（ ） 29T HE H ARRIS S CIENCE R EVIEW OF D OSHISHA U NIVERSITY, V OL . 59 , N O . 4 January 2019

ࠩʹහײͳ໰୊ʹର͢Δ਺஋ܭࢉͰ͸ɼ਺஋ޡ͕ࠩۃ

୺ʹ૿෯͞Εͯ਺஋ղ͕ৼಈ͠ɼݫີղΛۙࣅͰ͖ͳ

͍Ͳ͜Ζ͔Φʔόʔϑϩʔ͕ى͖ͯ਺஋ܭࢉ͕ഁ୼͢

Δ⁴⁾ɽզʑ͸͜ͷ໰୊఺Λղܾ͢΂͘ɼ਺஋ޡࠩΛ೚

ҙʹখ͘͢͞Δܭࢉख๏ΛఏҊͨ͠⁵⁾ɽଧͪ੾Γޡࠩ

Λ೚ҙʹখ͘͢͞Δʹ͸཭ࢄԽख๏ͱͯ͠εϖΫτϧ

๏Λ༻͍Δͷ͕࣮༻తͰ͋ΓɼؙΊޡࠩΛ೚ҙʹখ͞

͘͢Δʹ͸ଟഒ௕ԋࢉ⁶⁾Λ༻͍Ε͹Α͍ɽ͜ΕΒ͸ͦ

ΕͧΕ๲େͳܭࢉࢿݯΛඞཁͱ͢ΔͷͰɼ͠͹Β͘લ

·Ͱ͸͜ΕΒΛซ༻͢Δͱ͍͏ൃ૝͸ͳ͔ͬͨɽզʑ

͸ɼܭࢉػੑೳͷ޲্εϐʔυ͍ΘΏΔϜʔΞͷ๏ଇ Λߟྀͯ͠ɼ྆ख๏Λซ༻ͨ͠਺஋ܭࢉͷকདྷੑʹண

໨ͨ͠ɽͦͯ͜͠ͷ਺஋ܭࢉ๏Λແݶਫ਼౓਺஋γϛϡ Ϩʔγϣϯ๏ͱ໊෇͚ɼޡࠩʹහײͳٯ໰୊ͷ௚઀਺

஋ܭࢉΛؚΉɼ༷ʑͳ໰୊ɾํఔࣜʹରͯ͠਺஋ܭࢉ

Λߦ͖ͬͯͨ^7–10)ɽ

ຊݚڀͰ͸ɼεϖΫτϧ๏ͷਫ਼౓ʹؔ͢ΔಛੑΛར

༻͢Δɽmճ࿈ଓඍ෼Մೳͳؔ਺uʹର͢Δయܕత ͳޡࠩͷධՁࣜ͸

||u−uN||≦C N^−m (1) ͱͳΔɽ͜͜ͰɼN ͸εϖΫτϧ๏ʹΑΔۙࣅͷ࣍

਺Ͱ͋Γɼۙࣅؔ਺ΛuN ͱ͍ͯ͠ΔɽC͸͋Δਖ਼ఆ

਺ɼ|| · ||͸ద౰ͳϊϧϜͰ͋ΔɽNͷࢦ਺͸ɼϊϧ ϜͷऔΓํʹΑͬͯℓΛఆ਺ͱͯ͠ℓ−mͷΑ͏ʹগ

ͣ͠ΕΔ΋ͷͷͷɼuͷ׈Β͔͞ʹରԠ͢Δ͜ͱʹ͸

มΘΓͳ͍ɽ͜ͷ͔ࣜΒɼؔ਺͕ແݶճ࿈ଓඍ෼Մೳ

Ͱ͋Ε͹ɼεϖΫτϧ๏ʹΑΔۙࣅؔ਺͸ແݶ࣍ऩଋ

͢Δ͜ͱ͕Θ͔Δɽ͜ͷੑ࣭͸εϖΫτϧਫ਼౓ͱݺ͹

ΕΔ²⁾ɽ΋ؔ͠਺͕ղੳతͰ͋Ε͹ɼۙࣅؔ਺͸ࢦ਺

ؔ਺తऩଋΛࣔ͢͜ͱ͕͋Δɽզʑ͸ɼ͜ͷεϖΫτ ϧ๏ͷಛੑΛར༻ͯ͠ɼؔ਺΍ํఔࣜͷղͷ׈Β͔͞

ͷ਺஋ܭࢉΛߦͬͨ¹¹⁾ɽͦͷࡍɼؙΊޡࠩͷӨڹΛ ਖ਼֬ʹݟੵ΋Δඞཁ͕͋ΔͨΊɼଟഒ௕ԋࢉϥΠϒϥ Ϧ¹²⁾Λ༻͍ͨແݶਫ਼౓਺஋γϛϡϨʔγϣϯͰߦͬ

ͨɽ਺೥લɼ׈Β͔͕͞఻ൖ͢Δํఔࣜͱͯ͠஌ΒΕ

͍ͯΔ૒ۂܕํఔࣜʹ஫໨ͯ͠਺஋ܭࢉΛߦͬͨ¹³⁾ɽ

ͦͷࡍɼ׈Β͔͕͞ม਺ʹґଘ͢Δݫີղ͕ߏ੒Ͱ͖

ͨͷͰɼ׈Β͔͞ͷม਺ґଘͷ਺஋ܭࢉʹணखͨ͠ͱ

͜ΖɼղऍʹࠔΔ਺஋ܭࢉ݁Ռ(ະެද)͕ಘΒΕͯ

ݚڀ͕଺ͬͯ͠·ͬͨɽ

ͦ͜Ͱɼզʑ͸ɼࠞཚΛট͍ͨ਺஋ܭࢉ݁ՌΛղऍ

͢΂͘ɼ৽ͨͳ਺஋ղੳख๏ͷ։ൃΛ໨ࢦ͢͜ͱʹ͠

ͨɽղͷ׈Β͔͞ΛՄࢹԽͨ͠ਖ਼ଇੑͷ஍ਤͱ͍͏΂

͖΋ͷ͕਺஋తʹ࡞੒Ͱ͖Ε͹ɼ଺ͬͨݚڀΛਐΊΔ

͜ͱ͕Ͱ͖ΔͷͰ͸ͳ͍͔ͱߟ͑ͨɽ1ม਺ؔ਺f(x)

͕఺aͷ͋Δۙ๣ʹ͓͍ͯaΛத৺ͱ͢Δ΂͖ڃ਺ͷ Taylorڃ਺ʹల։͞ΕΔͱ͖ɼf(x)͸aͰղੳతͰ

͋Δͱ͍͏¹⁴⁾ɽຊ࿦จͰ͸ɼؔ਺͕ղੳతͰͳ͍఺

Λಛҟ఺ͱ͍͏͜ͱʹ͢Δɽ·ͨɼಛҟ఺Λ࣋ͭؔ਺

Λಛҟతͳؔ਺ͱ͍͏͜ͱʹ͢Δ¹⁵⁾ɽ௨ৗɼಛҟ఺

͸ؔ਺஋͕ແݶେʹͳΔͱ͜ΖͱࢥΘΕΔ͜ͱ͕ଟ͍

͕ɼεϖΫτϧ๏Λ༻͍ͯภඍ෼ํఔࣜͷղ͕രൃ͢

Δ(ղͷ஋͕༗ݶྖҬʹ͓͍ͯແݶେʹͳΔ)఺Λಛ

ఆͨ͠ݚڀ͕͋Δ¹⁶⁾ɽҰํɼຊ࿦จͰ͸ɼؔ਺஋͕

ແݶେʹͳΔΑ͏ͳݟͨ໨ʹ͸͖ͬΓͱ෼͔Δಛҟੑ

Λର৅ͱ͸͍ͯ͠ͳ͍ɽྫ͑͹ɼ3અͰ঺հ͢Δؔ਺

f4(x)ͷάϥϑΛFig.1ʹ͕ࣔ͢ɼಛҟ఺(ղੳతͰͳ

͍఺)ͷଘࡏ͕ݟͨ໨ʹΘ͔ΔͩΖ͏͔ɽ

0 0.01 0.02 0.03 0.04 0.05 0.06

-1 -0.5 0 0.5 1

Fig. 1. Graph off4(x).

