Taylor展開法による常微分方程式の数値計算の性能

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(1)2006−HPC−108（11） 2006／10／6. 社団法人情報処理学会研究報告 IPSJ SIG Technical Report. 6C[NQT ዷ㐿ᴺߦࠃࠆᏱᓸಽᣇ⒟ᑼߩᢙ୯⸘▚ߩᕈ⢻ *ᐔጊ ᒄ‫⷏*ޔ‬Ꮉ ⍍‫⊕**ޔ‬⍹ ᢥ⑲ *␹ᄹᎹᎿ⑼ᄢቇᎿቇㇱ **਻Ꮊᄢቇࡃࠗࠝࠕ࡯ࠠ࠹ࠢ࠴ࡖ࡯࠮ࡦ࠲࡯ ᦨㄭᄢ߈ߥࠪࠬ࠹ࡓߩ㐳ᤨ㑆㜞♖ᐲ⸘▚ᴺߪ㕖Ᏹߦ㊀ⷐߢ޽ࠆ‫ޕ‬છᗧᰴᢙนᄌࠬ࠹࠶ࡊߩ Taylor ዷ㐿ᴺߪߎߩࠃ߁ߥ໧㗴⸃᳿ߦㆡߔࠆᢙ୯⸘▚ᴺߢ޽ࠆ‫ޕ‬ ᧄ⺰ᢥߢߪ Taylor ዷ㐿ᴺࠍᄤᢥቇߦ⃻ࠇࠆࡇ࠲ࠧ࡜ࠬ㧟૕໧㗴ߦㆡ↪ߒ‫♖▚⸘ޔ‬ᐲ‫ᤨ▚⸘ޔ‬ 㑆ࠍ⺞ߴ‫ߩߎޔ‬ᣇᴺߩ᦭ലᕈࠍ⺞ߴߚ‫ޕ‬. Performance of Numerical Computation of the Differential Equations by Taylor Series *Hiroshi HIRAYAMA, *Shun NISHIKAWA, **Fumihide SHIRAISHI *Faculty of Engineering, Kanagawa Institute of Technology **Bio-Architecture Center, Kyushu University Recently, the highly-reliable calculation method for the time courses of the dependent variables in large-scale network systems is important. The arbitrary-order and variable-step Taylor series method is a good numerical calculation method for these problems. In the present paper, we apply the Taylor series method to the Pythagorean problem of three bodies, and examine the calculation accuracy and the computation time to elucidate the performance of the present method.. 1 ߪߓ߼ߦ. 䌹. ᦨㄭ‫ޔ‬Ᏹᓸಽᣇ⒟ᑼߢ⸥ㅀߐࠇߚᄢⷙᮨߥࠪ. 㪋. 㫄㪊䋨㪈㪃㪊䋩. 㪊. ࠬ࠹ࡓࠍ㐳ᤨ㑆ߦᷰࠅ㜞♖ᐲߢ⸘▚ߔࠆߎߣ. 㪉. ߇᳞߼ࠄࠇߡ޿ࠆ‫ߢ߹ࠇߎޕ‬Ᏹᓸಽᣇ⒟ᑼߩᢙ 㪈. ୯⸃ᴺߪ Euler ᴺ߿ Runge-Kutta ᴺߦࠃࠆᏅ ಽㄭૃࠍ೑↪ߒߚ⸘▚ᴺࠍ೑↪ߔࠆߎߣ߇৻. 䌸㪇㪄㪉. 㪄㪈. ⥸⊛ߣߐࠇߡ߈ߚ‫ߩࠄࠇߎޕ‬ᣇᴺߢߪ⸘▚ᰴᢙ. 㪇. ߊߥࠆ‫ߥ߁ࠃߩߎޕ‬໧㗴ߦኻߒ‫ޔ‬છᗧᰴᢙ‫ޔ‬น ᄌࠬ࠹࠶ࡊߩ⸘▚ᣇᴺߢ޽ࠆ Taylor ዷ㐿ᴺ߇. 1 −61−. 㪉. 㪄㪈. 㫄㪌㩿㪈㪃㪄㪈㪀. 㫄㪋㩿㪄㪈㪃㪄㪈㪀. ߇㒢ቯߐࠇ㐳ᤨ㑆㜞♖ᐲߩ⸘▚ߪ㕖Ᏹߦ㔍ߒ. 㪈. 㪄㪉. ࿑㧝㧚ࡇ࠲ࠧ࡜ࠬߩਃ૕໧㗴ߩೋᦼ⁁ᘒ.

