加速法と可積分方程式

Note about relations between convergence acceleration algorithms and discrete integrable systems.

Transcript

1. Ճ଎๏ͱՄੵ෼ํఔࣜ ۚઘେհ (ؙ໺ݚڀࣨ, ਫ਼౓อূ෇͖਺஋ܭࢉάϧʔϓ OB) 2019 ೥ 6 ݄, ˏദ໦ݚڀࣨ, ਺ֶԠ༻਺ཧઐ߈ 02017 ೥ޙظͷʮιϦτϯͷ਺ཧ Bʯ ʢߴڮେีઌੜ୲౰ʣͰͷൃදΛՃච͠·ͨ͠. ۚઘେհ (ؙ໺ݚڀࣨ OB) Ճ଎๏ͱՄੵ෼ํఔࣜ 2019 ೥ 6 ݄, ˏദ໦ݚڀࣨ 1 / 20
2. ຊൃදͷྲྀΕ 1 ݚڀഎܠ 2 Ճ଎๏ͷྺ࢙ 3 Ճ଎๏ͷ਺஋࣮ݧ 4 ·ͱΊ ۚઘେհ (ؙ໺ݚڀࣨ OB) Ճ଎๏ͱՄੵ෼ํఔࣜ 2019 ೥ 6 ݄, ˏദ໦ݚڀࣨ 2 / 20
3. ݚڀഎܠ ͸͡Ίʹ ຊࢿྉͰ͸Մੵ෼ܥΛѻ͏͕, Մੵ෼ܥʹ͸ݫີͳఆٛ͸ͳ͍ (Մੵ෼ܥΛఆٛ ͢Δ͜ͱΛ໨ඪͱ͍ͯ͠Δ). ͦͷͨΊ, ਓʹΑͬͯղऍ͕ҟͳΔ͜ͱʹཹҙ͢Δ ඞཁ͕͋Δ. ۚઘେհ (ؙ໺ݚڀࣨ OB) Ճ଎๏ͱՄੵ෼ํఔࣜ 2019 ೥ 6 ݄, ˏദ໦ݚڀࣨ 3 / 20
4. ݚڀഎܠ ݚڀഎܠ Մੵ෼ܥͱҟ෼໺ͷؔ܎ʹ͍༷ͭͯʑͳݚڀ͕ͳ͞Ε͍ͯΔ (Ԡ༻Մੵ෼ܥ). ਺஋ܭࢉ (ࢲ͕ڵຯ͋Δͷ͸͜͜) زԿֶ → Մੵ෼زԿֶ (integrable geometry)1 ηϧΦʔτϚτϯ 2 1ҪϊޱॱҰ. (2010). ۂઢͱιϦτϯ. ே૔ॻళ. 2࣌߂఩࣏. (2010). ശۄܥͷ਺ཧ. ே૔ॻళ. ۚઘେհ (ؙ໺ݚڀࣨ OB) Ճ଎๏ͱՄੵ෼ํఔࣜ 2019 ೥ 6 ݄, ˏദ໦ݚڀࣨ 4 / 20
5. ݚڀഎܠ ݚڀഎܠ ਺஋ܭࢉͱՄੵ෼ܥͷؔ܎ (Մੵ෼ΞϧΰϦζϜͳͲ) ʹ͍ͭͯ͸༷ʑͳݚڀ͕ͳ͞Ε͍ͯΔ. ιϦτϯํఔࣜͷ਺஋ղ๏ ࣗݾద߹Ҡಈ֨ࢠεΩʔϜ ʢFeng-Maruno-Ohta (2010), ߏ଄อଘܕ਺஋ղ๏ 3 ͷҰछʣ Մੵ෼ࠩ෼εΩʔϜ (Ablowitz-Ladik, Ablowitz-Taha, Hirota etc)4 ਺஋ٯࢄཚ๏ (numerical inverse scattering method) Painlev´ e ํఔࣜɾPainlev´ e ௒ӽؔ਺ͷ਺஋ղੳ (Novokshenov ͳͲ) Riemann-Hilbert ໰୊ͷ਺஋ղ (cf: Trogdon-Olver) ݻ༗஋ɾಛҟ஋໰୊ͷ਺஋ղ๏ͱ௚ަଟ߲ࣜɾ཭ࢄՄੵ෼ܥͷؔ܎ 5,6 Ճ଎๏ͱ཭ࢄՄੵ෼ܥͷؔ܎ 7,8→ ຊࢿྉͷओ୊ 3೔ຊͰ͸৿ਖ਼෢ઌੜΛத৺ͱͨ͠άϧʔϓ͕ݚڀΛߦ͍ͬͯΔ. 4தଜՂਖ਼. (2000). Մੵ෼ܥͷԠ༻਺ཧ, ী՚๪. 5தଜՂਖ਼ et al. (2018). Մੵ෼ܥͷ਺ཧ, ே૔ॻళ. 6཭ࢄՄੵ෼ܥ͔Β૑ग़͞Εͨݻ༗஋໰୊ͷ਺஋ղ๏ (dLV ΞϧΰϦζϜͳͲ) ͱݻ༗஋ʹؔ͢Δ ઁಈఆཧɾࣄޙධՁΛ࢖͑͹ݻ༗஋ͷਫ਼౓อূ͕Ͱ͖Δ. ઁಈఆཧɾࣄޙධՁʹ͍ͭͯ͸࣍Λࢀর: ࢁຊ఩࿕ (2003). ਺஋ղੳೖ໳, αΠΣϯεࣾ. େੴਐҰ (2000). ਫ਼౓อূ෇͖਺஋ܭࢉ, ίϩφࣾ. େੴਐҰ et al. (2018). ਫ਼౓อূ෇͖਺஋ܭࢉͷجૅ, ίϩφࣾ. 