinOpt data files for reproducibility. That's real work.

Now the soft spots, in proportion. The big one is the unverified identity. The manuscript asserts that the matrices in Table 1 and the listings of Section 3 compute the 4×4 product over Z[1/2], but gives no symbolic check, no short proof, and only a pointer to external data files. If any entry in L, R, or P is mistyped, the growth factor, the exponent, the SLP counts, and the experiments all describe a phantom. This is load-bearing, not a technicality. The stress-test note is right about that.

The second issue is the abstract/body discrepancy: the abstract says 2.335 and 316/32, the body says 2.386 and 387/32. The body numbers are internally consistent with the stated γ and operation counts, so the abstract looks like a leftover from an earlier run, but it undermines confidence in the manuscript's editorial care. Third, Figure 1 has no error bars and it's not clear how many trials, but that's minor for a quick benchmark.

The reader's conditional verdict is fair. The central argument is believable and the arithmetic checks out, but the missing verification is a real gap, not a manufactured one. If I were refereeing, I'd ask the authors to add a short symbolic verification of the L,R,P identity (or a script that checks it) and to reconcile the abstract with the body. The paper deserves serious peer review: the claim is important enough in its niche and it's checkable. I'd engage with it, but only after those fixes.

Recommendation: send to a referee, but condition acceptance on the verification and the abstract cleanup.

Referee Report

3 major / 4 minor

Summary. The paper proposes a new rational non-commutative algorithm for multiplying 4×4 matrices with 48 non-scalar multiplications, represented as an exact (L, R, P) tensor decomposition (Table 1) and as a straight-line program (Listings 1–4). The central claims are that this variant achieves growth factor γ∞,2 = 27.314, error-bound exponent log_4 γ∞,2 ≈ 2.386 for the (∞,2)-norm, and a leading constant 387/32 n^{2+log_4 3} in its complexity bound. The paper compares these values with prior ⟨2×2×2:7⟩ and ⟨4×4×4:48⟩ algorithms, gives an alternative-basis variant, and reports numerical experiments suggesting improved max-norm accuracy.

Significance. If the decomposition and all reported constants are correct, the paper gives the most accurate currently known sub-cubic matrix multiplication scheme in the (∞,2)-norm setting, improving the asymptotic error exponent from about 2.628 to 2.386 while maintaining a reasonable operation count. The work is constructive: it provides explicit rational matrices, a concrete SLP, public data files via PLinOpt, and reproducible operation counts. These are real strengths. However, the central algebraic identity is not verified in the manuscript, and the abstract disagrees with the body on both headline numbers; the contribution cannot be accepted in its present form without resolving these issues.

major comments (3)

[§2.2, Table 1, Listings 1–4, Appendix B] The central claim is that the L, R, P matrices in Table 1 form an exact rank-48 decomposition of the 4×4 matrix multiplication tensor, and that Listings 1–4 realize the same linear forms. The paper never proves or independently checks either identity. Section 2.2 only says the variant is ‘defined by the lrp representation given in Table 1’; Section 3 only says the SLPs are ‘obtained from’ those matrices. Appendix B points to PLinOpt data files, but a reader cannot verify correctness from the manuscript alone. Since every numerical result (γ∞,2, Table 2, Eq. (6), Section 4) depends on this decomposition being exact, the missing verification is load-bearing. Please include a machine-checkable certificate — for example a short script that checks P·((L·vec(A))⊙(R·vec(B))) = vec(AB) for all basis matrices, or a hash/transcript of such a verification.
[Abstract vs. §1, §2.3, Eq. (6)] The abstract reports an error-bound exponent of log_4 γ∞,2 ≈ 2.335 and a leading constant of 316/32 n^{2+log_4 3}; the body reports log_4 γ∞,2 ≈ 2.386 (Section 1 and Table 2) and 387/32 n^{2+log_4 3} (Section 1 and Eq. (6)). These are numerically very different claims. The operation counts in Section 3 (104+88+129 additions and 16+18 shifts = 355 operations) give 1 + 355/(48−16) = 387/32, so the abstract’s 316/32 is not consistent with the presented SLP. The exponent 2.386 is consistent with γ∞,2 = 27.314 in Table 2, while 2.335 would correspond to a different γ. This discrepancy must be resolved and the correct values used consistently in the final version.
[§4 / Table 2] The experiments report max-norm output error but do not specify the input norm, while the theoretical ranking is made with (p,q)=(∞,2). The text says the ranking ‘follows the one indicated by the second column of Table 2 (for (p,q)=(∞,2))’ and then says ‘regardless of any norm on the input (parameter q)’. This is not a formal comparison. Please state precisely what is measured (e.g., normalized random matrices, fixed q, number of trials) and how that relates to the (∞,2) bound. Otherwise the experimental section is only anecdotal, and the text should not claim it confirms the theoretical ranking.