͜ͷf4(x)ͷΑ͏ͳؔ਺ʹରͯ͠΋ɼಛҟ఺ͷଘࡏͱ

৔ॴΛಛఆ͢Δ؆୯ͳख๏Λ࣍અҎ߱ʹड़΂Δɽຊ

࿦จͷΑΓͲ͜Ζͱ͢Δ஌ࣝ͸Fourierڃ਺ͷੑ࣭Ͱ

͋ΔɽΑ͘஌ΒΕ͍ͯΔΑ͏ʹɼؔ਺ͷෆ࿈ଓ఺ͷۙ

๣ʹ͓͍ͯFourierڃ਺(ิؒؔ਺)͸Gibbsݱ৅Λ ى͜͠ɼؔ਺͕࿈ଓͰ͋Δ۠ؒͰ͸Fourierڃ਺͸Ұ

༷ऩଋ͢Δ¹⁷⁾ɽ͜Ε͸ɼFourierڃ਺(ิؒؔ਺)ͷ

Gibbsݱ৅͔Βෆ࿈ଓ఺ͷଘࡏ͕ਪఆͰ͖ͯɼ͞Βʹ

Gibbsݱ৅ͷ৔ॴ͔Βෆ࿈ଓ఺ͷ৔ॴ͕ਪఆͰ͖Δ͜

ͱΛҙຯ͍ͯ͠ΔɽεϖΫτϧ๏ͷதʹ͸Fourierڃ

਺Λ༻͍Δ΋ͷ΋͋Δ͠ɼͦ͏Ͱͳͯ͘΋εϖΫτϧ

๏΋ղΛۙࣅ͢Δิؒؔ਺Λߏ੒͢Δ΋ͷͰ͋ΔͨΊɼ

Fourierڃ਺ͱಉ͡ੑ࣭͕ظ଴͞ΕΔɽ࣮ࡍɼ3અͰ঺

հ͢Δෆ࿈ଓؔ਺f₋1(x)ʹର͢ΔεϖΫτϧ๏ͷۙ

ࣅิؒؔ਺ͷάϥϑΛFig.2ʹࣔ͢ɽෆ࿈ଓ఺ۙ๣ͷ

Gibbsݱ৅ͱ࿈ଓͰ͋Δ۠ؒʹ͓͚ΔҰ༷ऩଋͷ༷ࢠ

͕֬ೝͰ͖Δɽ͜͜ͰɼN͸ۙࣅͷ࣍਺Ͱ͋Δɽຊ࿦

จ͸ɼ͜ͷੑ࣭Λ༻͍ͯ1ม਺ؔ਺ͷಛҟੑʹؔ͢Δ جૅతͳ਺஋࣮ݧΛߦͬͨ΋ͷͰ͋Δɽ

N= 10 N= 50 f−1 (x)

-0.2 0 0.2 0.4 0.6 0.8 1 1.2

-1 -0.5 0 0.5 1

Fig. 2. Gibbs phenomenon forf₋1(x).

2. εϖΫτϧબ఺๏

εϖΫτϧ๏͸େผͯ͠ɼεϖΫτϧɾΨϨϧΩϯ

๏ͱεϖΫτϧબ఺๏͕͋ΔɽεϖΫτϧબ఺๏͸ࠩ

෼๏ͱಉ༷ͷ࢖͍ํʹͳΔͷͰ࢖͍΍͍͢ɽͱ͘ʹඇ ઢܗํఔࣜͷͱ͖ʹͦͷར఺͕͸͖ͬΓ͢Δɽ͜ͷཧ

༝͔Βզʑ͸΋ͬͺΒεϖΫτϧબ఺๏Λ༻͍͖ͯͨɽ εϖΫτϧબ఺๏Ͱ͸ؔ਺஋ΛఆΊΔ఺Λબ఺ͱݺͿɽ Ұํɼ࠷ྑۙࣅʹؔ࿈͢Δิؒଟ߲ࣜͱͯ͠Cheby- shevଟ߲͕ࣜ஌ΒΕ͍ͯΔͷͰ¹⁸⁾ɼChebyshevଟ߲

ࣜΛ༻͍ΔεϖΫτϧ๏͕ਫ਼౓తʹ༗རͱߟ͑ΒΕΔɽ

͜ͷΑ͏ͳཧ༝͔Βɼզʑ͸ChebyshevεϖΫτϧ બ఺๏Λଟ༻͖ͯͨ͠ɽ

ChebyshevεϖΫτϧબ఺๏ͷద༻๏Λ؆୯ʹ঺հ

͢Δɽ࣍ͷChebyshevల։Λߟ͑Δɽ

uN(x) =

∑N k=0

˜

uk Tk(x) (−1≦x≦1) (2)

͜͜ͰɼTk(x) = cos (k arccosx) ͸ k ࣍ Cheby- shevଟ߲ࣜͰ͋ΔɽChebyshevల։ʹؔ͢Δબ఺͸

CGL(Chebyshev Gauss-Lobatto)఺ɼCG(Chebyshev Gauss)఺ɼCGR(Chebyshev Gauss-Radau)఺͕͋Γɼ

ͦΕͧΕ࣍Ͱ༩͑ΒΕΔɽ

xj=















 cos j

Nπ (CGL఺) cos 2j+ 1

2N+ 2π (CG఺) cos 2j

2N+ 1π (CGR఺)

(j= 0,1,· · ·, N) (3)

ຊ࿦จͰ͸CGL఺Λ༻͍Δ͜ͱʹ͢ΔɽCGL఺্ͷ

ؔ਺஋uj=uN(xj)͕༩͑ΒΕΔͱల։܎਺͸࣍Ͱ༩

͑ΒΕΔɽ

˜ uk = 2

N ck

∑N j=0

1 cj

ujTk(xj) (4)

cj =

{2 (j= 0, N)

1 (ͦͷଞ) (5)

͜ͷؔ܎ࣜ͸൓సެࣜͱݺ͹ΕΔ¹⁾ɽ͜ͷ൓సެ͕ࣜ ҙຯ͢Δͱ͜Ζ͸ɼۙࣅͷ࣍਺͕બ఺ͷݸ਺ͱͯ͠ઃ ఆͰ͖Δͱ͍͏͜ͱͰ͋Δɽ͜ͷಛ௃͸ɼࠩ෼๏Ͱ௒ ߴ࣍ۙࣅྫ͑͹100࣍ۙࣅ͢Δ͜ͱΛߟ͑Ε͹༰қʹ

૝૾Ͱ͖Δ͕ɼߴਫ਼౓ܭࢉΛ໨ࢦ͢ͱ͖εϖΫτϧબ

఺๏͕͍͔ʹ࣮༻తͰ͋Δ͔Λ͍ࣔͯ͠ΔɽεϖΫτ ϧબ఺๏ʹΑΔۙࣅղ(ิؒؔ਺)ͷߏ੒͸ɼ໰୊Λ هड़͢Δํఔ͔ࣜΒબ఺্ʹ͓͚Δղͷ஋Λະ஌਺ͱ

͢Δ࿈ཱํఔࣜΛಋ͖ɼͦΕΛղ͍ͯબ఺্ͷղͷ஋ Λܾఆ͢Δɽͦͷޙɼ൓సެࣜΛ༻͍ͯల։܎਺Λܭ

ࢉ͠ɼิؒؔ਺Λߏ੒ͯ͠ํఔࣜͷۙࣅղͱ͢Δɽ

3. ਺஋ܭࢉ݁Ռ

ຊ࿦จͰ͸༩͑ΒΕͨؔ਺ʹରͯ͠ɼબ఺্ͷؔ਺

஋͔Β൓సެࣜΛ༻͍ͯิؒؔ਺Λߏ੒͠ɼ༩͑ΒΕ

ͨؔ਺ͷಛҟੑΛ਺஋తʹௐ΂Δɽ 3.1 ਺஋ܭࢉʹ༻͍ͨؔ਺

ಛҟੑΛ਺஋తʹௐ΂ΔͨΊʹɼಛҟੑ͕Θ͔ͬͯ

͍Δؔ਺Λ༻ҙ͢Δɽຊ࿦จ͸ɼجૅతͳ਺஋࣮ݧ͕

໨తͰ͋ΔͷͰ1ม਺ؔ਺Λߟ͑Δɽ·ͨɼCheby- shevબ఺๏Λద༻͢ΔͨΊɼؔ਺͕ఆٛ͞ΕΔ۠ؒ͸ [−1ɼ1]ͱ͢Δɽಛҟੑʹؔͯ͠ௐ΂͍ͨ͜ͱ͸ɼಛ ҟ఺ͷ৔ॴͱಛҟ఺ʹ͓͚Δ׈Β͔͞Ͱ͋Δɽ͜ͷ໨ తʹԊͬͨؔ਺Λ༻ҙ͢ΔͨΊʹɼ·ͣෆ࿈ଓؔ਺Λ