7ӬҪರ, ࡵຎॱ٢ (1995). Acceleration Methods and Discrete Soliton Equations, ਺ཧղੳݚڀॴߨڀ࿥ 933, 44-60. 8Papageorgiou, Grammaticos and Ramani (1993). Integrable Lattices and Convergence Acceleration Algorithms, Phys. Lett. A 179, 111-115. ۚઘେհ (ؙ໺ݚڀࣨ OB) Ճ଎๏ͱՄੵ෼ํఔࣜ 2019 ೥ 6 ݄, ˏദ໦ݚڀࣨ 5 / 20
6. ݚڀഎܠ ݚڀഎܠ ڃ਺ͷ਺஋ܭࢉ๏ͱͯ͠”Ճ଎๏”(ڃ਺ͷऩଋΛՃ଎ͤ͞Δํ๏) ͕͋Δ. Ճ଎๏ͷதʹ͸Մੵ෼ͳ཭ࢄιϦτϯํఔࣜʹॻ͖׵͑ΒΕΔ΋ͷ͕͋Δ. Ճ଎๏ͱՄੵ෼ํఔࣜͷܨ͕Γʹֶ͍ͭͯͿͷ͕ࠓ೔ͷ໨ඪ͕ͩ, Ճ଎๏͕ ਺஋ܭࢉʹͲ͏Өڹ͢Δ͔Λ਺஋࣮ݧΛ௨ͯ͠ݟ͍ͯ͘ͱ͜Ζ͔Β࢝ΊΑ͏. Definition (ڃ਺ͷऩଋՃ଎) ਺ྻ {Sm } ͕ S∞ ʹऩଋ͠, ਺ྻ {Sm } ʹม׵ T Λࢪͯ͠Ͱ͖Δ਺ྻ {Tm } ͕ T∞ ʹऩଋ͢Δͱ͢Δ. ͜ͷͱ͖, lim m→∞ Tm − T∞ Sm − S∞ = 0 ͕੒Γཱͭͱ͖, ม׵ T ͸਺ྻ {Sm } ͷऩଋΛՃ଎͍ͤͯ͞Δͱ͍͏. ۚઘେհ (ؙ໺ݚڀࣨ OB) Ճ଎๏ͱՄੵ෼ํఔࣜ 2019 ೥ 6 ݄, ˏദ໦ݚڀࣨ 6 / 20
7. Ճ଎๏ͷྺ࢙ Ճ଎๏ͷྺ࢙ Ճ଎๏ͷྺ࢙ 17 ੈل: Ϥʔϩούͱ೔ຊͰݚڀ͕࢝·Δ. ೔ຊͰ͸ؔ޹࿨ͳͲ, ϤʔϩούͰ͸ Newton ͳͲ͕ݚڀͨ͠ a. 1926 ೥: Aitken Ճ଎ b 1956 ೥: ϵ ΞϧΰϦζϜ, ρ ΞϧΰϦζϜ 1959 ೥: η ΞϧΰϦζϜ 1990 ೥୅Ҏ߱: Ճ଎๏ͱՄੵ෼ํఔࣜͷؔ܎͕໌Β͔ʹͳͬͨ. ·ͨ, Մੵ෼ͳ ํఔ͔ࣜΒՃ଎๏Λ࡞Δ͜ͱ͕ࢼΈΒΕΔ. 2000 ೥୅Ҏ߱: q-ϵ ΞϧΰϦζϜͳͲͷ q-Ճ଎๏ c,d· · · a௕ా௚थ, ऩଋͷՃ଎๏ͷྺ࢙, ਺ཧղੳݚڀॴߨڀ࿥ୈ 1787 ר, 2012 ೥, 88-104 bࢁຊ఩࿕ (2003). ਺஋ղੳೖ໳ [૿గ൛], αΠΣϯεࣾ. cHe, Y., Hu, X. B., Tam, H. W. (2009). A q-Difference Version of the ϵ-Algorithm. Journal of Physics A: Mathematical and Theoretical, 42(9), 095202. dSun, J. Q., He, Y., Hu, X. B., Tam, H. W. (2011). q-difference and Confluent Forms of the Lattice Boussinesq Equation and the Relevant Convergence Acceleration Algorithms. Journal of Mathematical Physics, 52(2), 023522. ຊൃදͰ͸ 1990 ೥୅Ҏ߱ͷݚڀʹண໨͢Δ. ۚઘେհ (ؙ໺ݚڀࣨ OB) Ճ଎๏ͱՄੵ෼ํఔࣜ 2019 ೥ 6 ݄, ˏദ໦ݚڀࣨ 7 / 20
