Symbolic Computation

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A Majorization Order on Monomials and Termination of a Successive Difference Substitution Algorithm

A Majorization Order on Monomials and Termination of a Successive Difference Substitution Algorithm

๋ณธ ๋…ผ๋ฌธ์€ ์„ฑ๊ณต์ ์ธ ์ฐจ๋ถ„ ๋Œ€์ฒด ์•Œ๊ณ ๋ฆฌ์ฆ˜(KSDS)์˜ ์ข…๊ฒฐ์„ฑ์— ๋Œ€ํ•œ ์‹ฌ๋„ ์žˆ๋Š” ์—ฐ๊ตฌ๋ฅผ ์ œ๊ณตํ•˜๋ฉฐ, ํŠนํžˆ ์ฃผ์š”ํ™” ์ˆœ์„œ๋ฅผ ์ด์šฉํ•˜์—ฌ ์ด ์•Œ๊ณ ๋ฆฌ์ฆ˜์ด ์–ธ์ œ ์ข…๋ฃŒ๋˜๋Š”์ง€ ๋ถ„์„ํ•œ๋‹ค. KSDS๋Š” ๋‹คํ•ญ์‹ ํ•จ์ˆ˜์—์„œ ํŠน์ • ์กฐ๊ฑด์„ ๋งŒ์กฑํ•˜๋Š” ๊ฒฝ์šฐ ์–‘์ ์œผ๋กœ ์ข…๋ฃŒ๋˜๋ฉฐ, ์ด๋Ÿฌํ•œ ์กฐ๊ฑด์€ ์ฃผ๋กœ ๋‹จํ•ญ์‹์˜ ๊ณ„์ˆ˜์™€ ๊ทธ๋“ค์˜ ๊ด€๊ณ„์— ๊ธฐ๋ฐ˜ํ•œ๋‹ค. 1. KSDS์˜ ์ •์˜ ๋ฐ ๋ฐฐ๊ฒฝ KSDS๋Š” ์ž…๋ ฅ ํ•จ์ˆ˜ f๋ฅผ ๋‹คํ•ญ์‹ ํ˜•ํƒœ๋กœ ํ‘œํ˜„ํ•˜๊ณ , ์ด ๋‹คํ•ญ์‹์—์„œ ํŠน์ •ํ•œ ๋Œ€์ฒด ๊ทœ์น™์„ ์ ์šฉํ•˜์—ฌ ์ƒˆ๋กœ์šด ๋‹คํ•ญ์‹์„ ์ƒ์„ฑํ•˜๋Š” ์•Œ๊ณ ๋ฆฌ์ฆ˜์ด๋‹ค. ๋…ผ๋ฌธ์—์„œ๋Š” ๋‹จํ•ญ์‹์ด ์–‘์ˆ˜ ๋˜๋Š” ์Œ์ˆ˜ ๊ณ„์ˆ˜๋ฅผ ๊ฐ€์งˆ ๋•Œ์˜ ํŠน์„ฑ์„ ์ •์˜ํ•œ๋‹ค. ํŠน

Symbolic Computation Mathematics Computer Science
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Note on fast division algorithm for polynomials using Newton iteration

: ์„œ๋ก  ๋…ผ๋ฌธ์€ ๋‹คํ•ญ์‹์ด ํ•„๋“œ ์œ„์—์„œ ์œ ํด๋ฆฌ๋“œ ์˜์—ญ์„ ํ˜•์„ฑํ•˜๋ฉฐ, ์ด๋Š” ๋ชจ๋“  a์™€ b์— ๋Œ€ํ•ด (b๊ฐ€ 0์ด ์•„๋‹Œ ๊ฒฝ์šฐ) ๊ณ ์œ ํ•œ q์™€ r์ด ์กด์žฌํ•˜์—ฌ a qb + r์ด๊ณ  r์˜ ์ฐจ์ˆ˜๊ฐ€ b๋ณด๋‹ค ์ž‘๋‹ค๋Š” ๊ฒƒ์„ ์„ค๋ช…ํ•œ๋‹ค. ์ „ํ†ต์ ์ธ ๋‹คํ•ญ์‹ ๋‚˜๋ˆ—์…ˆ ์•Œ๊ณ ๋ฆฌ์ฆ˜์€ O(n^2) ์—ฐ์‚ฐ์„ ํ•„์š”๋กœ ํ•˜์ง€๋งŒ, ๋‰ดํ„ด ๋ฐ˜๋ณต๊ณผ ๋ฐ˜์ „ ๊ธฐ๋ฒ•์„ ์‚ฌ์šฉํ•˜๋ฉด ์ด๋ฅผ M(n)์œผ๋กœ ๊ฐœ์„ ํ•  ์ˆ˜ ์žˆ๋‹ค. ๊ทธ๋Ÿฌ๋‚˜ ์ด ๋ฐฉ๋ฒ•์€ xl์˜ ์ฐจ์ˆ˜๊ฐ€ 2์˜ ๊ฑฐ๋“ญ์ œ๊ณฑ์ผ ๊ฒƒ์„ ์š”๊ตฌํ•œ๋‹ค. ๋‹คํ•ญ์‹ ๋‚˜๋ˆ—์…ˆ ์•Œ๊ณ ๋ฆฌ์ฆ˜ ๋…ผ๋ฌธ์—์„œ๋Š” ๋‹คํ•ญ์‹ a์™€ b์— ๋Œ€ํ•ด q์™€ r์„ ์ฐพ๋Š” ๋ฌธ์ œ๋ฅผ ๋‹ค๋ฃฌ๋‹ค. ์ด๋ฅผ ์œ„ํ•ด a qb + r ํ˜•ํƒœ๋กœ ํ‘œํ˜„