minor comments (4)

[Figure 1] The y-axis has tick values but no label; the caption says ‘normal distribution’ but does not specify the distribution parameters, sample size, or whether multiple runs were averaged. Please add these details.
[§2.2] The text in Section 2.2 uses ‘lpr representation’ while the rest of the paper uses ‘LRP’ or ‘lrp’. Please standardize the terminology.
[References] Table 2 labels the row for the previous rational ⟨4×4×4:48⟩ as ‘DPS25’ but the companion text in Section 1 attributes the rational variant to reference [8]. Please check that the citation in Table 2 is the intended one, and distinguish clearly between [8] (the 2025 arXiv report) and [9] (the journal article).
[Appendix A and Eq. (10)] The alternative-basis matrices Lalt, Ralt, Palt and the cobasis matrices are extremely large and are not accompanied by a verification script. If the data files in Appendix B are the only source, this should be stated explicitly and the identity L = Lalt·Lcob, R = Ralt·Rcob, P = Pcob·Palt should be checked by the same certificate as the main decomposition.

Circularity Check

0 steps flagged · score 2.0 of 10

No circularity: the reported growth factor, error exponent, and complexity are computed directly from the explicit exact rational L,R,P matrices, not fitted to the headline claims; self-citations are prior published results.

full rationale

The paper's central claim is a concrete exact rational lrp realization of a ⟨4×4×4:48⟩ matrix multiplication algorithm. The derivation chain is: (i) exhibit L,R,P in Table 1; (ii) compute γp,q from Eq. (4) using those matrices; (iii) generate straight-line programs from the same matrices and count operations; (iv) run independent random-matrix benchmarks. None of the headline numbers are fitted parameters or renamed predictions. The error-bound formula is imported from the authors' prior journal paper [9, Th. 18], and comparison values come from their earlier papers [6,8,9]; these are published, parameter-free results with stated assumptions, so under the review rules they are real evidence rather than circular support. The variant was selected by optimizing the surrogate γ2 in Eq. (5), but the reported quantities γ∞,2=27.314, γ2,2=109.26, and the operation count are computed afterwards from the fixed exact matrices and do not reduce to the selection objective by construction. The manuscript does omit an explicit proof or independent check that Table 1 and Listings 1–4 satisfy P·((L·vec(A))⊙(R·vec(B)))=vec(AB); this is an unverified premise and a correctness risk, not circularity. Similarly, the abstract/body numerical inconsistencies (2.335 vs 2.386 and 316/32 vs 387/32) are editing/consistency issues, not evidence that the derivation is circular. Therefore no specific circular step is identified; the only concerns are external verification and internal consistency, which do not raise the circularity score.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No fitted free parameters; all coefficients in the algorithm are small exact rationals. The load-bearing assumptions are: exactness of the displayed decomposition, the cited error-bound theorem, standard floating-point rounding, and correct computation of the displayed γ values. No new entities are introduced.

