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    "title": "Coppersmith–Winograd: using progression-free sets to bound matrix multiplication",
    "summary": "Coppersmith and Winograd combined trilinear tensor constructions with progression-free hashing and pruning to obtain an asymptotic matrix-multiplication exponent below2.376, extending earlier Strassen and Schönhage machinery.\n\nMultiplying two n-by-n matrices by the elementary method takes cubic arithmetic work. Coppersmith and Winograd sought a smaller asymptotic exponent by starting from a compact trilinear form rather than a small ordinary matrix product. Tensor powers produce many interacting blocks. They hash block indices into a set without nontrivial three-term arithmetic progressions, then remove conflicting blocks so that independent matrix products remain. Schönhage’s asymptotic-sum theorem converts those products into an upper bound on the exponent. The1990 article develops successive constructions near2.404,2.388 and finally below2.376, with a displayed refined estimate below2.375477. This is an upper bound on field-arithmetic complexity, not a measured wall-clock speed or proof that the exponent equals that number. The authors explicitly describe the construction as impractical for conceivable applications at the time. Their final conditional route to exponent2 depends on an unresolved combinatorial construction and is not a solution. Earlier1986 and1987 versions are acknowledged; the assessed source is the full1990 journal article. Later improvements and implementations are separate contributions.",
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      "statement": "# Coppersmith–Winograd: using progression-free sets to bound matrix multiplication\n\nCoppersmith and Winograd combined trilinear tensor constructions with progression-free hashing and pruning to obtain an asymptotic matrix-multiplication exponent below2.376, extending earlier Strassen and Schönhage machinery.\n\nMultiplying two n-by-n matrices by the elementary method takes cubic arithmetic work. Coppersmith and Winograd sought a smaller asymptotic exponent by starting from a compact trilinear form rather than a small ordinary matrix product. Tensor powers produce many interacting blocks. They hash block indices into a set without nontrivial three-term arithmetic progressions, then remove conflicting blocks so that independent matrix products remain. Schönhage’s asymptotic-sum theorem converts those products into an upper bound on the exponent. The1990 article develops successive constructions near2.404,2.388 and finally below2.376, with a displayed refined estimate below2.375477. This is an upper bound on field-arithmetic complexity, not a measured wall-clock speed or proof that the exponent equals that number. The authors explicitly describe the construction as impractical for conceivable applications at the time. Their final conditional route to exponent2 depends on an unresolved combinatorial construction and is not a solution. Earlier1986 and1987 versions are acknowledged; the assessed source is the full1990 journal article. Later improvements and implementations are separate contributions.\n\n## Scientific scope\n\nThe result is an asymptotic bound on matrix multiplication in its field-arithmetic model, using inherited tensor and progression-free-set tools. It is not a measured practical runtime or a proof that the exponent equals two. The paper explicitly describes practicality limits, and its exponent-two discussion is conditional. Later improvements and implementations receive separate credit.\n\n## People and systems\n\nDon Coppersmith; Shmuel Winograd\n\n\n\n## Original sources\n\n- [D.Coppersmith and S.Winograd, Matrix Multiplication via Arithmetic Progressions (1990) — complete journal article](https://deltadbu.github.io/UCAS_algorithm_course/Lectures/Lec5-Matrix-Multiplication-Coppersmith-Winograd1990.pdf)\n- [IBM Research publication identity: Matrix multiplication via arithmetic progressions](https://research.ibm.com/publications/matrix-multiplication-via-arithmetic-progressions)\n",
      "summary": "Coppersmith and Winograd combined trilinear tensor constructions with progression-free hashing and pruning to obtain an asymptotic matrix-multiplication exponent below2.376, extending earlier Strassen and Schönhage machinery.\n\nMultiplying two n-by-n matrices by the elementary method takes cubic arithmetic work. Coppersmith and Winograd sought a smaller asymptotic exponent by starting from a compact trilinear form rather than a small ordinary matrix product. Tensor powers produce many interacting blocks. They hash block indices into a set without nontrivial three-term arithmetic progressions, then remove conflicting blocks so that independent matrix products remain. Schönhage’s asymptotic-sum theorem converts those products into an upper bound on the exponent. The1990 article develops successive constructions near2.404,2.388 and finally below2.376, with a displayed refined estimate below2.375477. This is an upper bound on field-arithmetic complexity, not a measured wall-clock speed or proof that the exponent equals that number. The authors explicitly describe the construction as impractical for conceivable applications at the time. Their final conditional route to exponent2 depends on an unresolved combinatorial construction and is not a solution. Earlier1986 and1987 versions are acknowledged; the assessed source is the full1990 journal article. Later improvements and implementations are separate contributions.",
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