Computer Science Symbolic Computation
A Note on the Group-theoretic Approach to Fast Matrix Multiplication

A Note on the Group-theoretic Approach to Fast Matrix Multiplication

: ์ด ๋…ผ๋ฌธ์€ ํ–‰๋ ฌ ๊ณฑ์…ˆ ์•Œ๊ณ ๋ฆฌ์ฆ˜์˜ ๋ณต์žก๋„๋ฅผ ์ค„์ด๋Š” ๋ฐ ์žˆ์–ด ๊ทธ๋ฃน ์ด๋ก ์  ์ ‘๊ทผ ๋ฐฉ์‹์˜ ์ค‘์š”์„ฑ์— ๋Œ€ํ•ด ํƒ๊ตฌํ•˜๊ณ  ์žˆ๋‹ค. ์ „ํ†ต์ ์ธ ํ–‰๋ ฌ ๊ณฑ์…ˆ ์•Œ๊ณ ๋ฆฌ์ฆ˜์€ O(nยณ) ์‹œ๊ฐ„ ๋ณต์žก๋„๋ฅผ ๊ฐ€์ง„๋‹ค. ๊ทธ๋Ÿฌ๋‚˜, ๋ณด๋‹ค ํšจ์œจ์ ์ธ ์•Œ๊ณ ๋ฆฌ์ฆ˜์ด ์กด์žฌํ•˜๋ฉฐ, ๊ฐ€์žฅ ๋น ๋ฅธ ์•Œ๋ ค์ง„ ์•Œ๊ณ ๋ฆฌ์ฆ˜์€ Don Coppersmith์™€ Shmuel Winograd์— ์˜ํ•ด ์ œ์‹œ๋œ O(nยฒ.376) ์‹œ๊ฐ„ ๋ณต์žก๋„์˜ ์•Œ๊ณ ๋ฆฌ์ฆ˜์ด๋‹ค. ๋…ผ๋ฌธ์—์„œ๋Š” ํ–‰๋ ฌ ๊ณฑ์…ˆ ์•Œ๊ณ ๋ฆฌ์ฆ˜์˜ ์„ฑ๋Šฅ์„ ํ–ฅ์ƒ์‹œํ‚ค๋Š” ๋ฐ ์žˆ์–ด ๊ทธ๋ฃน ์ด๋ก ์  ์ ‘๊ทผ ๋ฐฉ์‹์˜ ์ค‘์š”์„ฑ์„ ๊ฐ•์กฐํ•œ๋‹ค. ํŠนํžˆ, 2003๋…„์— ์ฝ”ํ—จ(Cohn)๊ณผ ์šฐ๋งŒ์Šค(Umans)์ด

Symbolic Computation Mathematics Computer Science
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Structure of lexicographic Groebner bases in three variables of ideals of dimension zero

: ์„œ๋ก  ๋ฐ ์—ฐ๊ตฌ ๋ฐฐ๊ฒฝ ๋…ผ๋ฌธ์€ ๋‹ค๋ณ€์ˆ˜ ๋‹คํ•ญ์‹ ๋ฐ˜ํ™˜์˜ ์ œ๋กœ ์ฐจ์› ์ด์ƒ์ ๊ณผ ๊ทธ๋กœ๋ธŒ๋„ˆ ๊ธฐ์ €์— ์ดˆ์ ์„ ๋งž์ถ”๊ณ  ์žˆ๋‹ค. ํŠนํžˆ, 3๊ฐœ ๋ณ€์ˆ˜ x , y , z ๋ฅผ ์‚ฌ์šฉํ•˜๋Š” ๋‹คํ•ญ์‹ ๋ฐ˜ํ™˜ R

Symbolic Computation Computer Science

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