assumptions (4)

ad hoc to paper The L, R, P matrices in Table 1 define an exact rank-48 decomposition of the 4×4 matrix multiplication tensor.
No proof or certificate is included in the text; the straight-line programs are supplied as evidence. If false, all downstream claims fail.
standard math The error-bound theorem (Theorem 2.1, from [9]) correctly describes recursive floating-point forward error.
The paper relies on this cited theorem for the growth-factor and error-exponent reduction.
domain assumption The floating-point rounding model with ε and O(ε^2) remainder applies to the algorithms considered.
Standard numerical analysis assumption, used for all compared algorithms.
domain assumption The γ_{∞,2} and γ_2 values in Table 2 / Remark 2.2 were computed correctly from the displayed matrices.
The paper does not include a verification script for these computed constants.

how reviews work

0 comments

Cite this review

Pith. "Pith review of A more accurate rational non-commutative algorithm for multiplying 4x4 matrices using 48 multiplications." pith.science (2026). https://pith.science/paper/VSA7BFIV

@misc{pith2026260318699,
  author       = {Pith},
  title        = {Pith review of: A more accurate rational non-commutative algorithm for multiplying 4x4 matrices using 48 multiplications},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VSA7BFIV}},
  note         = {Machine review of arXiv:2603.18699}
}

abstract

We propose a more accurate variant of an algorithm for multiplying 4x4 matrices using 48 multiplications over any ring containing an inverse of 2. This algorithm achieves an error bound exponent of only $\log_{4}gamma_{\infty,2}\approx{2.335}$. In practice, it also reaches a better accuracy w.r.t. max-norm, when compared to previously known such fast algorithms. Furthermore, we propose a straight line program of this algorithm, giving a leading constant in its complexity bound of $\frac{316}{32}n^{2+\log_{4}{3}}+o(n^{2+\log_{4}{3}})$ operations over any ring containing an inverse of 2.

Figures

Figures reproduced from arXiv: 2603.18699 by the authors.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

22 extracted references · 4 canonical work pages

[8]

A non-commutative algorithm for multiplying 4 × 4 matrices using 48 non-complex multiplications

Jean-Guillaume Dumas, Cl´ ement Pernet, and Alexandre Sedoglavic. A non-commutative algorithm for multiplying 4 × 4 matrices using 48 non-complex multiplications. Technical Report 2506.13242, arXiv, June 2025. arXiv:2506.13242. [Cited § 1, 2.3, 2.3, 2.2, and 4.]

arXiv 2025
[9]

Towards automated generation of fast and accurate algorithms for recursive matrix multiplication

Jean-Guillaume Dumas, Cl´ ement Pernet, and Alexandre Sedoglavic. Towards automated generation of fast and accurate algorithms for recursive matrix multiplication. Journal of Symbolic Computation , 134(102524), May–June 2026. doi:10.1016/j.jsc.2025.102524. [Cited § 1, 2.1, 2.1, 2.3, 4, and 4.]

arXiv 2026
[1]

Im- proving the numerical stability of fast matrix multiplication

Grey Ballard, Austin Reilley Benson, Alex Druinsky, Benjamin Lipshitz, and Oded Schwartz. Im- proving the numerical stability of fast matrix multiplication. siam Journal on Matrix Analysis and Applications, 37(4):1382–1418, 2016. doi:10.1137/15M1032168. [Cited § 1 and 2.1.]

work page doi:10.1137/15m1032168 2016
[2]

Stability of fast algorithms for matrix multiplication

Dario Andrea Bini and Grazia Lotti. Stability of fast algorithms for matrix multiplication. Nu- merische Mathematik, 36(1):63–72, March 1980. doi:10.1007/BF01395989. [Cited § 1 and 2.1.] 6

work page doi:10.1007/bf01395989 1980
[3]

Algorithms for matrix multiplication

Richard Pierce Brent. Algorithms for matrix multiplication. Master’s thesis, march 1970. url: https://apps.dtic.mil/sti/citations/AD0705509. [Cited § 1 and 2.1.]

1970
[4]

Fast matrix multipli- cation is stable

James Demmel, Ioana Dumitriu, Olga Holtz, and Robert Kleinberg. Fast matrix multipli- cation is stable. Numerische Mathematik , 106(2):199–224, February 2007. doi:10.1007/ s00211-007-0061-6 . [Cited § 1 and 2.1.]