(2) ᦨ߽ㆡߒߡ޿ࠆߣᕁࠊࠇࠆ‫⺰ᧄޕ‬ᢥߢߪ‫ߣ߭ޔ‬. ߢ޽ࠆ‫ޕ‬. ߟߩ଀ߣߒߡ‫ޔ‬ᄤᢥቇ਄ߩਃ૕໧㗴ࠍᓥ᧪ߩ. ૶↪ߒߚ⸘▚ᣇᴺߪ‫ޔ‬ㅢᏱࠃߊ૶ࠊࠇࠆ 4. Runge-Kutta ᴺߣ Taylor ዷ㐿ᴺࠍ૶ߞߡ⸃߈‫ ޔ‬ᰴ ߩ Runge-Kutta ᴺ [6] ‫ ޔ‬8 Ბ 6 ᰴ ߩ ⸘▚♖ᐲ‫ᤨ▚⸘ޔ‬㑆ࠍ⺞ߴ‫ޔ‬Taylor ዷ㐿ᴺߩ. Runge-Kutta ᴺ[2]߅ࠃ߮ Taylor ዷ㐿ᴺ[4]ߢ. ᕈ⢻ࠍ⺞ߴߚ‫ޕ‬. ޽ࠆ‫ߡߒ↪૶ࠍࠄࠇߎޕ‬᳞߼ߚ⸘▚⚿ᨐࠍᲧセ. ⸘▚ߔࠆᏱᓸಽᣇ⒟ᑼߣߒߡ‫߇ߩߊ⸃ޔ‬㔍ߒ. ᬌ⸛ߒߚ‫ޕ‬. ޿ߣߐࠇࠆࡇ࠲ࠧ࡜ࠬߩਃ૕໧㗴[5]ࠍㆬᛯߒ. 2 ⸘▚ᣇᴺ. ߚ‫ߩࠬ࡜ࠧ࠲ࡇޕ‬ਃ૕໧㗴ߣߪ‫ޔ‬ㄝ㐳 3,4,5 ߩ. 2.1 Runge-Kutta ᴺ. ⋥ⷺਃⷺᒻߩ㗂ὐߩ૏⟎ߦ‫ߩࠇߙࠇߘޔ‬ኻㄝ㐳. 㧠ᰴߩ Runge-Kutta ᴺ㧔એ㒠 RK ᴺߣ⇛ߔ㧕. ߦᲧ଀ߔࠆ⾰㊂ 3,4,5 ߩ⾰ὐࠍ࿑㧝ߩࠃ߁ߦ㕒. ߢߪߤߩ⒟ᐲࠬ࠹࠶ࡊ᏷ࠍขࠆߴ߈߆ࠍ੐೨. ᱛ⁁ᘒߢ㈩⟎ߒ‫⁁ߩߘޔ‬ᘒࠍೋᦼ᧦ઙߣߒߡ‫ޔ‬. ߦ್ᢿߔࠆߎߣ߇㔍ߒ޿ߩߢ‫࠹ࠬߩ߆ߟߊ޿ޔ‬. ߎࠇࠄߩ⾰ὐ߇⋧੕ߩᒁജߦࠃߞߡ‫ߩߎޔ‬ᓟߤ. ࠶ࡊ᏷ࠍ↪ᗧߒߘࠇߙࠇ⸘▚ߒ‫♖ߩߘޔ‬ᐲࠍ⏕. ߁ㆇേߔࠆ߆ࠍㅊ෸ߔࠆ໧㗴ߢ޽ࠆ‫ޕ‬. ⹺ߒߚ‫ޕ‬. ⾰ὐ m3 ‫ ޔ‬m 4 㑆ߩ〒㔌ࠍ r34 ߣ޿߁ࠃ߁ߦቯ. ૶↪ߒߚࠬ࠹࠶ࡊ᏷ 't ߪ‫ޔ‬. ⟵ߔࠆߣ‫ޔ‬. r34. ( x3 x 4 ) ( y 3 y 4 ). r35. ( x3 x5 ) 2 ( y 3 y 5 ) 2. r45. ( x 4 x5 ) 2 ( y 4 y 5 ) 2. 2. 't. 2. (1.1). ߢ޽ࠆ‫ޕ‬ 8 Ბ 6 ᰴߩ RK ᴺߢ߽ 4 ᰴߩ႐วߣหߓࠬ࠹ ࠶ࡊ᏷ࠍ૶↪ߒߚ‫ޕ‬ 2.2 Taylor ዷ㐿ᴺ. ߎࠇࠄࠍ↪޿ߡㆇേᣇ⒟ᑼࠍᦠߊߣ‫ޔ‬. 4( x3 x 4 ) 5( x3 x5 ) r343 r353. Taylor ዷ㐿ᴺߢߪ૗ᰴ߹ߢዷ㐿ߒߡ߽ࠃ޿. 4( y 3 y 4 ). છᗧᰴᢙߩᢙ୯⸘▚ᴺࠍ૶ߞߡ߽୚♖ᐲߩ⸘. x3". . y 3". . x 4". 3( x3 x 4 ) 5( x 4 x5 ) r343 r453. r. 3 34. . y 4". 3( y 3 y 4 ) 5( y 4 y 5 ) r343 r453. x5". 3( x3 x5 ) 4( x 4 x5 ) r353 r453. y 5". 3( y 3 y 5 ) 4( y 4 y 5 ) r353 r453. x3. 1‫ޔ‬x. x4. 2‫ޔ‬x 4'. x5. 1‫ޔ‬x. ' 3. ' 5. ߇‫ ߪߢߎߎޔ‬20 ᰴߩ Taylor ዷ㐿ᑼࠍ૶ߞߚ‫ޕ‬. 5( y 3 y 5 ) r353. ߣߥࠆ‫ ޕ‬ೋᦼ୯ߪ t. . 10 4 ,10 5 ,10 6 ,10 7. ࠇࠆߎߣ߇ᄙ޿ߩߢ‫ ߪߢ▚⸘ᧄޔ‬20 ᰴߩ (1.2). . 0‫ޔ‬y 4. 3‫ޔ‬y. Taylor ዷ㐿ᑼࠍ೑↪ߔࠆߎߣߦߒߚ‫ޕ‬. f (t ). a 0 a1t a 2 t 2 㨯㨯㨯a 20 t 20. (2.1). ⸘▚ࠬ࠹࠶ࡊ᏷ 't ߪ‫ޔ‬Taylor ዷ㐿ᑼߩᦨᓟߩ 㗄߇୚♖ᐲߩ⸘▚ߢήⷞߢ߈ࠆ߶ߤዊߐ޿ߣ ߒߡ‫ࡊ࠶࠹ࠬޔ‬᏷ߩᄢ߈ߐࠍ᳿ቯߔࠆ‫ߣߩߎޕ‬ ߈ᚑࠅ┙ߟᑼߪ. 0 ߩߣ߈‫ޔ‬. 0‫ޔ‬y 3 0‫ޔ‬y 5. ▚ߢߪ 20 ᰴ⒟ᐲߩ౏ᑼߣ߶߷หߓ⚿ᨐ߇ᓧࠄ. ' 3. 0‫ޔ‬. 1‫ޔ‬y 4' 1‫ޔ‬y. a 20 ('t ) 20 10 17 a0. ' 5. 0‫ޔ‬ 0‫ޔ‬. 2 −62−. ߣߥࠆ‫ߩߎޕ‬ᑼ߆ࠄࠬ࠹࠶ࡊ᏷ 't ߩ୯ߪ. (2.2).