8. Ճ଎๏ͷྺ࢙ (ࢀߟ) ͦͷଞͷՃ଎๏ ຊൃදͰѻ͏ Aitken Ճ଎, ϵ ΞϧΰϦζϜ, ρ ΞϧΰϦζϜ, η ΞϧΰϦζϜҎ֎ ʹ΋࣍ͷՃ଎๏͕஌ΒΕ͍ͯΔ 9 . Ұ࣍༗ཧม׵ y = a + bx c + dx . Euler ม׵ an = 1 2n+1 n ∑ j=0 ( n j ) aj . 9෱ౡొࢤ෉, ਺஋ఱจֶೖ໳ -ఱจֶͰ༻͍Δ਺஋ٕ๏- chiron.mtk.nao.ac.jp/˜ toshio/education/numerical mono 2006.pdf ۚઘେհ (ؙ໺ݚڀࣨ OB) Ճ଎๏ͱՄੵ෼ํఔࣜ 2019 ೥ 6 ݄, ˏദ໦ݚڀࣨ 8 / 20
9. Ճ଎๏ͷ਺஋࣮ݧ η ΞϧΰϦζϜ มܗલͷ਺ྻΛ {cm } ͱͯ͠, η(m) 0 = ∞ (i.e. 1/η(m) 0 = 0), η(m) 1 = cm , m = 0, 1, 2, · · · ,      η(m) 2n+1 + η(m) 2n = η(m+1) 2n + η(m+1) 2n−1 1 η(m) 2n+2 + 1 η(m) 2n+1 = 1 η(m+1) 2n+1 + 1 η(m+1) 2n ͱ͍͏نଇʹैͬͨ਺ྻͷม׵Λ η ΞϧΰϦζϜͱ͍͏. c′ n = η(0) n (n = 1, 2, · · · ) ͕ม׵ޙͷ਺ྻͰ͋Δ. ۚઘେհ (ؙ໺ݚڀࣨ OB) Ճ଎๏ͱՄੵ෼ํఔࣜ 2019 ೥ 6 ݄, ˏദ໦ݚڀࣨ 9 / 20
10. Ճ଎๏ͷ਺஋࣮ݧ ߦྻࣜղͷಋग़      η(m) 2n+1 + η(m) 2n = η(m+1) 2n + η(m+1) 2n−1 1 η(m) 2n+2 + 1 η(m) 2n+1 = 1 η(m+1) 2n+2 + 1 η(m+1) 2n ͜ͷํఔࣜ͸ߦྻࣜղΛ࣋ͭ. ߦྻࣜղ͕͜ͷํఔࣜΛຬͨ͢͜ͱͷ֬ೝʹ͸࣍ ͷ߃౳ࣜΛ༻͍Δ. Jacobi ߃౳ࣜ தଜՂਖ਼. (2000). Մੵ෼ܥͷԠ༻਺ཧ, ী՚๪. ೚ҙͷߦྻࣜ D ʹରͯ͠ҎԼ͕੒Γཱͭ. D × D [ i j k l ] − D [ i k ] × D [ j l ] + D [ i l ] × D [ j k ] = 0. ۚઘେհ (ؙ໺ݚڀࣨ OB) Ճ଎๏ͱՄੵ෼ํఔࣜ 2019 ೥ 6 ݄, ˏദ໦ݚڀࣨ 10 / 20
11. Ճ଎๏ͷ਺஋࣮ݧ Ճ଎๏ͷ਺஋࣮ݧ log 2 ∈ [0.69314718055994528, 0.69314718055994529] Ͱ͋Δ 10. Sm = 1 − 1/2 + 1/3 − · · · + (−1)m/(m + 1) Λม׵͢Δͱ, 1 − 1/3 + 1 30 − 1 130 + 1 975 − 1 4725 + 1 32508 · · · ͱͳΔ. ࣮ݧ݁Ռ ม׵લͷ࿨ (࠷ॳͷ 7 ߲):0.7595 · · · ม׵ޙͷ࿨ (࠷ॳͷ 7 ߲):0.693152 · · · ม׵ޙʢՃ଎ޙʣͷํ͕ਅ஋ͷଘࡏൣғʹ͍͍ۙͮͯΔ͜ͱ͕Θ͔Δ. 10ദ໦խӳ, kv - C++ʹΑΔਫ਼౓อূ෇͖਺஋ܭࢉϥΠϒϥϦ http://verifiedby.me/kv/index.html ۚઘେհ (ؙ໺ݚڀࣨ OB) Ճ଎๏ͱՄੵ෼ํఔࣜ 2019 ೥ 6 ݄, ˏദ໦ݚڀࣨ 11 / 20