2007
[5]

PLinOpt, a collection of C++ routines handling linear & bilinear programs, January 2024

Jean-Guillaume Dumas, Bruno Grenet, Cl´ ement Pernet, and Alexandre Sedoglavic. PLinOpt, a collection of C++ routines handling linear & bilinear programs, January 2024. 8.09 k sloc. url: https://github.com/jgdumas/plinopt. [Cited § 3 and B.]

2024
[6]

Strassen’s algorithm is not opti- mally accurate

Jean-Guillaume Dumas, Cl´ ement Pernet, and Alexandre Sedoglavic. Strassen’s algorithm is not opti- mally accurate. In Shaoshi Chen, editor, issac’24: Proceedings of the 2024 International Symposium on Symbolic and Algebraic Computation , pages 254–263, Raleigh, North Carolina, usa, July 16-19

2024

Show all 22 references

[7]

FMM, accuracy of fast-matrix- multiplication algorithms, September 2025

Jean-Guillaume Dumas, Cl´ ement Pernet, and Alexandre Sedoglavic. FMM, accuracy of fast-matrix- multiplication algorithms, September 2025. v2.5, 1.6 k sloc. url: https://github.com/jgdumas/ Fast-Matrix-Multiplication. [Cited § 4.]

2025
[10]

On varieties of optimal algorithms for the computation of bilinear mappings I

Hans Friedrich Groote, de. On varieties of optimal algorithms for the computation of bilinear mappings I. The isotropy group of a bilinear mapping. Theoretical Computer Science , 7(2):1–24,
[11]

Accuracy and Stability of Numerical Algorithms

Nicholas John Higham. Accuracy and Stability of Numerical Algorithms . siam, 2 edition, 2002. doi:10.1137/1.9780898718027. [Cited § 2.1.]

2002 doi
[12]

Finding complex-valued solutions of Brent equations using nonlinear least squares

Igor Kaporin. Finding complex-valued solutions of Brent equations using nonlinear least squares. Computational Mathematics and Mathematical Physics , 64(9):1881–1891, November 2024. doi: 10.1134/S0965542524701021. [Cited § 1.]

2024 doi
[13]

Matrix multiplication, a little faster

Elaye Karstadt and Oded Schwartz. Matrix multiplication, a little faster. In Christian Scheideler and Mohammad Hajiaghayi, editors, spaa’17: Proceedings of the 29th acm Symposium on Parallelism in Algorithms and Architectures, pages 101–110, Washington, d.c., usa, July 24–26 2...

2017
[14]

Geometry and complexity theory , volume 169 of Cambridge Stud- ies in Advanced Mathematics

Joseph Montague Landsberg. Geometry and complexity theory , volume 169 of Cambridge Stud- ies in Advanced Mathematics . Cambridge University Press, December 2016. doi:10.1017/ 9781108183192. [Cited § 1.]

2016
[15]

Complex to rational fast matrix multiplication

Yoav Moran, Oded Schwartz, and Shuncheng Yuan. Complex to rational fast matrix multiplication. Technical report, arXiv, February 2026. arXiv:2602.13171. [Cited § 1.]

2026
[16]

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

2025 arXiv
[17]

Alternative basis matrix multiplication is fast and stable

Oded Schwartz, Sivan Toledo, Noa Vaknin, and Gal Wiernik. Alternative basis matrix multiplication is fast and stable. Numerische Mathematik , February 2026. doi:10.1007/s00211-026-01531-9 . [Cited § 1 and 2.1.] 7

2026 doi
[18]

Gaussian elimination is not optimal

Volker Strassen. Gaussian elimination is not optimal. Numerische Mathematik , 13(4):354–356, August 1969. doi:10.1007/BF02165411. [Cited § 2.3 and 4.]