༻ҙͯ͠ɼͦΕΛ࣍ʑੵ෼͢Δ͜ͱͰɼಛҟ఺ʹ͓͚

ࠩʹහײͳ໰୊ʹର͢Δ਺஋ܭࢉͰ͸ɼ਺஋ޡ͕ࠩۃ

୺ʹ૿෯͞Εͯ਺஋ղ͕ৼಈ͠ɼݫີղΛۙࣅͰ͖ͳ

͍Ͳ͜Ζ͔Φʔόʔϑϩʔ͕ى͖ͯ਺஋ܭࢉ͕ഁ୼͢

Δ⁴⁾ɽզʑ͸͜ͷ໰୊఺Λղܾ͢΂͘ɼ਺஋ޡࠩΛ೚

ҙʹখ͘͢͞Δܭࢉख๏ΛఏҊͨ͠⁵⁾ɽଧͪ੾Γޡࠩ

Λ೚ҙʹখ͘͢͞Δʹ͸཭ࢄԽख๏ͱͯ͠εϖΫτϧ

๏Λ༻͍Δͷ͕࣮༻తͰ͋ΓɼؙΊޡࠩΛ೚ҙʹখ͞

͘͢Δʹ͸ଟഒ௕ԋࢉ⁶⁾Λ༻͍Ε͹Α͍ɽ͜ΕΒ͸ͦ

ΕͧΕ๲େͳܭࢉࢿݯΛඞཁͱ͢ΔͷͰɼ͠͹Β͘લ

·Ͱ͸͜ΕΒΛซ༻͢Δͱ͍͏ൃ૝͸ͳ͔ͬͨɽզʑ

͸ɼܭࢉػੑೳͷ޲্εϐʔυ͍ΘΏΔϜʔΞͷ๏ଇ Λߟྀͯ͠ɼ྆ख๏Λซ༻ͨ͠਺஋ܭࢉͷকདྷੑʹண

໨ͨ͠ɽͦͯ͜͠ͷ਺஋ܭࢉ๏Λແݶਫ਼౓਺஋γϛϡ Ϩʔγϣϯ๏ͱ໊෇͚ɼޡࠩʹහײͳٯ໰୊ͷ௚઀਺

஋ܭࢉΛؚΉɼ༷ʑͳ໰୊ɾํఔࣜʹରͯ͠਺஋ܭࢉ

Λߦ͖ͬͯͨ^7–10)ɽ

ຊݚڀͰ͸ɼεϖΫτϧ๏ͷਫ਼౓ʹؔ͢ΔಛੑΛར

༻͢Δɽmճ࿈ଓඍ෼Մೳͳؔ਺uʹର͢Δయܕత ͳޡࠩͷධՁࣜ͸

||u−uN||≦C N^−m (1) ͱͳΔɽ͜͜ͰɼN ͸εϖΫτϧ๏ʹΑΔۙࣅͷ࣍

਺Ͱ͋Γɼۙࣅؔ਺ΛuN ͱ͍ͯ͠ΔɽC͸͋Δਖ਼ఆ

਺ɼ|| · ||͸ద౰ͳϊϧϜͰ͋ΔɽNͷࢦ਺͸ɼϊϧ ϜͷऔΓํʹΑͬͯℓΛఆ਺ͱͯ͠ℓ−mͷΑ͏ʹগ

ͣ͠ΕΔ΋ͷͷͷɼuͷ׈Β͔͞ʹରԠ͢Δ͜ͱʹ͸

มΘΓͳ͍ɽ͜ͷ͔ࣜΒɼؔ਺͕ແݶճ࿈ଓඍ෼Մೳ

Ͱ͋Ε͹ɼεϖΫτϧ๏ʹΑΔۙࣅؔ਺͸ແݶ࣍ऩଋ

͢Δ͜ͱ͕Θ͔Δɽ͜ͷੑ࣭͸εϖΫτϧਫ਼౓ͱݺ͹

ΕΔ²⁾ɽ΋ؔ͠਺͕ղੳతͰ͋Ε͹ɼۙࣅؔ਺͸ࢦ਺

ؔ਺తऩଋΛࣔ͢͜ͱ͕͋Δɽզʑ͸ɼ͜ͷεϖΫτ ϧ๏ͷಛੑΛར༻ͯ͠ɼؔ਺΍ํఔࣜͷղͷ׈Β͔͞

ͷ਺஋ܭࢉΛߦͬͨ¹¹⁾ɽͦͷࡍɼؙΊޡࠩͷӨڹΛ ਖ਼֬ʹݟੵ΋Δඞཁ͕͋ΔͨΊɼଟഒ௕ԋࢉϥΠϒϥ Ϧ¹²⁾Λ༻͍ͨແݶਫ਼౓਺஋γϛϡϨʔγϣϯͰߦͬ

ͨɽ਺೥લɼ׈Β͔͕͞఻ൖ͢Δํఔࣜͱͯ͠஌ΒΕ

͍ͯΔ૒ۂܕํఔࣜʹ஫໨ͯ͠਺஋ܭࢉΛߦͬͨ¹³⁾ɽ

ͦͷࡍɼ׈Β͔͕͞ม਺ʹґଘ͢Δݫີղ͕ߏ੒Ͱ͖

ͨͷͰɼ׈Β͔͞ͷม਺ґଘͷ਺஋ܭࢉʹணखͨ͠ͱ

͜ΖɼղऍʹࠔΔ਺஋ܭࢉ݁Ռ(ະެද)͕ಘΒΕͯ

ݚڀ͕଺ͬͯ͠·ͬͨɽ

ͦ͜Ͱɼզʑ͸ɼࠞཚΛট͍ͨ਺஋ܭࢉ݁ՌΛղऍ

͢΂͘ɼ৽ͨͳ਺஋ղੳख๏ͷ։ൃΛ໨ࢦ͢͜ͱʹ͠

ͨɽղͷ׈Β͔͞ΛՄࢹԽͨ͠ਖ਼ଇੑͷ஍ਤͱ͍͏΂

͖΋ͷ͕਺஋తʹ࡞੒Ͱ͖Ε͹ɼ଺ͬͨݚڀΛਐΊΔ

͜ͱ͕Ͱ͖ΔͷͰ͸ͳ͍͔ͱߟ͑ͨɽ1ม਺ؔ਺f(x)

͕఺aͷ͋Δۙ๣ʹ͓͍ͯaΛத৺ͱ͢Δ΂͖ڃ਺ͷ Taylorڃ਺ʹల։͞ΕΔͱ͖ɼf(x)͸aͰղੳతͰ

͋Δͱ͍͏¹⁴⁾ɽຊ࿦จͰ͸ɼؔ਺͕ղੳతͰͳ͍఺

Λಛҟ఺ͱ͍͏͜ͱʹ͢Δɽ·ͨɼಛҟ఺Λ࣋ͭؔ਺

Λಛҟతͳؔ਺ͱ͍͏͜ͱʹ͢Δ¹⁵⁾ɽ௨ৗɼಛҟ఺

͸ؔ਺஋͕ແݶେʹͳΔͱ͜ΖͱࢥΘΕΔ͜ͱ͕ଟ͍

͕ɼεϖΫτϧ๏Λ༻͍ͯภඍ෼ํఔࣜͷղ͕രൃ͢

Δ(ղͷ஋͕༗ݶྖҬʹ͓͍ͯແݶେʹͳΔ)఺Λಛ

ఆͨ͠ݚڀ͕͋Δ¹⁶⁾ɽҰํɼຊ࿦จͰ͸ɼؔ਺஋͕

ແݶେʹͳΔΑ͏ͳݟͨ໨ʹ͸͖ͬΓͱ෼͔Δಛҟੑ

Λର৅ͱ͸͍ͯ͠ͳ͍ɽྫ͑͹ɼ3અͰ঺հ͢Δؔ਺

f4(x)ͷάϥϑΛFig.1ʹ͕ࣔ͢ɼಛҟ఺(ղੳతͰͳ

͍఺)ͷଘࡏ͕ݟͨ໨ʹΘ͔ΔͩΖ͏͔ɽ

0 0.01 0.02 0.03 0.04 0.05 0.06

-1 -0.5 0 0.5 1

Fig. 1. Graph off4(x).