12. Ճ଎๏ͷ਺஋࣮ݧ ϵ ΞϧΰϦζϜͱ ρ ΞϧΰϦζϜ ϵ(m) 0 = 0, ϵ(m) 1 = Sm (given), m = 0, 1, 2, · · · , ( ϵm n+1 − ϵm+1 n−1 ) ( ϵm+1 n − ϵm n ) = 1 ͱ͍͏نଇʹैͬͨ਺ྻͷม׵Λ ϵ ΞϧΰϦζϜͱ͍͏. ϵ(m) 2n+1 (m = 0, 1, · · · ) ͕ม׵ޙͷ਺ྻͰ ͋Δ. Mathematica ͷ NSum, NLimit Ͱ΋࢖ΘΕ͍ͯΔ. Weisstein, Eric W. ”Wynn’s Epsilon Method.” From MathWorld–A Wolfram Web Resource. http://mathworld.wolfram.com/WynnsEpsilonMethod.html ρ(m) 0 = 0, ρ(m) 1 = Sm (given), m = 0, 1, 2, · · · , ( ρm n+1 − ρm+1 n−1 ) ( ρm+1 n − ρm n ) = n ͱ͍͏਺ྻͷม׵Λ ρ ΞϧΰϦζϜͱ͍͏. ρ(m) 2n+1 (m = 0, 1, · · · ) ͕ม׵ޙͷ਺ྻͰ͋Δ. ϓϩάϥϜྫ͸ԼهΛࢀর (௕ాઌੜˏ౦ژঁࢠେֶͷϖʔδ): http://www.cis.twcu.ac.jp/ osada/rikei/2009-1.html 10ϵ ΞϧΰϦζϜΛ࡞ͬͨ Wynn ͷݚڀۀ੷ʹ͍ͭͯ͸࣍Λࢀর: Brezinski, C. (2019). Reminiscences of Peter Wynn, Numerical Algorithms. ۚઘେհ (ؙ໺ݚڀࣨ OB) Ճ଎๏ͱՄੵ෼ํఔࣜ 2019 ೥ 6 ݄, ˏദ໦ݚڀࣨ 12 / 20
13. Ճ଎๏ͷ਺஋࣮ݧ ϵ ΞϧΰϦζϜͱ཭ࢄ potential KdV ํఔࣜ ϵ ΞϧΰϦζϜ ϵ(m) 0 = 0, ϵ(m) 1 = Sm (given), m = 0, 1, 2, · · · , ( ϵm n+1 − ϵm+1 n−1 ) ( ϵm+1 n − ϵm n ) = 1 ཭ࢄ potential KdV ํఔࣜ Hietarinta, J., Joshi, N., Nijhoff, F. W. (2016). Discrete Systems and Integrability (Vol. 54). Cambridge University Press. ( wm+1 n+1 − wm n ) ( wm+1 n − wm n+1 ) = 1 ϵ ΞϧΰϦζϜͱ཭ࢄ potential KdV ํఔࣜͷैଐม਺͸ͦΕͧΕਖ਼ํܗ, ඛܕ ͷ֨ࢠΛ੒͕͢, ద੾ͳม਺ม׵Λߟ͑Ε͹͜ͷ 2 ͭΛಉҰࢹͰ͖Δ. ۚઘେհ (ؙ໺ݚڀࣨ OB) Ճ଎๏ͱՄੵ෼ํఔࣜ 2019 ೥ 6 ݄, ˏദ໦ݚڀࣨ 13 / 20