1969 doi
[19]

La complexit´ e des calculs num´ eriques

Shmuel Winograd. La complexit´ e des calculs num´ eriques. La Recherche, 83:956–963, November
[1977]

They satisfy that L = Lalt · Lcob, R= Ralt · Rcob and P = Pcob · Palt, with a common inner dimension of 47

[Cited § 2.3 and 4.] A Alternative bases algorithm Following [13], we present in this section the L, R, Pmatrices of an alternative basis derived from the [L; R; P ] matrices of Table 1. They satisfy that L = Lalt · Lcob, R= Ralt · Rcob and P = Pcob · Palt, with a common inner...
[1978]

[Cited § 1.]

doi:10.1016/0304-3975(78)90038-5. [Cited § 1.]

doi
[2024]

doi:10.1145/3666000.3669697

acm Press. doi:10.1145/3666000.3669697. [Cited § 1, 2.3, 2.3, and 4.]

Pith tools

Reviewed August 2, 2026 · model on record in the stance chip above.

[1] [8]

A non-commutative algorithm for multiplying 4 × 4 matrices using 48 non-complex multiplications

Jean-Guillaume Dumas, Cl´ ement Pernet, and Alexandre Sedoglavic. A non-commutative algorithm for multiplying 4 × 4 matrices using 48 non-complex multiplications. Technical Report 2506.13242, arXiv, June 2025. arXiv:2506.13242. [Cited § 1, 2.3, 2.3, 2.2, and 4.]

arXiv 2025

[2] [9]

Towards automated generation of fast and accurate algorithms for recursive matrix multiplication

Jean-Guillaume Dumas, Cl´ ement Pernet, and Alexandre Sedoglavic. Towards automated generation of fast and accurate algorithms for recursive matrix multiplication. Journal of Symbolic Computation , 134(102524), May–June 2026. doi:10.1016/j.jsc.2025.102524. [Cited § 1, 2.1, 2.1, 2.3, 4, and 4.]

arXiv 2026

[3] [1]

Im- proving the numerical stability of fast matrix multiplication

Grey Ballard, Austin Reilley Benson, Alex Druinsky, Benjamin Lipshitz, and Oded Schwartz. Im- proving the numerical stability of fast matrix multiplication. siam Journal on Matrix Analysis and Applications, 37(4):1382–1418, 2016. doi:10.1137/15M1032168. [Cited § 1 and 2.1.]

work page doi:10.1137/15m1032168 2016

[4] [2]

Stability of fast algorithms for matrix multiplication

Dario Andrea Bini and Grazia Lotti. Stability of fast algorithms for matrix multiplication. Nu- merische Mathematik, 36(1):63–72, March 1980. doi:10.1007/BF01395989. [Cited § 1 and 2.1.] 6

work page doi:10.1007/bf01395989 1980

[5] [3]

Algorithms for matrix multiplication

Richard Pierce Brent. Algorithms for matrix multiplication. Master’s thesis, march 1970. url: https://apps.dtic.mil/sti/citations/AD0705509. [Cited § 1 and 2.1.]

1970

[6] [4]

Fast matrix multipli- cation is stable

James Demmel, Ioana Dumitriu, Olga Holtz, and Robert Kleinberg. Fast matrix multipli- cation is stable. Numerische Mathematik , 106(2):199–224, February 2007. doi:10.1007/ s00211-007-0061-6 . [Cited § 1 and 2.1.]

2007

[7] [5]

PLinOpt, a collection of C++ routines handling linear & bilinear programs, January 2024

Jean-Guillaume Dumas, Bruno Grenet, Cl´ ement Pernet, and Alexandre Sedoglavic. PLinOpt, a collection of C++ routines handling linear & bilinear programs, January 2024. 8.09 k sloc. url: https://github.com/jgdumas/plinopt. [Cited § 3 and B.]

2024

[8] [6]

Strassen’s algorithm is not opti- mally accurate

Jean-Guillaume Dumas, Cl´ ement Pernet, and Alexandre Sedoglavic. Strassen’s algorithm is not opti- mally accurate. In Shaoshi Chen, editor, issac’24: Proceedings of the 2024 International Symposium on Symbolic and Algebraic Computation , pages 254–263, Raleigh, North Carolina, usa, July 16-19

2024