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Fast-Matrix-Multiplication

Accuracy of Fast matrix multiplication Algorithms via 2x2 recursion

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Authors: Jean-Guillaume Dumas, Clément Pernet, Alexandre Sedoglavic

• J-G. Dumas, C. Pernet, A. Sedoglavic; Strassen's algorithm is not optimally accurate, Feb. 2024

About: This is a Fork of the Complex-Matrix-Multiplication repository, created for the numerical experiments of the paper:

• Dai, Z., Lim, LH; Numerical stability and tensor nuclear norm. Numer. Math. 155, 345--376 (2023).

• We add more accurate variants of fast matrix multiplication via 2x2 recursion

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FMM-matlab-benchmarks, Accuracy benchmarks with Matlab:

Program	Growth Factor	Description
strassenw.m	17.8530	original Winograd's algorithm
strassen.m	14.8284	original Strassen's algorithm
DPS_evenpow.m	12.2034	using only powers of 2
DPS_smallrat.m	12.2034	small rational coefficients
DPS_intermediate.m	12.0695	fast rational
DPS_integral.m	12.0662	large rational coefficients
DPS.m	12.0660	most accurate, with sqrt(3)

Faster Algorithms, Faster variants, using an alternative basis from:

• G. Beniamini, N. Cheng, O. Holtz, E. Karstadt, O. Schwartz; Sparsifying the Operators of Fast Matrix Multiplication Algorithms.

Program	Complexity bound	Description
Strassen_aternative.m	$5n^{\log_2(7)}$	Strassen's algorithm, sparsified
Winograd_aternative.m	$5n^{\log_2(7)}$	Winograd's algorithm, sparsified
DPS_aternative.m	$5n^{\log_2(7)}$	DPS's algorithm, sparsified

All "aternative" variants use left/right and inverse change of basis (*CoB* files) together with an inner sparse multiplication (*mul* files).

Benchmarks, Accuracy comparison with symbolic matrix multiplication:

• accuracy_2x2_real.m: comparing algorithms accuracy
• accuracy_alternative_real.m: comparing alternative change of basis
• gallery_alternative_real.m: change of basis on large condition number

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FMM-plinopt-codegen, Progam generation via the PLinOpt library:

Command-line scritps generating optimized matlab programs.

Examples: DPS.plo, DPS_evenpow.plo, DPS_smallrat.plo, DPS_intermediate.plo, DPS_integral.plo, DPS_CoB.plo.

• PLinOpt library: Routines handling linear, bilinear & trilinear programs.

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FMM-maple-proofs, Proofs of accuracy bounds in Maple:

• Minimal2Norm: Proof of the Frobenius norm minimal point of Fast Matrix Multiplication in Strassen's orbit
• LowerBoundGamma: Proof of a lower bound on the growth factor of Fast Matrix Multiplication in Strassen's orbit