14. Ճ଎๏ͷ਺஋࣮ݧ Ճ଎๏͸ສೳ͔? ऩଋ཰ͱऩଋͷ෼ྨ தଜՂਖ਼. (2000). Մੵ෼ܥͷԠ༻਺ཧ, ী՚๪. S∞ ʹऩଋ͢Δ਺ྻ {Sm} ͕ lim m→∞ Sm+1 − S∞ Sm − S∞ = κ Λຬͨ͢ͱ͖, κ Λऩଋ཰ͱ͍͏. κ ∈ [−1, 1)\{0} ͷͱ͖ઢܗऩଋ, .κ = 1 ͷͱ͖ର਺ऩଋ͢Δͱ͍͏. Aitken Ճ଎͸ઢܗऩଋ਺ྻΛՃ଎Ͱ͖Δ͕, .p(> 1) ࣍ऩଋ਺ྻʹద༻͢΂͖Ͱͳ͍. ࢁຊ఩࿕ (2003). ਺஋ղੳೖ໳ [૿గ൛], αΠΣϯεࣾ. Reich, S. (1970). On Aitken’s ∆3-Method. The American Mathematical Monthly, 77(3), 283-284. ϵ ΞϧΰϦζϜ͸ઢܗऩଋ਺ྻΛՃ଎Ͱ͖Δ͕, ର਺ऩଋ਺ྻΛՃ଎Ͱ͖ͳ͍. தଜՂਖ਼. (2000). Մੵ෼ܥͷԠ༻਺ཧ, ী՚๪. Aitken Ճ଎, ϵ ΞϧΰϦζϜͰՃ଎Ͱ͖ͳ͍਺ྻΛՃ଎͢ΔͨΊʹ͸, ଞͷՃ଎๏͕ඞཁʹͳΔ. ۚઘେհ (ؙ໺ݚڀࣨ OB) Ճ଎๏ͱՄੵ෼ํఔࣜ 2019 ೥ 6 ݄, ˏദ໦ݚڀࣨ 14 / 20
15. Ճ଎๏ͷ਺஋࣮ݧ ର਺ऩଋ਺ྻʹ͍ͭͯ Theorem (Delahaye, Germain-Bonne) ೚ҙͷର਺ऩଋ਺ྻΛՃ଎ͤ͞ΔՃ଎๏͸ͳ͍. → ର਺ऩଋҎ֎ͷ৚͕݅ඞཁ. Theorem (Gray-Clark, ର਺ऩଋͷ൑ఆ๏) ࣮୯ௐ਺ྻ Sn ͕ s ʹऩଋͯ͠, lim n→∞ ∆Sn+1 ∆Sn = 1 (∆: લਐࠩ෼) Λຬͨ͢ͳΒ, Sn ͸ s ʹର਺ऩଋ͢Δ. Gray, H. L., Clark, W. D., On a Class of Nonlinear Transformations and their Applications to the Evaluation of Infinite Series, Journal of Research of the National Bureau of Standards, 73B (1969), 251-274. ۚઘେհ (ؙ໺ݚڀࣨ OB) Ճ଎๏ͱՄੵ෼ํఔࣜ 2019 ೥ 6 ݄, ˏദ໦ݚڀࣨ 15 / 20
16. Ճ଎๏ͷ਺஋࣮ݧ Ճ଎๏ͷ਺஋࣮ݧ x(m) 0 = 0, x(m) 1 = Sm (given), m = 0, 1, 2, · · · , ( xm n+1 − xm+1 n−1 ) ( xm+1 n − xm n ) = σ(n + m) − σ(m) ͱ͍͏نଇʹैͬͨ਺ྻͷม׵Λ Thiele ͷ ρ ΞϧΰϦζϜͱ͍͏. x(0) 2n+1 (n = 1, 2, · · · ) ͕ม׵ޙͷ਺ྻͰ͋Δ. σ(x) = x ͱஔ͚͹͜Ε͸௨ৗ ͷ ρ ΞϧΰϦζϜʹ໭Δ. ∞ ∑ k=1 1 k3/2 , ∫ 1 0 1 (0.05 + t)1/2 dt ͷ 2 ͭʹ͍ͭͯ, ρ ΞϧΰϦζϜΛ࢖ͬͨ࣌ͱ Thiele ͷ ρ ΞϧΰϦζϜΛ࢖ͬͨ ࣌ͷൺֱ࣮ݧΛߦ͏. ۚઘେհ (ؙ໺ݚڀࣨ OB) Ճ଎๏ͱՄੵ෼ํఔࣜ 2019 ೥ 6 ݄, ˏദ໦ݚڀࣨ 16 / 20
17. Ճ଎๏ͷ਺஋࣮ݧ ࣮ݧख๏ Thiele ͷ ρ ΞϧΰϦζϜͰ͸࣍ͷΑ͏ʹͳΔڃ਺ΛՃ଎͢Δ. Sm ∼ S∞ + ∑ k≥1 ck (σ(m))k Sm = ∑ m k=1 1 k3/2 ʹ͍ͭͯͷ࣮ݧ ͜ͷ৔߹, Sm ∼ S∞ + ∑ k≥1 ck ( √ m)k ͱͳΔͷͰ, Thiele ͷ ρ ΞϧΰϦζϜʹ͓͍ͯ σ(x) = √ x ͱ͓͍࣮ͯݧ͢Δ. Mathematica ʹΑΔͱۙࣅ஋͸ҎԼͷͱ͓Γ: ∞ ∑ k=1 1 k3/2 = 2.612375348685488 . ۚઘେհ (ؙ໺ݚڀࣨ OB) Ճ଎๏ͱՄੵ෼ํఔࣜ 2019 ೥ 6 ݄, ˏദ໦ݚڀࣨ 17 / 20