͜ͷf4(x)ͷΑ͏ͳؔ਺ʹରͯ͠΋ɼಛҟ఺ͷଘࡏͱ

৔ॴΛಛఆ͢Δ؆୯ͳख๏Λ࣍અҎ߱ʹड़΂Δɽຊ

࿦จͷΑΓͲ͜Ζͱ͢Δ஌ࣝ͸Fourierڃ਺ͷੑ࣭Ͱ

͋ΔɽΑ͘஌ΒΕ͍ͯΔΑ͏ʹɼؔ਺ͷෆ࿈ଓ఺ͷۙ

๣ʹ͓͍ͯFourierڃ਺(ิؒؔ਺)͸Gibbsݱ৅Λ ى͜͠ɼؔ਺͕࿈ଓͰ͋Δ۠ؒͰ͸Fourierڃ਺͸Ұ

༷ऩଋ͢Δ¹⁷⁾ɽ͜Ε͸ɼFourierڃ਺(ิؒؔ਺)ͷ

Gibbsݱ৅͔Βෆ࿈ଓ఺ͷଘࡏ͕ਪఆͰ͖ͯɼ͞Βʹ

Gibbsݱ৅ͷ৔ॴ͔Βෆ࿈ଓ఺ͷ৔ॴ͕ਪఆͰ͖Δ͜

ͱΛҙຯ͍ͯ͠ΔɽεϖΫτϧ๏ͷதʹ͸Fourierڃ

਺Λ༻͍Δ΋ͷ΋͋Δ͠ɼͦ͏Ͱͳͯ͘΋εϖΫτϧ

๏΋ղΛۙࣅ͢Δิؒؔ਺Λߏ੒͢Δ΋ͷͰ͋ΔͨΊɼ

Fourierڃ਺ͱಉ͡ੑ࣭͕ظ଴͞ΕΔɽ࣮ࡍɼ3અͰ঺

հ͢Δෆ࿈ଓؔ਺f₋1(x)ʹର͢ΔεϖΫτϧ๏ͷۙ

ࣅิؒؔ਺ͷάϥϑΛFig.2ʹࣔ͢ɽෆ࿈ଓ఺ۙ๣ͷ

Gibbsݱ৅ͱ࿈ଓͰ͋Δ۠ؒʹ͓͚ΔҰ༷ऩଋͷ༷ࢠ

͕֬ೝͰ͖Δɽ͜͜ͰɼN͸ۙࣅͷ࣍਺Ͱ͋Δɽຊ࿦

จ͸ɼ͜ͷੑ࣭Λ༻͍ͯ1ม਺ؔ਺ͷಛҟੑʹؔ͢Δ جૅతͳ਺஋࣮ݧΛߦͬͨ΋ͷͰ͋Δɽ

N= 10 N= 50 f−1 (x)

-0.2 0 0.2 0.4 0.6 0.8 1 1.2

-1 -0.5 0 0.5 1

Fig. 2. Gibbs phenomenon forf₋1(x).

2. εϖΫτϧબ఺๏

εϖΫτϧ๏͸େผͯ͠ɼεϖΫτϧɾΨϨϧΩϯ

๏ͱεϖΫτϧબ఺๏͕͋ΔɽεϖΫτϧબ఺๏͸ࠩ

෼๏ͱಉ༷ͷ࢖͍ํʹͳΔͷͰ࢖͍΍͍͢ɽͱ͘ʹඇ ઢܗํఔࣜͷͱ͖ʹͦͷར఺͕͸͖ͬΓ͢Δɽ͜ͷཧ

༝͔Βզʑ͸΋ͬͺΒεϖΫτϧબ఺๏Λ༻͍͖ͯͨɽ εϖΫτϧબ఺๏Ͱ͸ؔ਺஋ΛఆΊΔ఺Λબ఺ͱݺͿɽ Ұํɼ࠷ྑۙࣅʹؔ࿈͢Δิؒଟ߲ࣜͱͯ͠Cheby- shevଟ߲͕ࣜ஌ΒΕ͍ͯΔͷͰ¹⁸⁾ɼChebyshevଟ߲

ࣜΛ༻͍ΔεϖΫτϧ๏͕ਫ਼౓తʹ༗རͱߟ͑ΒΕΔɽ

͜ͷΑ͏ͳཧ༝͔Βɼզʑ͸ChebyshevεϖΫτϧ બ఺๏Λଟ༻͖ͯͨ͠ɽ

ChebyshevεϖΫτϧબ఺๏ͷద༻๏Λ؆୯ʹ঺հ

͢Δɽ࣍ͷChebyshevల։Λߟ͑Δɽ

uN(x) =

∑N k=0

˜

uk Tk(x) (−1≦x≦1) (2)

͜͜ͰɼTk(x) = cos (k arccosx) ͸ k ࣍ Cheby- shevଟ߲ࣜͰ͋ΔɽChebyshevల։ʹؔ͢Δબ఺͸

CGL(Chebyshev Gauss-Lobatto)఺ɼCG(Chebyshev Gauss)఺ɼCGR(Chebyshev Gauss-Radau)఺͕͋Γɼ

ͦΕͧΕ࣍Ͱ༩͑ΒΕΔɽ

xj=















 cos j

Nπ (CGL఺) cos 2j+ 1

2N+ 2π (CG఺) cos 2j

2N+ 1π (CGR఺)

(j= 0,1,· · ·, N) (3)

ຊ࿦จͰ͸CGL఺Λ༻͍Δ͜ͱʹ͢ΔɽCGL఺্ͷ

ؔ਺஋uj=uN(xj)͕༩͑ΒΕΔͱల։܎਺͸࣍Ͱ༩

͑ΒΕΔɽ

˜ uk = 2

N ck

∑N j=0

1 cj

ujTk(xj) (4)

cj =

{2 (j= 0, N)

1 (ͦͷଞ) (5)

͜ͷؔ܎ࣜ͸൓సެࣜͱݺ͹ΕΔ¹⁾ɽ͜ͷ൓సެ͕ࣜ

ҙຯ͢Δͱ͜Ζ͸ɼۙࣅͷ࣍਺͕બ఺ͷݸ਺ͱͯ͠ઃ

ఆͰ͖Δͱ͍͏͜ͱͰ͋Δɽ͜ͷಛ௃͸ɼࠩ෼๏Ͱ௒

ߴ࣍ۙࣅྫ͑͹100࣍ۙࣅ͢Δ͜ͱΛߟ͑Ε͹༰қʹ

૝૾Ͱ͖Δ͕ɼߴਫ਼౓ܭࢉΛ໨ࢦ͢ͱ͖εϖΫτϧબ

఺๏͕͍͔ʹ࣮༻తͰ͋Δ͔Λ͍ࣔͯ͠ΔɽεϖΫτ ϧબ఺๏ʹΑΔۙࣅղ(ิؒؔ਺)ͷߏ੒͸ɼ໰୊Λ هड़͢Δํఔ͔ࣜΒબ఺্ʹ͓͚Δղͷ஋Λະ஌਺ͱ

͢Δ࿈ཱํఔࣜΛಋ͖ɼͦΕΛղ͍ͯબ఺্ͷղͷ஋

Λܾఆ͢Δɽͦͷޙɼ൓సެࣜΛ༻͍ͯల։܎਺Λܭ

ࢉ͠ɼิؒؔ਺Λߏ੒ͯ͠ํఔࣜͷۙࣅղͱ͢Δɽ

3. ਺஋ܭࢉ݁Ռ

ຊ࿦จͰ͸༩͑ΒΕͨؔ਺ʹରͯ͠ɼબ఺্ͷؔ਺

஋͔Β൓సެࣜΛ༻͍ͯิؒؔ਺Λߏ੒͠ɼ༩͑ΒΕ

ͨؔ਺ͷಛҟੑΛ਺஋తʹௐ΂Δɽ 3.1 ਺஋ܭࢉʹ༻͍ͨؔ਺

ಛҟੑΛ਺஋తʹௐ΂ΔͨΊʹɼಛҟੑ͕Θ͔ͬͯ

͍Δؔ਺Λ༻ҙ͢Δɽຊ࿦จ͸ɼجૅతͳ਺஋࣮ݧ͕

໨తͰ͋ΔͷͰ1ม਺ؔ਺Λߟ͑Δɽ·ͨɼCheby- shevબ఺๏Λద༻͢ΔͨΊɼؔ਺͕ఆٛ͞ΕΔ۠ؒ͸

[−1ɼ1]ͱ͢Δɽಛҟੑʹؔͯ͠ௐ΂͍ͨ͜ͱ͸ɼಛ ҟ఺ͷ৔ॴͱಛҟ఺ʹ͓͚Δ׈Β͔͞Ͱ͋Δɽ͜ͷ໨

తʹԊͬͨؔ਺Λ༻ҙ͢ΔͨΊʹɼ·ͣෆ࿈ଓؔ਺Λ

༻ҙͯ͠ɼͦΕΛ࣍ʑੵ෼͢Δ͜ͱͰɼಛҟ఺ʹ͓͚

Δ׈Β͔͞Λม͍͑ͯ͘ɽຊ࿦จͰ͸ɼx= 0.5Λෆ

࿈ଓ఺ͱ͢Δ࣍ͷf₋1(x)Λجຊʹͯ͠ɼͦΕΛ࣍ʑ

ੵ෼͢Δ͜ͱͰಛҟ఺͕ৗʹx= 0.5Ͱɼಛҟ఺ʹ͓

͚Δඍ෼Մೳ֊਺͕ੵ෼͢Δͨͼʹ1ͭͣͭ૿͍͑ͯ

ؔ͘਺Λ༻ҙ͢Δɽf₋1(x)Ҏ֎͸ɼؔ਺ͷԼ෇਺ࣈ

͕ಛҟ఺ʹ͓͚Δ࿈ଓඍ෼Ͱ͖Δճ਺Λද͢ɽͳ͓ɼ f1(x)ʙf3(x)͸লུͨ͠ɽ

f₋1(x) =

{ 0, −1≦x < ¹₂

1, ¹₂ ≦x≦1 (6)

f0(x) = { ₁

2, −1≦x < ¹₂

x, ¹₂ ≦x≦1 (7)

f4(x) =





x⁴

48 −^x48³ +^x₉₆²−384^x +₃₈₄₀¹ , −1≦x < ¹₂

x⁵

120, ¹₂ ≦x≦1

(8)