(3) ߘࠇ߶ߤᄢ߈ߥ♖ᐲᡷༀߪ⷗ࠄࠇߥ߆ߞߚ‫੹ޕ‬. a0 20 10 ࡮ a20. 't. 17. ࿁ߩ໧㗴ߢߪ‫ޔ‬Padé ዷ㐿ࠍ૶ࠊߥ޿ߢ߽ห⒟. (2.3). . ᐲߩ♖ᐲߢ⸘▚ߢ߈ࠆߩߢ‫ޔ‬Padé ዷ㐿ࠍ੹࿁. ߣᓧࠄࠇࠆ‫(ޕ‬2.2)ᑼߪ⋧ኻ⹏ଔᑼߢ޽ࠆ߇‫ޔ‬. ߩ⸘▚ߢߪ૶ࠊߥ߆ߞߚ‫ޕ‬. 㧔2.2㧕ߩಽᲣߩ a 0 ࠍ㧝ߦᄌ߃ߚ߽ߩ߇⛘ኻ୯. 3 ᢙ୯⸘▚⚿ᨐ. ⹏ଔᑼߦߥࠆ‫ޕ‬୚♖ᐲߢ⸘▚ߒߡ޿ࠆߩߢ‫⸘ޔ‬. 3.1 Runge-Kutta ᴺ. ▚♖ᐲߩ㒢⇇ࠃࠅዋߒዊߐ޿୯ 10. 17. ࠍ૶ߞߚ‫ޕ‬. Taylor ዷ㐿ߩ⸘▚ߢߪ(2.1)ߩዷ㐿ᑼࠍ A ቟. RK ᴺߢߩ⸃ᨆߪ 4 ᰴ‫ޔ‬8 Ბ 6 ᰴߣ߽ߦࠬ࠹ ࠶ࡊ᏷ࠍᄢ߈޿ᣇ߆ࠄ㗅ߦ߅ߎߥߞߚ‫ޕ‬. ቯൻ[3]ߒ‫ޔ‬㜞♖ᐲൻࠍ⸘ࠆߚ߼ߦ Padé ዷ㐿ࠍ. ࠬ࠹࠶ࡊ᏷ࠍᄢ߈ߊขߞߚ႐ว‫▚⸘ޔ‬ㅜਛߢ. ૶ߞߡ⸘▚ߔࠆߎߣ߇ߢ߈ࠆ‫ޕ‬. ⾰ὐ m 4 ‫ޔ‬m5 ߩᐳᮡ୯߇ᕆỗߦჇടߒ⛯ߌࠆߣ. Padé ዷ㐿ߣߪ‫ޔ‬Taylor ዷ㐿ᑼࠍ‫᦭ޔ‬ℂ㑐ᢙ. ޿߁ਇ⥄ὼߥ⚿ᨐࠍ␜ߒߚ‫߽ߣߞ߽ޕ‬ᄢ߈޿ࠬ. ߦᄌᒻߒߚ߽ߩߢ޽ࠆ‫ޕ‬. ࠹࠶ࡊ᏷ 't. p 0 p1 x 㨯㨯㨯 p M x M 1 q1 x 㨯㨯㨯 q L x L. a 0 a1 x 㨯㨯㨯. 㑆t. (2.4). 10 4 ߩ႐วߩ 4 ᰴߩ RK ᴺߢᤨ. (0䌾10) ߩ⸘▚ߒߚ⚿ᨐࠍࠣ࡜ࡈߢ⴫ߔ. ߣ࿑ 2 ߩࠃ߁ߦߥࠆ‫ޕ‬. (2.4)ᑼߩਔㄝߦฝㄝߩಽᲣࠍដߌ‫ ޔ‬M L ᰴ ߩଥᢙ߹ߢ৻⥌ߔࠆࠃ߁ߦ‫᦭ޔ‬ℂ㑐ᢙߩଥᢙࠍ. 㪊㪉㪅㪌. ᳿ቯߔࠆߎߣߦࠃߞߡ Padé ዷ㐿ᑼ߇ᓧࠄࠇࠆ‫ޕ‬. 㪉㪈㪅㪌. ߎߩߣ߈ߩ᧦ઙߪ‫ޔ‬ᰴߩࠃ߁ߦ⴫ߐࠇࠆ‫ޕ‬ m. al ¦ al k q k k 1. pl (l. l ߇ L ࠍ⿧߃ࠆߥࠄ‫ޔ‬m. 㪈㪇㪅㪌. 0,㨯㨯㨯, M ) (2.5) 㪄㪊. ߚߛߒ‫ޔ‬m ߪ‫ޔ‬l ߇ L એਅߥࠄ߫‫ޔ‬m. 0(l. 㪄㪈. l ߣߒ‫ޔ‬. 㪇㪄㪇㪅㪌㪇. 㪈. 㪉. 㪊. 