18. Ճ଎๏ͷ਺஋࣮ݧ ࠓͷ৔߹, ఆ਺ ck ͸࣍ͷنଇʹΑܾͬͯ·Δ. ҰൠԽௐ࿨਺ͷ઴ۙల։ Gourdon, X., Sebah, P. (2003). Numerical Evaluation of the Riemann Zeta-Function. http://numbers.computation.free.fr/Constants/Miscellaneous/zetaevaluations.pdf ζ(s) ∼ Hs m + ∑ k≥−1 Bk+1(s)k (k + 1)!ms+k as m → ∞ Hs m ͸ҰൠԽௐ࿨਺Ͱ, Hs m := m ∑ k≥1 k−s Ͱ͋Δ. ·ͨ, ζ(s) = ∑ k≥1 k−s (Zeta ؔ਺) Ͱ͋Δ. (s)k := s(s + 1) · · · (s + k − 1), (Pochhammer ه߸). Definition (Bernoulli ਺) Bn ͸ Bernoulli ਺ͱΑ͹Ε, ࣍ͷ઴ԽࣜͰܾ·Δ. B0 := 1, Bn := − 1 n + 1 n−1 ∑ k=0 ( n + 1 k ) Bk (n ≥ 1). Weisstein, Eric W. ”Bernoulli Number.” From MathWorld–A Wolfram Web Resource. http://mathworld.wolfram.com/BernoulliNumber.html ۚઘେհ (ؙ໺ݚڀࣨ OB) Ճ଎๏ͱՄੵ෼ํఔࣜ 2019 ೥ 6 ݄, ˏദ໦ݚڀࣨ 18 / 20
19. Ճ଎๏ͷ਺஋࣮ݧ ∫ 1 0 1 (0.05 + t)1/2 dt Λܭࢉ͢Δࡍ, Sm = 1 m { 1 2 g(0) + m−1 ∑ k=1 g ( k m ) + 1 2 g(1) } , g(t) = 1 (0.05 + t)1/2 ͱ͓͘ͱ, Euler-Maclaurin ͷެࣜʹΑΓ, Sm ∼ S∞ + ∑ k≥1 dk m2k ͱ͍͏ܗʹͳΔͷͰ Thiele ͷ ρ ΞϧΰϦζϜͰ σ(x) = x2 ͱ࣮ͯ͠ݧ͢Δ. Euler-Maclaurin ͷެࣜ ؔ਺ f(x) ͕۠ؒ [a, b] Ͱ C2 ڃͱ͠, h := (a − b)/n ͱ͓͘. ͜ͷͱ͖, ∫ b a f(x)dx ∼ h { 1 2 f(a) + m−1 ∑ k=1 f(a + kh) + 1 2 f(b) } + m−1 ∑ k=1 B2k (2k)! h2k { f(2k−1)(a)−f(2k−1)(b) } ࢁຊ఩࿕ (2003). ਺஋ղੳೖ໳ [૿గ൛], αΠΣϯεࣾ. kv ϥΠϒϥϦʹΑΔͱਅ஋ͷଘࡏൣғ͸࣍ͷ௨Γ: ∫ 1 0 1 (0.05 + t)1/2 dt ∈ [1.6021765576808946, 1.6021765577247085]. ۚઘେհ (ؙ໺ݚڀࣨ OB) Ճ଎๏ͱՄੵ෼ํఔࣜ 2019 ೥ 6 ݄, ˏദ໦ݚڀࣨ 19 / 20