ؔ਺ f4(x) ͷάϥϑ͸طʹ Fig.1 ʹ͍ࣔͯ͠Δɽ f₋1(x)ɼf0(x)ͷάϥϑΛFig.3ʹࣔ͢ɽ

0 0.2 0.4 0.6 0.8 1

-1 -0.5 0 0.5 1

(1) Graph off₋1(x)

0.4 0.5 0.6 0.7 0.8 0.9 1 1.1

-1 -0.5 0 0.5 1

(2) Graph off0(x) Fig. 3. Graphs of functions.

3.2 ਺஋ܭࢉ݁Ռ

f₋1(x)ɼf0(x)ɼf4(x)ʹର͢Δ਺஋ܭࢉ݁ՌΛࣔ͢ɽ f1(x)ʙf3(x)ʹର͢Δ਺஋ܭࢉ΋ߦ͕ͬͨɼಉ༷ͷ܏

޲Ͱ͋ͬͨͷͰলུ͢Δɽ਺஋ܭࢉ͸ഒਫ਼౓Ͱߦͬͨɽ

·ͣɼؔ਺ͷ׈Β͔͞Λௐ΂ͨͷ͕࣍ͷFig.4Ͱ

͋Δɽ

Err

N

Err O(N0 )

0.01 0.1 1 10

10 100

(1)f₋₁

Err

N

Err O(N−1 ) O(N−2 )

1e-005 0.0001 0.001 0.01 0.1

10 100

(2)f0

Err

N

Err O(N−4 ) O(N−5 )

1e-012 1e-011 1e-010 1e-009 1e-008 1e-007 1e-006

10 100

(3)f4

Fig. 4. Behaviors ofErr.

ݫີղ͕Θ͔Βͳ͍ํఔࣜͷղͷಛҟੑΛ਺஋తʹ ௐ΂Δͷ͕࠷ऴ໨తͰ͋ΔͷͰɼͦΕΛҙࣝͯ͠ɼޡ

ࠩ͸༩͑ΒΕͨؔ਺ͱۙࣅؔ਺(ิؒؔ਺)ͷࠩͰ͸

ͳ͘ɼۙࣅͷ౓߹͍(࣍਺ɿN)͕ҟͳΔ਺஋ղ(ิ

ؒؔ਺)ͷࠩͰܭΔ͜ͱʹ͢Δɽ͜ͷޡࠩ͸ɼۙࣅ

ؔ਺(ิؒؔ਺)͕ݩͷؔ਺ʹऩଋ͢Δͱ͖͸ۙࣅؔ

਺ͱݩͷؔ਺ͷޡࠩͱಉ஋ʹͳΔ͕ɼͦ͏Ͱͳ͍ͱ

͖͸஫ҙ͕ඞཁͱͳΔɽ༩͑ΒΕͨؔ਺f(x)ʹର͢

Δ ChebyshevεϖΫτϧબ఺๏ͰN ࣍ۙࣅͨؔ͠

਺(ิؒؔ਺)Λfˆ^N(x)Ͱද͢ɽ·ͨɼޡࠩΛଌఆ͢

Δ఺Λx = ˜xi ≡ 2i/640−1 (i = 0 ∼ 640)ͱ͠ɼ fˆ_i^N ≡fˆ^N(˜xi)ͱͯ۠ؒ͠I = [−1ɼ1]ͷઈର஋࠷େ

ͷޡࠩErrΛ

Err= max

i,x˜i∈I|fˆ_i^N+2−fˆ_i^N| (9) Ͱఆٛ͢ΔɽFig.4͸ɼͦΕͧΕͷؔ਺ʹରͯ͠ɼ͜ͷ ޡࠩErrͷৼΔ෣͍Λ͍ࣔͯ͠ΔɽεϖΫτϧ๏ͷಛ

ੑ͔Βɼ͜ͷޡࠩͷऩଋ࣍਺͔Βݩͷؔ਺ͷ׈Β͔͞

͕ਪఆͰ͖ΔɽFig.4(1)ͷऩଋͷ༷ࢠ͸O(N⁰)Ͱ͋

ΔͷͰɼGibbsݱ৅͕ى͖͍ͯΔͱਪఆ͞ΕΔɽଈͪɼ f₋1͸ෆ࿈ଓؔ਺Ͱ͋Δ͜ͱ͕ਪఆ͞ΕΔɽFig.4(2) ͷऩଋͷ༷ࢠ͸O(N⁻¹)Ͱ͋ΔͷͰɼ࿈ଓؔ਺Ͱ͋Δ

͕࿈ଓඍ෼͸Ͱ͖ͳ͍ؔ਺ͱਪఆ͞ΕΔɽFig.4(3)ͷ ऩଋͷ༷ࢠ͸O(N⁻⁵)Ͱ͋ΔͷͰɼ4ճ࿈ଓඍ෼Մೳ

Ͱ5ճ࿈ଓඍ෼͸ෆՄೳͳؔ਺ͱਪఆ͞ΕΔɽFig.1 ʹ͓͍ͯf4(x)ͷಛҟ఺(ղੳతͰͳ͍఺)ͷଘࡏ͕ݟ

ͨ໨Ͱ෼͔Βͳ͔͕ͬͨɼ਺஋ܭࢉͷ݁Ռ͔Βಛҟ఺ ͷଘࡏͱಛҟੑ(׈Β͔͞)͕਺஋తʹਪఆͰ͖ͨɽ

࣍ʹɼಛҟ఺ͷ৔ॴΛಛఆ͢ΔͨΊͷ؆୯ͳํ๏Λ ఏҊ͢Δɽ|fˆ_i^N+2−fˆ_i^N|Λ࠷େʹ͢ΔxiΛx^N_s ͱ͢ Δɽx^N_s ͸ಛҟ఺ͷީิͱͳΔ͕NʹΑͬͯ஋͕มԽ

͢ΔͷͰɼx^N_s ͔Βಛҟ఺ͷ৔ॴΛਪఆ͢Δͷ͸༰қ Ͱͳ͍ɽͦ͜Ͱɼಛҟ఺ΛؚΉ۠ؒΛಛఆ͢Δ͜ͱΛ ߟ͑ΔɽͦͷͨΊʹɼؔ਺͕ఆٛ͞ΕΔ۠ؒI= [−1ɼ 1]ΛM ౳෼ʹ෼ׂͨ͠খ۠ؒIk (k= 1∼M)ɿ

Ik ={x: 2(k−1)/M−1≦x≦2k/M−1} (10) Λߟ͑Δɽ͞Βʹɼ

Errk = max

i,x˜i∈Ik|fˆ_i^N+2−fˆ_i^N| (11) Errk =|Err−Errk| (12) ͱ͢ΔͱɼErrk ͸খ۠ؒ Ik ʹ͓͚ΔޡࠩͰ͋Γɼ Errk͸ɼx^N_s ͕ଘࡏ͢Δ۠ؒͰ͋Ε͹0ͱͳΔɽIkʹ

͓͍ͯɼErrͱErrkͷ஋͕ܻҧ͍ʹҟͳΔΑ͏Ͱ͋ Ε͹ɼErrk͸Errͱ΄΅ಉ͡஋ΛͱΔɽ

Err1

N

Err1 O(N0 )

0.01 0.1 1 10

10 100

Err2

N

Err2 O(N0 )

0.01 0.1 1 10

10 100

(1)Err1 (2)Err2

Err3

N

Err3 O(N0 )

1e-007 1e-006 1e-005 0.0001 0.001 0.01 0.1 1

10 100

Err4

N

Err4 O(N0 )

1e-007 1e-006 1e-005 0.0001 0.001 0.01 0.1 1

10 100

(3)Err3 (4)Err4

Fig. 5. Behaviors of errors forf₋1(x) (M = 4).