㪄㪈㪅㪌. 䌴䋽㪈㪇㪄㪋. L ߣߔࠆ‫(ޕ‬2.5)ߣห. ࿑㧞㧚⾰ὐߩㆇേ ࿑㧞߇␜ߔࠃ߁ߦ‫⾰ޔ‬ὐ m 4 ‫ޔ‬m5 ߇ធㄭߒߚᤨ. ߦߪ‫ޔ‬ᰴߩ㑐ଥᑼ߇ᓧࠄࠇࠆ‫ޕ‬ l. 㪄㪉. 㪄㪈. ߓ㑐ଥᑼߢ޽ࠆ߇‫ ޔ‬l ߇ M ࠍ⿧߃ߡ޿ࠆ႐ว. al ¦ al k q k. 㫄㪊㫄㪋㫄㪌㫄㪊㐿ᆎὐ 㫄㪋㐿ᆎὐ 㫄㪌㐿ᆎὐ 㫄㪊⚳ੌὐ 㫄㪋⚳ੌὐ 㫄㪌⚳ੌὐ. 㪊㪅㪌. M 1,㨯㨯㨯, M L) (2.6). k 1. ὐࠍႺߦ‫⾰ߩࠇߙࠇߘޔ‬ὐߩㆇേ߇ਇ⥄ὼߦᄌ ൻߒߡ޿ࠆ‫ߩߎޕ‬ਃ૕໧㗴ߦ߅޿ߡ߽㊀ᔃ‫ޔ‬ㆇ. (2.6)ߩㅪ┙৻ᰴᣇ⒟ᑼࠍ⸃߈‫᦭ޔ‬ℂᑼߩಽᲣߩ. േ㊂‫ޔ‬ㆇേࠛࡀ࡞ࠡ࡯ߪ଻ሽߐࠇ‫৻ޔ‬ቯ୯ߩߪ. ଥ ᢙ ( q1 , q 2 ,㨯㨯㨯, q L ) ࠍ ᳿ ቯ ߒ ‫ ߩ ߘ ޔ‬ଥ ᢙ ࠍ. ߕߢ޽ࠆ߇‫⾰ޔ‬ὐ m 4 ‫ޔ‬m5 ߩᢙ୯߇ਇ⥄ὼߦᄌ. (2.5) ᑼ ઍ ౉ ߒ ߡ ‫ ޔ‬ಽ ሶ ߩ ଥ ᢙ. ൻߒᆎ߼ߚὐ t. ( p 0 , p1 , p 2 ,㨯㨯㨯, p M ) ࠍ᳞߼ࠆߎߣ߇ߢ߈ࠆ‫ޕ‬. ࠆߪߕߩㆇേࠛࡀ࡞ࠡ࡯ߩ୯߇‫ޔ‬t. Padé ዷ㐿ߪ‫ߦ⥸৻ޔ‬หߓᰴᢙߩ Taylor ዷ㐿. ߢ E=-12.817 ߆ࠄ E=-13.181 ߦᄌൻߒ‫ߎߎޔ‬. ᑼࠃࠅ㜞♖ᐲߢ‫ޔ‬෼᧤ߩ⦟޿ᑼࠍਈ߃ࠆߎߣ߇. ߹ߢ E=-13.181 ߢ৻ቯߢ଻ߚࠇߡ޿ߚ߽ߩ߇. ᄙ޿ߩߢ‫ޔ‬Ᏹᓸಽᣇ⒟ᑼߩ⸃ߩ Taylor ዷ㐿ᑼ. ߎߩᤨὐࠍႺߦ E=77763 ߦߥࠆߣ޿߁ਇ⥄ὼ. ࠍ Padé ዷ㐿ߔࠆߎߣߪ‫ޔ‬Ᏹᓸಽᣇ⒟ᑼࠍ㜞♖. ߥ୯ߦᄢ߈ߊᄌൻߒߡ޿ߚ‫ࠄ߆ߣߎޕ‬੐ታ਄ᢙ. ᐲߢ቟ቯ⊛ߦ⸘▚ߢ߈ࠆߣᦼᓙߢ߈ࠆ‫ޕ‬. ୯⸘▚ߦᄬᢌߒߡ޿ࠆߣ್ᢿߒ‫' ޔ‬t. ᧄ⸘▚ࠍ‫ޔ‬Padé ዷ㐿ࠍ૶ߞߡ⸘▚ߔࠆߣ⸘. ߩᢙ୯⸘▚ߪߎߎߢᛂߜಾߞߚ‫ߦ߁ࠃߩߎޕ‬ᄢ. ▚ᤨ㑆ߪ⚂㧞୚ߦߥࠆ‫♖▚⸘ޕ‬ᐲߪ⦟ߊߥࠆ߇. ߈ߊࠛࡀ࡞ࠡ࡯߇ᄌൻߒߚߦ߽߆߆ࠊࠄߕ‫ޔ‬㊀. 3 −63−. 8.6 ઃㄭߦ߅޿ߡ‫৻ޔ‬ቯߢ޽ 1.88 ઃㄭ. 10 4 ߢ.