20. Ճ଎๏ͷ਺஋࣮ݧ ࣮ݧख๏ (ਫ਼౓อূ෇͖਺஋ੵ෼) ਫ਼౓อূ෇͖਺஋ܭࢉ ۙࣅ஋ܭࢉͱಉ࣌ʹܭࢉ݁ՌͷޡࠩධՁ΋ߦ͏਺஋ܭࢉͷ͜ͱ. ਅ஋ΛؚΉ۠ؒΛग़ྗ͢Δ. େੴਐҰ et al. (2018). ਫ਼౓อূ෇͖਺஋ܭࢉͷجૅ, ίϩφࣾ. େੴਐҰ (2000). ਫ਼౓อূ෇͖਺஋ܭࢉ, ίϩφࣾ. Tucker, W. (2011). Validated numerics: a short introduction to rigorous computations. Princeton University Press. ”kv ϥΠϒϥϦ”ʹΑΔਫ਼౓อূ෇͖਺஋ੵ෼ͷྲྀΕ ദ໦խӳ, ϕΩڃ਺ԋࢉʹ͍ͭͯ, http://verifiedby.me/kv/psa/psa.pdf ੵ෼۠ؒΛ෼ׂ͢Δ (࣮ݧͰ͸ 20 ݸʹ෼ׂ) ⇓ ඃੵ෼ؔ਺ f ʹରͯ͠৒༨߲෇͖ Taylor ల։Λߦ͏ ⇓ ֤۠෼Ͱ f ͷ૾Λ܎਺͕۠ؒͰ͋Δଟ߲ࣜͱͯ͠ಘΔ (࣮ݧͰ͸ 20 ࣍ʹࢦఆ) ⇓ ֤۠෼ͰಘΒΕͨଟ߲ࣜΛෆఆੵ෼ͯ͠ݪ࢝ؔ਺ΛಘΔ ⇓ ֤۠෼Ͱ۠ؒ୺ͷ஋Λ୅ೖͯ͠ఆੵ෼ͷ஋Λ۠ؒͱͯ͠ಘΔ ۚઘେհ (ؙ໺ݚڀࣨ OB) Ճ଎๏ͱՄੵ෼ํఔࣜ 2019 ೥ 6 ݄, ˏദ໦ݚڀࣨ 20 / 20
21. Ճ଎๏ͷ਺஋࣮ݧ PGR ๏ PGR ๏͸ singularity confinement ͔Βಋग़͞ΕΔ. Definition (singularity confinement) ॳظ஋ʹґଘ͢Δಛҟ఺͕༗ݶճͷεςοϓͰଧͪফ͞Εͯͳ͘ͳΓ, ಛҟ఺Λ ௨աͯ͠΋ॳظ஋ͷ৘ใ͕࢒Δ͜ͱΛ singularity confinement ͱ͍͏. → ͔ͭͯ͸཭ࢄํఔࣜͷՄੵ෼ੑ൑ఆʹ࢖ΘΕ͍ͯͨ. Grammaticos B., Ramani A., Papageorgiou V. (1991). Do Integrable Mappings Have the Painlev´ e Property ?, Physical Review Letters, 67, 1825. ஫ҙ (Hietarinta-Viallet, 1998) Singularity confinement Ͱ͋ͬͯ΋ํఔ͕ࣜՄੵ෼ੑΛ࣋ͭͱ͸ݶΒͳ͍. Hietarinta, J., Viallet, C. (1998). Singularity Confinement and Chaos in Discrete Systems. Physical Review Letters, 81(2), 325. 10Hietarinta-Viallet (1998) Λܖػʹ୅਺తΤϯτϩϐʔ (Falqui-Viallet, 1993) Ͱ཭ࢄํఔࣜͷ Մੵ෼ੑΛ൑ఆ͠Α͏ͱ͍͏ݚڀ͕ΑΓ੝Μʹͳͬͨ. ͳ͓୅਺తΤϯτϩϐʔ͸۠ؒԋࢉʹ͓͚Δ ۠ؒ෯ͷ૿େͱؔ܎͕͋ΔͷͰ͸ͳ͍͔ͱ͍ΘΕ͍ͯΔ. ۚઘେհ (ؙ໺ݚڀࣨ OB) Ճ଎๏ͱՄੵ෼ํఔࣜ 2019 ೥ 6 ݄, ˏദ໦ݚڀࣨ 21 / 20