Δ׈Β͔͞Λม͍͑ͯ͘ɽຊ࿦จͰ͸ɼx= 0.5Λෆ

࿈ଓ఺ͱ͢Δ࣍ͷf₋1(x)Λجຊʹͯ͠ɼͦΕΛ࣍ʑ

ੵ෼͢Δ͜ͱͰಛҟ఺͕ৗʹx= 0.5Ͱɼಛҟ఺ʹ͓

͚Δඍ෼Մೳ֊਺͕ੵ෼͢Δͨͼʹ1ͭͣͭ૿͍͑ͯ

ؔ͘਺Λ༻ҙ͢Δɽf₋1(x)Ҏ֎͸ɼؔ਺ͷԼ෇਺ࣈ

͕ಛҟ఺ʹ͓͚Δ࿈ଓඍ෼Ͱ͖Δճ਺Λද͢ɽͳ͓ɼ f1(x)ʙf3(x)͸লུͨ͠ɽ

f₋1(x) =

{ 0, −1≦x < ¹₂

1, ¹₂ ≦x≦1 (6)

f0(x) = { ₁

2, −1≦x < ¹₂

x, ¹₂ ≦x≦1 (7)

f4(x) =





x⁴

48 −^x48³ +^x₉₆² −384^x +₃₈₄₀¹ , −1≦x < ¹₂

x⁵

120, ¹₂ ≦x≦1

(8)

ؔ਺ f4(x) ͷάϥϑ͸طʹ Fig.1 ʹ͍ࣔͯ͠Δɽ f₋1(x)ɼf0(x)ͷάϥϑΛFig.3ʹࣔ͢ɽ

0 0.2 0.4 0.6 0.8 1

-1 -0.5 0 0.5 1

(1) Graph off₋1(x)

0.4 0.5 0.6 0.7 0.8 0.9 1 1.1

-1 -0.5 0 0.5 1

(2) Graph off0(x) Fig. 3. Graphs of functions.

3.2 ਺஋ܭࢉ݁Ռ

f₋1(x)ɼf0(x)ɼf4(x)ʹର͢Δ਺஋ܭࢉ݁ՌΛࣔ͢ɽ f1(x)ʙf3(x)ʹର͢Δ਺஋ܭࢉ΋ߦ͕ͬͨɼಉ༷ͷ܏

޲Ͱ͋ͬͨͷͰলུ͢Δɽ਺஋ܭࢉ͸ഒਫ਼౓Ͱߦͬͨɽ

·ͣɼؔ਺ͷ׈Β͔͞Λௐ΂ͨͷ͕࣍ͷFig.4Ͱ

͋Δɽ

Err

N

Err O(N0 )

0.01 0.1 1 10

10 100

(1)f₋₁

Err

N

Err O(N−1 ) O(N−2 )

1e-005 0.0001 0.001 0.01 0.1

10 100

(2)f0

Err

N

Err O(N−4 ) O(N−5 )

1e-012 1e-011 1e-010 1e-009 1e-008 1e-007 1e-006

10 100

(3)f4

Fig. 4. Behaviors ofErr.

ݫີղ͕Θ͔Βͳ͍ํఔࣜͷղͷಛҟੑΛ਺஋తʹ ௐ΂Δͷ͕࠷ऴ໨తͰ͋ΔͷͰɼͦΕΛҙࣝͯ͠ɼޡ

ࠩ͸༩͑ΒΕͨؔ਺ͱۙࣅؔ਺(ิؒؔ਺)ͷࠩͰ͸

ͳ͘ɼۙࣅͷ౓߹͍ (࣍਺ɿN)͕ҟͳΔ਺஋ղ(ิ

ؒؔ਺)ͷࠩͰܭΔ͜ͱʹ͢Δɽ͜ͷޡࠩ͸ɼۙࣅ

ؔ਺(ิؒؔ਺)͕ݩͷؔ਺ʹऩଋ͢Δͱ͖͸ۙࣅؔ

਺ͱݩͷؔ਺ͷޡࠩͱಉ஋ʹͳΔ͕ɼͦ͏Ͱͳ͍ͱ

͖͸஫ҙ͕ඞཁͱͳΔɽ༩͑ΒΕͨؔ਺f(x)ʹର͢

Δ ChebyshevεϖΫτϧબ఺๏ͰN ࣍ۙࣅͨؔ͠

਺(ิؒؔ਺)Λfˆ^N(x)Ͱද͢ɽ·ͨɼޡࠩΛଌఆ͢

Δ఺Λx = ˜xi ≡ 2i/640−1 (i = 0 ∼ 640)ͱ͠ɼ fˆ_i^N ≡fˆ^N(˜xi)ͱͯ۠ؒ͠I = [−1ɼ1]ͷઈର஋࠷େ

ͷޡࠩErrΛ

Err= max

i,x˜i∈I|fˆ_i^N+2−fˆ_i^N| (9) Ͱఆٛ͢ΔɽFig.4͸ɼͦΕͧΕͷؔ਺ʹରͯ͠ɼ͜ͷ ޡࠩErrͷৼΔ෣͍Λ͍ࣔͯ͠ΔɽεϖΫτϧ๏ͷಛ

ੑ͔Βɼ͜ͷޡࠩͷऩଋ࣍਺͔Βݩͷؔ਺ͷ׈Β͔͞

͕ਪఆͰ͖ΔɽFig.4(1)ͷऩଋͷ༷ࢠ͸O(N⁰)Ͱ͋

ΔͷͰɼGibbsݱ৅͕ى͖͍ͯΔͱਪఆ͞ΕΔɽଈͪɼ f₋1͸ෆ࿈ଓؔ਺Ͱ͋Δ͜ͱ͕ਪఆ͞ΕΔɽFig.4(2) ͷऩଋͷ༷ࢠ͸O(N⁻¹)Ͱ͋ΔͷͰɼ࿈ଓؔ਺Ͱ͋Δ

͕࿈ଓඍ෼͸Ͱ͖ͳ͍ؔ਺ͱਪఆ͞ΕΔɽFig.4(3)ͷ ऩଋͷ༷ࢠ͸O(N⁻⁵)Ͱ͋ΔͷͰɼ4ճ࿈ଓඍ෼Մೳ

Ͱ5ճ࿈ଓඍ෼͸ෆՄೳͳؔ਺ͱਪఆ͞ΕΔɽFig.1 ʹ͓͍ͯf4(x)ͷಛҟ఺(ղੳతͰͳ͍఺)ͷଘࡏ͕ݟ

ͨ໨Ͱ෼͔Βͳ͔͕ͬͨɼ਺஋ܭࢉͷ݁Ռ͔Βಛҟ఺

ͷଘࡏͱಛҟੑ(׈Β͔͞)͕਺஋తʹਪఆͰ͖ͨɽ

࣍ʹɼಛҟ఺ͷ৔ॴΛಛఆ͢ΔͨΊͷ؆୯ͳํ๏Λ ఏҊ͢Δɽ|fˆ_i^N+2−fˆ_i^N|Λ࠷େʹ͢ΔxiΛx^N_s ͱ͢

Δɽx^N_s ͸ಛҟ఺ͷީิͱͳΔ͕NʹΑͬͯ஋͕มԽ

͢ΔͷͰɼx^N_s ͔Βಛҟ఺ͷ৔ॴΛਪఆ͢Δͷ͸༰қ Ͱͳ͍ɽͦ͜Ͱɼಛҟ఺ΛؚΉ۠ؒΛಛఆ͢Δ͜ͱΛ ߟ͑ΔɽͦͷͨΊʹɼؔ਺͕ఆٛ͞ΕΔ۠ؒI= [−1ɼ 1]ΛM ౳෼ʹ෼ׂͨ͠খ۠ؒIk (k= 1∼M)ɿ

Ik ={x: 2(k−1)/M −1≦x≦2k/M−1} (10) Λߟ͑Δɽ͞Βʹɼ

Errk = max

i,x˜i∈Ik|fˆ_i^N+2−fˆ_i^N| (11) Errk =|Err−Errk| (12) ͱ͢ΔͱɼErrk ͸খ۠ؒIk ʹ͓͚ΔޡࠩͰ͋Γɼ Errk͸ɼx^N_s ͕ଘࡏ͢Δ۠ؒͰ͋Ε͹0ͱͳΔɽIkʹ

͓͍ͯɼErrͱErrkͷ஋͕ܻҧ͍ʹҟͳΔΑ͏Ͱ͋

Ε͹ɼErrk͸Errͱ΄΅ಉ͡஋ΛͱΔɽ

Err1

N

Err1 O(N0 )

0.01 0.1 1 10

10 100

Err2

N

Err2 O(N0 )

0.01 0.1 1 10

10 100

(1)Err1 (2)Err2

Err3

N

Err3 O(N0 )

1e-007 1e-006 1e-005 0.0001 0.001 0.01 0.1 1

10 100

Err4

N

Err4 O(N0 )

1e-007 1e-006 1e-005 0.0001 0.001 0.01 0.1 1

10 100

(3)Err3 (4)Err4

Fig. 5. Behaviors of errors forf₋1(x) (M = 4).