(4) 10 7 ߩ႐วߩ⚿ᨐߢ޽ࠆ‫ޕ‬40 ᰴߩ Taylor. ᔃ߿ㆇേ㊂ߪ߶߷৻ቯߩ୯ࠍ଻ߞߡ޿ߚ‫ޕ‬. 't. ห ᭽ ߦ 㧤 Ბ 㧢 ᰴ ߩ RK ᴺ ߦ ߅ ޿ ߡ ߽ ‫ޔ‬. ዷ㐿ᑼࠍ૶޿‫ޔ‬㧠୚♖ᐲ㧔10 ㅴᢙߢ⚂ 35 ᩴ㧕. 't. ߩ ႐ ว ߢ ߪ t. 4. 1.88 ઃ ㄭ ߢ E=-12.817 ߆ࠄ E=-12.783 ߦᄌൻߒ‫ޔ‬t 15.8. ߢ⸘▚ߒߚ߽ߩࠍ⌀ߩ୯ߣߒߡ૶ߞߡ޿ࠆ‫ޕ‬. ઃㄭ߹ߢ E=-12.783 ߢ߶߷৻ቯߦ଻ߚࠇߡ޿. ᴺ‫ޔ‬Taylor ዷ㐿ᴺߣ߽ߦ⌀୯ߣߩ᦭ലᩴᢙ߇ 5. ߚ߽ߩ߇‫ᤨߩߎޔ‬ὐߢ E=100073 ߹ߢჇടߒ‫ޔ‬. ᩴએ਄ࠍ଻ߞߡ޿ࠆ߇ߘߩᓟ t ߇ 10 Ⴧടߔࠆ. ᣿ࠄ߆ߦ⸘▚ߦᄬᢌߒߡ޿ߚ‫߆ߣߎߩࠄࠇߎޕ‬. ߏߣߦ᦭ലᩴᢙ߇㧝ߟᷫࠆߣ޿ߞߚ⚿ᨐ߇⷗. ࠄ RK ᴺߢࡇ࠲ࠧ࡜ࠬߩਃ૕໧㗴ࠍ⸘▚ߔࠆ. ࠄࠇ‫ ޔ‬t. ႐ว‫ࡊ࠶࠹ࠬޔ‬᏷ࠍᄢ߈ߊߒߚ႐ว‫ޔ‬ᱜ⏕ߥ୯. Taylor ዷ㐿ᴺߤߜࠄߩ႐วߢ⸘▚ߒߚ⚿ᨐ߽. ࠍᓧࠆߎߣ߇࿎㔍ߢ޽ࠆߎߣ߇ࠊ߆ࠆ‫ࠃߩߎޕ‬. ⌀୯ߣߪ߆ߥࠅ⺋Ꮕ߇޽ࠆ‫ޕ‬RK ᴺߦࠃࠆ⚿ᨐ. ߁ߥ႐วߢ߽㊀ᔃ‫ޔ‬ㆇേ㊂ߪ߶߷৻ቯߦ଻ߞߡ. ߣ Taylor ዷ㐿ᴺߦࠃࠆ⚿ᨐࠍᲧߴߡߺࠆߣ. ޿ߚ‫ޕ‬. RK ᴺߦࠃߞߡ⸘▚ߒߚ⚿ᨐߩ߶߁߇⌀୯ߣ. ࿑㧞߆ࠄ߽ࠊ߆ࠆࠃ߁ߦ‫߇࡯ࠡ࡞ࡀࠛޔ‬ᄢ߈. ߩ⺋Ꮕ߇ᄢ߈޿ߎߣ߇ࠊ߆ࠆ‫ޕ‬. 10. ߊᄌൻߔࠆߩߪ‫ޔ‬㧞ߟߩ‛૕㧔ᄤ૕㧕߇ធㄭߔ ࠆᤨߢ޽ࠆ‫ޕ‬ t. 5. ߎߩ⚿ᨐࠍ⷗ࠆ㒢ࠅߢߪ, t. 80 ߩᤨὐߢߪߘߩ⺋Ꮕߪ RK ᴺ‫ޔ‬. ⸘▚ᤨ㑆ߪ⴫㧞ߩࠃ߁ߦߥߞߚ‫ޕ‬RK ᴺߩ႐ ว‫ࡊ࠶࠹ࠬޔ‬᏷ࠍ 10. 6. 40 ߹ߢߪ RK. 7. ߣ㕖Ᏹߦዊߐߊߣߞߡ. 10 ,10 ߩߣ߈߽ࠛࡀ࡞ࠡ࡯߇଻ሽߐ. ޽ࠆߚ߼‫▚⸘ޔ‬࿁ᢙ߇ᄙߊߥࠅ⸘▚ᤨ㑆߽㕖Ᏹ. ࠇߕ᣿ࠄ߆ߦ⸘▚߇⎕✋ߒߡ޿ࠆߣ⠨߃ࠄࠇ. ߦᄙߊߥࠆ‫ޕ‬RK ᴺߢ߽ࠬ࠹࠶ࡊ᏷ࠍนᄌߦߢ. ࠆᢙ୯߇⷗ࠄࠇ‫ߦ⊛⚳ᦨޔ‬㧠ᰴߩ RK ᴺߢߪ. ߈ࠆࠃ߁ߦ૞ᚑߔࠇ߽߫߁ዋߒᤨ㑆⍴❗ࠍⴕ. 