22. Ճ଎๏ͷ਺஋࣮ݧ Ճ଎๏ͷԠ༻ Steffensen ൓෮ Aitken Ճ଎ͷԠ༻Ͱ, x = g(x) ͷࠜΛٻΊΔ൓෮๏Ͱ͋Δ. ॳظ஋Λ͏·͘ͱΕ͹ 1 ࣍ऩଋ͢Δ (g(x) ͕ C1 ڃͳΒ௒̍࣍ऩଋ, C2 ڃͳΒ 2 ࣍ऩଋ͢Δ.). ࢁຊ఩࿕ (2003). ਺஋ղੳೖ໳ [૿గ൛], αΠΣϯεࣾ. Romberg ੵ෼ (Romberg, 1955) ୆ܗެࣜͱՃ଎๏Λ૊Έ߹Θͤͨ਺஋ੵ෼๏Ͱ͋Δ. Bauer-Rutishauser-Stiefel (1963) ʹΑΓৄࡉ ͳղੳ͕ͳ͞Ε͍ͯΔ. Romberg, W. (1955). Vereinfachte numerische integration. Norske Vid. Selsk. Forh., 28, 30-36. ࢁຊ఩࿕ (2003). ਺஋ղੳೖ໳ [૿గ൛], αΠΣϯεࣾ. େੴਐҰ et al.(2018) ਫ਼౓อূ෇͖਺஋ܭࢉͷجૅ, ίϩφࣾ. ಛघؔ਺΁ͷԠ༻ ڃ਺ͷऩଋՃ଎ʹΑͬͯෳૉؔ਺ΛΑΓ޿͍ྖҬͰܭࢉͰ͖Δ (1 छͷղੳ઀ଓ) → ޡࠩؔ਺, ෆ׬શ gamma ؔ਺ͳͲͷಛघؔ਺ͷ਺஋ܭࢉʹԠ༻͞ΕΔ. ৿ਖ਼෢ (2000), ղੳ઀ଓͱڃ਺ͷऩଋͷՃ଎, ਺ཧղੳݚڀॴߨڀ࿥, 1155, 104-119. ۚઘେհ (ؙ໺ݚڀࣨ OB) Ճ଎๏ͱՄੵ෼ํఔࣜ 2019 ೥ 6 ݄, ˏദ໦ݚڀࣨ 22 / 20
23. ·ͱΊ ·ͱΊ ·ͱΊ Ճ଎๏͔ΒՄੵ෼ํఔ͕ࣜಘΒΕΔ͜ͱ͕෼͔ͬͨ. Մੵ෼ํఔ͕ࣜऩଋΛՃ଎ͤ͞Δͱ͸ݶΒͳ͍ͱ͍͏͜ͱ͕෼͔ͬͨ Ճ଎๏ʹؔ͢Δࢀߟจݙ N. Osada (1993) Acceleration Methods for Slowly Convergent Sequences and their Applications, PhD Thesis. Brezinski, C., & Redivo-Zaglia, M. (2019). The genesis and early developments of Aitken’s process, Shanks transformation, the ϵ-algorithm, and related fixed point methods. Numerical Algorithms, 80(1), 11-133. (ࢀߟ) ࠷ۙͷݚڀ Brezinski, C., He, Y., Hu, X. B., Sun, J. Q., & Tam, H. W. (2011). Confluent Form of the Multistep ϵ-Algorithm, and the Relevant Integrable System. Studies in Applied Mathematics, 127(2), 191-209. Chang, X. K., He, Y., Hu, X. B., & Li, S. H. (2018). A new integrable convergence acceleration algorithm for computing Brezinski-Durbin-Redivo-Zaglia’s sequence transformation via Pfaffians. Numerical Algorithms, 1-20. ۚઘେհ (ؙ໺ݚڀࣨ OB) Ճ଎๏ͱՄੵ෼ํఔࣜ 2019 ೥ 6 ݄, ˏദ໦ݚڀࣨ 23 / 20