Err1

N

Err1 O(N0 )

0.01 0.1 1 10

10 100

(1)Err1

Err2

N

Err2 O(N0 )

0.01 0.1 1 10

10 100

Err3

N

Err3 O(N0 )

0.01 0.1 1 10

10 100

(2)Err2 (3)Err3

Err4

N

Err4 O(N0 )

0.01 0.1 1

10 100

Err5

N

Err5 O(N0 )

0.01 0.1 1 10

10 100

(4)Err4 (5)Err5

Fig. 6. Behaviors of errors forf₋₁(x) (M = 5).

f₋1(x)ʹରͯ͠M = 4ɼ5ͱͨ͠ͱ͖ͷܭࢉ݁ՌΛ Figs.5ɼ6ʹࣔ͢ɽ·ͣFigs.5(1)ɼ5(2)͔ΒɼErr1ɼ Err2ͷৼΔ෣͍͕Fig.4(1)ͷErrͱେࠩͳ͍͜ͱ͔

Βɼಛҟ఺͸͜ͷ۠ؒʹͳ͍ͱ൑அ͞ΕΔɽFigs.5(3)ɼ 5(4)͔ΒɼErr3ɼErr4ͷάϥϑ͕N ͕૿Ճ͢Δͱ

͖ަޓʹԼʹಥ͖ൈ͚͍ͯΔ(0ʹͳ͍ͬͯΔ)͜ͱ

͔Βɼಛҟ఺x^N_s ͕྆۠ؒI3 ͱI4Λߦ͖དྷ͍ͯ͠

Δ͜ͱ͕෼͔ΔɽଈͪɼGibbsݱ৅ΛҾ͖ى͜͢ಛҟ

఺(ෆ࿈ଓ఺)͕྆۠ؒI3 = [0, 0.5]ͱI4 = [0.5,1]

ͷڞ௨෦෼x= 0.5ͷۙ๣ʹ͋Δ͜ͱ͕ਪଌ͞ΕΔɽ Figs.6(1)ʙ6(3)ɼ6(5)͔ΒɼErr1ʙErr3ɼErr5ͷৼ

Δ෣͍͸Fig.4(1)ͷErrͱେࠩͳ͘ɼFig.6(4)ͷErr4

ͷάϥϑ͸஋͕খ͗ͯ͢͞(0ͳͷͰ)දࣔ͞Ε͍ͯͳ

͍ɽҎ্͔Βɼಛҟ఺(ෆ࿈ଓ఺)͸x= 0.5ΛؚΉ۠

ؒI4= [0.2,0.6]ʹଘࡏ͢Δͱਪఆ͞ΕΔɽ࣍ʹɼಛ ҟ఺ͷ৔ॴΛΑΓݶఆ͢ΔͨΊʹɼM = 62ɼ64ͱ͠

ͨͱ͖ͷܭࢉ݁ՌΛFigs.7ɼ8ʹࣔ͢ɽͲͪΒʹ͓͍

ͯ΋ɼάϥϑ͕Fig.4(1)ͷErrͱେࠩͳ͍༷ࢠΛࣔ

۠ؒ͢ͰڬΉΑ͏ʹਤΛ഑ஔ͍ͯ͠ΔɽN͕े෼େ͖

͍ͱ͖͕৴པੑ͕ߴ͍ͷͰɼͦͷΑ͏ͳNͷͱ͖ͷά ϥϑ͕Լʹಥ͖ൈ͚͍ͯΔ͔Ͳ͏͔ΛݟΔɽFig.7(2)

͔Βɼಛҟ఺͸۠ؒI47= [0.483· · · ,0.516· · ·]ʹ͋

Δͱਪఆ͞ΕΔɽFigs.8(2)ɼ8(3)͔Βɼಛҟ఺͸྆۠

ؒI48= [0.46875,0.5]ͱI49= [0.5,0.53125]ͷڞ௨

ू߹ͷx= 0.5ͷۙ๣ʹ͋Δ͜ͱ͕ਪଌ͞ΕΔɽ

Err46

N

Err46 O(N0 )

1e-007 1e-006 1e-005 0.0001 0.001 0.01 0.1 1

10 100

(1)Err46

Err47

N

Err47 O(N0 )

1e-007 1e-006 1e-005 0.0001 0.001 0.01 0.1 1

10 100

Err48

N

Err48 O(N0 )

1e-007 1e-006 1e-005 0.0001 0.001 0.01 0.1 1

10 100

(2)Err47 (3) Err48

Fig. 7. Behaviors of errors forf₋₁(x) (M = 62).

Err47

N

Err47 O(N0 )

1e-007 1e-006 1e-005 0.0001 0.001 0.01 0.1 1 10

10 100

Err48

N

Err48 O(N0 )

1e-007 1e-006 1e-005 0.0001 0.001 0.01 0.1 1

10 100

(1)Err47 (2) Err48

Err49

N

Err49 O(N0 )

1e-007 1e-006 1e-005 0.0001 0.001 0.01 0.1 1

10 100

Err50

N

Err50 O(N0 )

1e-007 1e-006 1e-005 0.0001 0.001 0.01 0.1 1 10

10 100

(3)Err49 (4) Err50

Fig. 8. Behaviors of errors forf₋1(x) (M = 64).

Err1

N

Err1 O(N0 )

0.01 0.1 1 10

10 100

(1)Err1

Err2

N

Err2 O(N0 )

0.01 0.1 1 10

10 100

Err3

N

Err3 O(N0 )

0.01 0.1 1 10

10 100

(2)Err2 (3)Err3

Err4

N

Err4 O(N0 )

0.01 0.1 1

10 100

Err5

N

Err5 O(N0 )

0.01 0.1 1 10

10 100

(4)Err4 (5)Err5

Fig. 6. Behaviors of errors forf₋₁(x) (M = 5).

f₋1(x)ʹରͯ͠M = 4ɼ5ͱͨ͠ͱ͖ͷܭࢉ݁ՌΛ Figs.5ɼ6ʹࣔ͢ɽ·ͣFigs.5(1)ɼ5(2)͔ΒɼErr1ɼ Err2ͷৼΔ෣͍͕Fig.4(1)ͷErrͱେࠩͳ͍͜ͱ͔

Βɼಛҟ఺͸͜ͷ۠ؒʹͳ͍ͱ൑அ͞ΕΔɽFigs.5(3)ɼ 5(4)͔ΒɼErr3ɼErr4ͷάϥϑ͕N ͕૿Ճ͢Δͱ

͖ަޓʹԼʹಥ͖ൈ͚͍ͯΔ(0ʹͳ͍ͬͯΔ)͜ͱ

͔Βɼಛҟ఺x^N_s ͕྆۠ؒI3ͱI4Λߦ͖དྷ͍ͯ͠

Δ͜ͱ͕෼͔ΔɽଈͪɼGibbsݱ৅ΛҾ͖ى͜͢ಛҟ

఺(ෆ࿈ଓ఺)͕྆۠ؒI3 = [0,0.5]ͱI4 = [0.5,1]

ͷڞ௨෦෼x= 0.5ͷۙ๣ʹ͋Δ͜ͱ͕ਪଌ͞ΕΔɽ Figs.6(1)ʙ6(3)ɼ6(5)͔ΒɼErr1ʙErr3ɼErr5ͷৼ

Δ෣͍͸Fig.4(1)ͷErrͱେࠩͳ͘ɼFig.6(4)ͷErr4

ͷάϥϑ͸஋͕খ͗ͯ͢͞(0ͳͷͰ)දࣔ͞Ε͍ͯͳ

͍ɽҎ্͔Βɼಛҟ఺(ෆ࿈ଓ఺)͸x= 0.5ΛؚΉ۠

ؒI4= [0.2,0.6]ʹଘࡏ͢Δͱਪఆ͞ΕΔɽ࣍ʹɼಛ ҟ఺ͷ৔ॴΛΑΓݶఆ͢ΔͨΊʹɼM = 62ɼ64ͱ͠

ͨͱ͖ͷܭࢉ݁ՌΛFigs.7ɼ8ʹࣔ͢ɽͲͪΒʹ͓͍

ͯ΋ɼάϥϑ͕Fig.4(1)ͷErrͱେࠩͳ͍༷ࢠΛࣔ

۠ؒ͢ͰڬΉΑ͏ʹਤΛ഑ஔ͍ͯ͠ΔɽN͕े෼େ͖

͍ͱ͖͕৴པੑ͕ߴ͍ͷͰɼͦͷΑ͏ͳNͷͱ͖ͷά ϥϑ͕Լʹಥ͖ൈ͚͍ͯΔ͔Ͳ͏͔ΛݟΔɽFig.7(2)