't. 10 ߩ႐วߢ߽ t 7. 41 ઃㄭ߆ࠄࠛࡀ࡞ࠡ. ߁ߎߣ߇ߢ߈ࠆߣᕁࠊࠇࠆ‫ޕ‬. ࡯߇ᄢ߈ߊᄌൻߒߡᱜ⏕ߦ⸘▚ࠍ⚳ੌߔࠆߎ. Taylor ዷ㐿ᴺߢߪ‫ޔ‬㧝࿁ߩࠬ࠹࠶ࡊࠍㅴ߼. ߣ߇ߢ߈ߥ߆ߞߚ‫ޕ‬㧤Ბ㧢ᰴߩ RK ᴺߦ߅޿ߡ. ࠆߩߦ⸘▚ᰴᢙ߇㜞޿ߎߣ߽޽ࠅ‫ᤨߩࠅߥ߆ޔ‬. ߪ‫' ޔ‬t. 10 7 ߩࠬ࠹࠶ࡊ᏷ߢ޽ࠇ߫ t. 80 ߹. 㑆ࠍᔅⷐߣߔࠆ‫▚⸘ޕ‬ᰴᢙ߇㜞޿ߚ߼‫࠶࠹ࠬޔ‬. ߢ‫ޔ‬㊀ᔃ‫ޔ‬ㆇേ㊂‫৻߷߶ࠍ࡯ࠡ࡞ࡀࠛޔ‬ቯߦ଻. ࡊ᏷ࠍᄢ߈ߊขࠇࠆߚ߼⸘▚࿁ᢙ߇ዋߥߊߥ. ߞߚ߹߹⸘▚ࠍቢੌߔࠆߎߣ߇ߢ߈ߚ‫ޕ‬. ࠅ‫ోޔ‬૕ߣߒߡ⸘▚ᤨ㑆ߩ⍴❗ߦߥࠆ႐ว߇ᄙ. 3.2 Taylor ዷ㐿ᴺ. ޿‫߽ߢ଀▚⸘ߩߎޕ‬ᄢ᏷ߦ⸘▚ᤨ㑆ࠍ⍴❗ߔࠆ. 20 ᰴߩ Taylor ዷ㐿ᴺࠍ↪޿ߡ⸘▚ߒߚ႐ว. ߎߣ߇ߢ߈ߚ‫ޕ‬. ߪ໧㗴ߥߊ t. 80 ߹ߢ⸘▚ࠍቢੌߔࠆߎߣ߇. RK ᴺߢߪ‫ߩࡊ࠶࠹ࠬ▚⸘ޔ‬૏⟎ߩ⸘▚୯ߪ. ߢ߈ߚ‫ޔ࡯ࠡ࡞ࡀࠛޕ‬ㆇേ㊂‫ޔ‬㊀ᔃߪ㧤Ბ㧢ᰴ. ኈᤃߦࠊ߆ࠆ߇‫ࡊ࠶࠹ࠬޔ‬ㅜਛߩὐߩ㑐ᢙ୯ߪ. ߩ RK ᴺࠃࠅዋߒߛߌ♖ᐲࠃ޿⚿ᨐࠍᓧࠆߎ. ⵬ቢᴺࠍ೑↪ߒߥߌࠇ߫ߥࠄߥ޿߇‫ޔ‬Taylor. ߣ߇ߢ߈ߚ‫(߈ߣߩߎޕ‬2.3)ᑼߦࠃߞߡዉ߆ࠇߚ. ዷ㐿ᴺߪዷ㐿ᑼ߇ᓧࠄࠇࠆߩߢ‫ߩߘޔ‬ᑼߦ⸘▚. ࠬ࠹࠶ࡊ᏷ߩᦨᄢ୯ߪ 't. ߒߚ޿૏⟎ߩᐳᮡࠍઍ౉ߔࠆߎߣߦࠃߞߡኈ. ߪ 't. 0.35528 ‫ᦨޔ‬ዊ୯. 7. 6.7728 u 10 ߢ޽ߞߚ‫ޕ‬. ᤃߦ⸘▚ߢ߈ࠆ‫ޕ‬. 3.3 ♖ᐲߩ⏕⹺ߣᲧセ. ⴫㧞㧚⸘▚ᤨ㑆. ߘࠇߙࠇᣇᴺߢߩᢙ୯⸘▚ߩ⚿ᨐࠍ 10 Ფߩ. ⸘▚ᣇᴺ. t ߦߟ޿ߡઃ㍳㧝ߩ⴫㧝ߦ␜ߒߚ‫ࠍ▚⸘ޕ‬ᱜᏱ. Rnge-Kutta㧔8 Ბ 6 ᰴ㧕 1347.5. ߦ⚳ੌߢ߈ߥ߆ߞߚ RK ᴺߩ⚿ᨐߪ㒰ᄖߒߡ. Taylor. ޽ࠆߚ߼‫ ߩ⴫ޔ‬Runge ߩ㗄⋡ߪ 8 Ბ 6 ᰴߩ. −64− 4. ታⴕᤨ㑆㧔⑽㧕. 3.078. ߎߩᢙ୯⸘▚ࠍⴕߞߚⅣႺߪ OS ߣߒߡ.

(5) Windows XP‫ޔ‬CPU ߣߒߡ Pentium4 3.0GHz. ዷ㐿ᴺࠍ↪޿ߚ߶߁ߪนᄌ㐳ߢ޽ࠆߚ߼౏ᐔ. ߢ ޽ ࠆ ‫ ߡ ߒ ߣ ࡯ ࡜ ࠗ ࡄ ࡦ ࠦ ޕ‬Visual. ߥᲧセߣߪ޿߃ߥ޿߇‫ࠍࠇߎޔ‬⠨ᘦߒߡ߽‫ޔ‬. Studio .NET2003 ߩ C++⸒⺆[1]ࠍ૶↪ߒߚ‫ޕ‬. Taylor ዷ㐿ᴺ߇߆ߥࠅㅦߊ⸘▚ߢ߈ࠆߣᕁࠊ. ౒ㅢ⸒⺆࡜ࡦ࠲ࠗࡓ㧔CLR㧕ࠍ૶ࠊߥ޿ࡀ࡯. ࠇࠆ‫ޕ‬. ࠹ࠖࡉࠦ࡯࠼ࠍ಴ߔࠃ߁ߦߒߡࠦࡦࡄࠗ࡞ߒ. RK ᴺߦ߅޿ߡ 4 ᰴߢߪ⸘▚ࠍᱜᏱߦታⴕ಴. ߚ߽ߩࠍ૶↪ߒߚ‫ޕ‬. ᧪ߥ޿߇‫ޔ‬8 Ბ 6 ᰴߢࠃ߁߿ߊታⴕ಴᧪ߚߎߣ. 4 ߅ࠊࠅߦ. ߆ࠄ⷗ߡ߽‫ޔ‬ᰴᢙߩ㜞޿⸘▚ᴺ߇⸘▚♖ᐲ߇᰼ ߥࠅ‫ࡊ࠶࠹ࠬޔ‬᏷߽ᄢ߈ߊขࠇࠆ‫ޕ‬ᰴᢙࠍ਄ߍ. 㪌㪇. ࠆߎߣ߇ኈᤃߥ Taylor ዷ㐿ᴺߪ‫ޔ‬Ᏹᓸಽᣇ⒟ 㪋㪇. ᑼࠍ⸃ߊ਄ߢ᦭↪ߢ޽ࠆߣ޿߃ࠆ‫ޕ‬ ⸘▚⚿ᨐࠍ⷗ࠆߣࠛࡀ࡞ࠡ࡯߇଻ሽߐࠇࠆ. 㪊㪇. ߩߢ‫଻߇࡯ࠡ࡞ࡀࠛޔ‬ሽߐࠇࠆߣ޿߁ 㪉㪇. 㪈㪇. 㪇㪄㪈㪇. 㪄㪌. 㪇㪄㪈㪇. 㪌. 㪈㪇. Sympletic ᴺ[3]ࠍ೑↪ߒߥ߆ߞߚ‫੹ߪࠇߎޕ‬ᓟ. 㫄㪊㫄㪊㐿ᆎὐ 㫄㪊⚳ੌὐ 㫄㪋㫄㪋㐿ᆎὐ 㪈㪌㫄㪋⚳ੌὐ 㫄㪌㫄㪌㐿ᆎὐ 㫄㪌⚳ੌὐ. ߩ⺖㗴ߣߒߚ޿‫ޕ‬. ෳ⠨ᢥ₂. 㪄㪉㪇. ࿑㧟㧚ਃ૕ߩᦨ⚳ㆇേ㧔㨠㧩60㨪80㧕 RK ᴺ‫ޔ‬Taylor ዷ㐿ᴺߤߜࠄߩᣇᴺߢ⸘▚ߒ ߡ߽ᦨ⚳⊛ߦ⚂㧟ᩴߩ♖ᐲߩ⚿ᨐߒ߆ᓧࠄࠇ ߆ߞߚ‫੍ޕ‬ᗐࠃࠅ‫ࠅߥ߆ޔ‬ᄢ߈ߥ⺋Ꮕ߇ߢࠆ⚿ ᨐߣߥߞߚ‫⾰߽ࠄߜߤޕ‬ὐ m3 ߇ xy ᐔ㕙਄ߩ ╙ 1 ⽎㒢ᣇะ߳ߪߓ߈ߛߐࠇ‫ ޔ‬m 4 ‫ ޔ‬m5 ߇⋧ ੕ߦ࿁ォߒ޽߁ኻࠍ૞ࠅ‫ ╙ޔ‬3 ⽎㒢ᣇะ߳㆙ߑ ߆ߞߡ޿ߊߣ޿߁ࡇ࠲ࠧ࡜ࠬߩਃ૕໧㗴ߩᦨ ⚳ㆇേ㧔࿑㧟㧕ߦ㆐ߒߡ޿ࠆߩߢ‫ࡊ࠶࠹ࠬޔ‬᏷ ࠍ 't. 10 7 ߦߣߞߚ 8 Ბ 6 ᰴߩ RK ᴺ‫࠹ࠬޔ‬. ࠶ࡊ᏷ࠍ(2.2)ᑼߩ♖ᐲߢߣߞߚ 20 ᰴߩ Taylor ዷ㐿ᴺߢߩ⸘▚⚿ᨐߪ৻ᔕᱜߒ޿ߣ⠨߃ࠄࠇ ࠆ‫ޕ‬ ࡇ࠲ࠧ࡜ࠬߩਃ૕໧㗴ࠍ Taylor ዷ㐿ᴺߢ⸃ ߊߎߣߢ Taylor ዷ㐿ᴺߦࠃࠆᏱᓸಽᣇ⒟ᑼߩ ᢙ୯⸘▚ߩᕈ⢻ࠍ⺞ߴߚ⚿ᨐ‫ޔ‬RK ᴺࠍ↪޿ߚ ႐วࠃࠅ߽ዋߒߢ޽ࠆ߇⸘▚♖ᐲ߇ࠃߊ‫ࡠࡊޔ‬ ࠣ࡜ࡓߢߩ⸘▚ㅦᐲߪ㕖Ᏹߦㅦ޿ߣ޿߃ࠆ‫ޕ‬ RK ᴺߩ⸘▚ߩࠬ࠹࠶ࡊ᏷ߪ࿕ቯ㐳ߢ‫ޔ‬Taylor. −65− 5. [1] Ellis M.A. and Stroustrup B 㧦 The Annotated C++ Reference Manual, Addison-Wesley, 1990 [2] Gisela Engeln-Müllges Frank Uhlig 㧦 Numerical Algorithms with Fortran, Springer(1996) [3] Hairer E., Wanner G., Solving Ordinary Differential Equations II, Springer-Verlag, 1991 [4] ᐔጊ ᒄ , ዊች⡛ม‫ޔ‬૒⮮ഃᄥ㇢‫ޔ‬ Taylor ⚖ᢙᴺߦࠃࠆᏱᓸಽᣇ⒟ᑼߩ ⸃ᴺ,ᣣᧄᔕ↪ᢙℂቇળ‫ޔ‬Vol 12. No.1, pp.1-8,(2002) [5] 㐳ᴛ Ꮏ‫ޔ‬᪍ጊ Ẵሶ㧦ࡄ࠰ࠦࡦߢ⷗ ࠆᄤ૕ߩേ߈‫ޔ‬࿾ੱᦠ㙚(1992) [6] ጊᧄ ື㇢㧦ᢙ୯⸃ᨆ౉㐷‫ࡦࠛࠗࠨޔ‬ ࠬ␠(1959).

(6) ઃ㍳㧝㧚㨠ߦኻߔࠆਃኻߩᐳᮡ⸘▚⚿ᨐ⴫ . 0.0. 10.0. 20.0. 30.0. 40.0. 50.0. 60.0. 70.0. 80.0. t. x3. y3. x4. y4. x5. y5. Runge 1.00000. 3.00000. -2.00000 -1.00000 1.00000. -1.00000. Taylor 1.00000. 3.00000. -2.00000 -1.00000 1.00000. -1.00000. ⌀୯. 1.00000. 3.00000. -2.00000 -1.00000 1.00000. -1.00000. Runge 0.77848. 0.14139. -2.02509 0.09722. 1.15299. -0.16261. Taylor 0.77848. 0.14139. -2.02509 0.09722. 1.15299. -0.16261. ⌀୯. 0.77848. 0.14139. -2.02509 0.09722. 1.15299. -0.16261. Runge 3.00429. 0.51193. -1.38863 -0.47048 -0.69167 0.06923. Taylor 3.00429. 0.51193. -1.38863 -0.47048 -0.69167 0.06923. ⌀୯. 3.00429. 0.51193. -1.38863 -0.47048 -0.69167 0.06923. Runge 0.85634. 2.28709. -0.88798 -0.86596 0.18858. -0.67949. Taylor 0.85634. 2.28709. -0.88798 -0.86596 0.18858. -0.67949. ⌀୯. 2.28709. -0.88798 -0.86596 0.18858. -0.67949. Runge -0.62201 1.85831. 0.17355. -2.36842 0.23437. 0.77974. Taylor -0.62200 1.85831. 0.17354. -2.36841 0.23437. 0.77974. ⌀୯. 0.17354. -2.36841 0.23437. 0.77974. Runge -2.70102 -3.79575 1.50436. 0.96132. 0.41712. 1.50840. Taylor -2.70145 -3.79717 1.50588. 0.96083. 0.41616. 1.50964. ⌀୯. 0.96081. 0.41613. 1.50968. 0.85634. -0.62200 1.85831. -2.70146 -3.79722 1.50594. Runge 0.77353. 2.01724. 0.26684. -0.76305 -0.67759 -0.59990. Taylor 0.74483. 1.94267. 0.26436. -0.73282 -0.65838 -0.57935. ⌀୯. 0.74381. 1.93995. 0.26401. -0.73163 -0.65749 -0.57867. Runge 7.48949. 21.4635. -2.43528 -7.14237 -2.54547 -7.16423. Taylor 7.10026. 20.6254. -1.89601 -6.94495 -2.74334 -6.94495. ⌀୯. 6.93346. 20.2618. -2.00301 -6.87246 -2.55767 -6.65911. Runge 13.5850. 39.1561. -4.01787 -13.1847 -4.93671 -12.9460. Taylor 12.7897. 37.4056. -3.86475 -12.5161 -4.58199 -12.5161. ⌀୯. 36.6423. -3.55587 -12.3548 -4.62377 -12.1016. 12.4474. −66− 6.

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