͔Βɼಛҟ఺͸۠ؒI47= [0.483· · ·,0.516· · ·]ʹ͋

Δͱਪఆ͞ΕΔɽFigs.8(2)ɼ8(3)͔Βɼಛҟ఺͸྆۠

ؒI48 = [0.46875,0.5]ͱI49= [0.5,0.53125]ͷڞ௨

ू߹ͷx= 0.5ͷۙ๣ʹ͋Δ͜ͱ͕ਪଌ͞ΕΔɽ

Err46

N

Err46 O(N0 )

1e-007 1e-006 1e-005 0.0001 0.001 0.01 0.1 1

10 100

(1)Err46

Err47

N

Err47 O(N0 )

1e-007 1e-006 1e-005 0.0001 0.001 0.01 0.1 1

10 100

Err48

N

Err48 O(N0 )

1e-007 1e-006 1e-005 0.0001 0.001 0.01 0.1 1

10 100

(2)Err47 (3)Err48

Fig. 7. Behaviors of errors forf₋₁(x) (M = 62).

Err47

N

Err47 O(N0 )

1e-007 1e-006 1e-005 0.0001 0.001 0.01 0.1 1 10

10 100

Err48

N

Err48 O(N0 )

1e-007 1e-006 1e-005 0.0001 0.001 0.01 0.1 1

10 100

(1)Err47 (2)Err48

Err49

N

Err49 O(N0 )

1e-007 1e-006 1e-005 0.0001 0.001 0.01 0.1 1

10 100

Err50

N

Err50 O(N0 )

1e-007 1e-006 1e-005 0.0001 0.001 0.01 0.1 1 10

10 100

(3)Err49 (4)Err50

Fig. 8. Behaviors of errors forf₋1(x) (M = 64).

f0(x)ʹରͯ͠M = 62ɼ64ͱͨ͠ͱ͖ͷܭࢉ݁ՌΛ Figs.9ɼ10ʹࣔ͢ɽಉ༷ͷߟ࡯͔Βɼಛҟ఺͸M = 64

ͷͱ͖ͷ2۠ؒI48∪I49= [0.46875,0.53125]ʹ͋Δ ͱਪଌ͞ΕΔɽ

Err45

N

Err45 O(N−1 ) O(N−2 )

1e-008 1e-007 1e-006 1e-005 0.0001 0.001 0.01 0.1

10 100

Err46

N

Err46 O(N−1 ) O(N−2 )

1e-008 1e-007 1e-006 1e-005 0.0001 0.001 0.01 0.1

10 100

(1)Err45 (2)Err46

Err47

N

Err47 O(N−1 ) O(N−2 )

1e-008 1e-007 1e-006 1e-005 0.0001 0.001 0.01 0.1

10 100

Err48

N

Err48 O(1/N1 ) O(1/N2 )

1e-008 1e-007 1e-006 1e-005 0.0001 0.001 0.01 0.1

10 100

(3)Err47 (4)Err48

Fig. 9. Behaviors of errors forf0(x) (M = 62).

Err47

N

Err47 O(N−1 ) O(N−2 )

1e-008 1e-007 1e-006 1e-005 0.0001 0.001 0.01 0.1

10 100

Err48

N

Err48 O(N−1 ) O(N−2 )

1e-008 1e-007 1e-006 1e-005 0.0001 0.001 0.01 0.1

10 100

(1)Err47 (2)Err48

Err49

N

Err49 O(N−1 ) O(N−2 )

1e-008 1e-007 1e-006 1e-005 0.0001 0.001 0.01 0.1

10 100

Err50

N

Err50 O(N−1 ) O(N−2 )

1e-008 1e-007 1e-006 1e-005 0.0001 0.001 0.01 0.1

10 100

(3)Err49 (4)Err50

Fig. 10. Behaviors of errors for f0(x) (M = 64).

f4(x)ʹରͯ͠M = 62ɼ64ͱͨ͠ͱ͖ͷܭࢉ݁

ՌΛFigs.11ɼ12ʹࣔ͢ɽಉ༷ͷߟ࡯͔Βɼಛҟ఺͸

M = 64ͷͱ͖ͷ2۠ؒI48∪I49= [0.46875,0.53125] ʹ͋Δͱਪఆ͞ΕΔɽ

Err45

N

Err45 O(N−4 ) O(N−5 )

1e-016 1e-014 1e-012 1e-010 1e-008 1e-006

10 100

Err46

N

Err46 O(N−4 ) O(N−5 )

1e-016 1e-014 1e-012 1e-010 1e-008 1e-006

10 100

(1)Err45 (2) Err46

Err47

N

Err47 O(N−4 ) O(N−5 )

1e-016 1e-014 1e-012 1e-010 1e-008 1e-006

10 100

Err48

N

Err48 O(N−4 ) O(N−5 )

1e-016 1e-014 1e-012 1e-010 1e-008 1e-006

10 100

(3)Err47 (4) Err48

Fig. 11. Behaviors of errors forf4(x) (M = 62).

Err47

N

Err47 O(N−4 ) O(N−5 )

1e-016 1e-014 1e-012 1e-010 1e-008 1e-006

10 100

Err48

N

Err48 O(N−4 ) O(N−5 )

1e-016 1e-014 1e-012 1e-010 1e-008 1e-006

10 100

(1)Err47 (2) Err48

Err49

N

Err49 O(N−4 ) O(N−5 )

1e-016 1e-014 1e-012 1e-010 1e-008 1e-006

10 100

Err50

N

Err50 O(N−4 ) O(N−5 )

1e-016 1e-014 1e-012 1e-010 1e-008 1e-006

10 100

(3)Err49 (4) Err50

Fig. 12. Behaviors of errors forf4(x) (M = 64).

f0(x)ʹରͯ͠M = 62ɼ64ͱͨ͠ͱ͖ͷܭࢉ݁ՌΛ Figs.9ɼ10ʹࣔ͢ɽಉ༷ͷߟ࡯͔Βɼಛҟ఺͸M = 64

ͷͱ͖ͷ2۠ؒI48∪I49= [0.46875,0.53125]ʹ͋Δ ͱਪଌ͞ΕΔɽ

Err45

N

Err45 O(N−1 ) O(N−2 )

1e-008 1e-007 1e-006 1e-005 0.0001 0.001 0.01 0.1

10 100

Err46

N

Err46 O(N−1 ) O(N−2 )

1e-008 1e-007 1e-006 1e-005 0.0001 0.001 0.01 0.1

10 100

(1)Err45 (2)Err46

Err47

N

Err47 O(N−1 ) O(N−2 )

1e-008 1e-007 1e-006 1e-005 0.0001 0.001 0.01 0.1

10 100

Err48

N

Err48 O(1/N1 ) O(1/N2 )

1e-008 1e-007 1e-006 1e-005 0.0001 0.001 0.01 0.1

10 100

(3)Err47 (4)Err48

Fig. 9. Behaviors of errors forf0(x) (M = 62).

Err47

N

Err47 O(N−1 ) O(N−2 )

1e-008 1e-007 1e-006 1e-005 0.0001 0.001 0.01 0.1

10 100

Err48

N

Err48 O(N−1 ) O(N−2 )

1e-008 1e-007 1e-006 1e-005 0.0001 0.001 0.01 0.1

10 100

(1)Err47 (2)Err48

Err49

N

Err49 O(N−1 ) O(N−2 )

1e-008 1e-007 1e-006 1e-005 0.0001 0.001 0.01 0.1

10 100

Err50

N

Err50 O(N−1 ) O(N−2 )

1e-008 1e-007 1e-006 1e-005 0.0001 0.001 0.01 0.1

10 100

(3)Err49 (4)Err50

Fig. 10. Behaviors of errors for f0(x) (M = 64).

f4(x)ʹରͯ͠M = 62ɼ64ͱͨ͠ͱ͖ͷܭࢉ݁

ՌΛFigs.11ɼ12ʹࣔ͢ɽಉ༷ͷߟ࡯͔Βɼಛҟ఺͸

M = 64ͷͱ͖ͷ2۠ؒI48∪I49= [0.46875,0.53125]

ʹ͋Δͱਪఆ͞ΕΔɽ

Err45

N

Err45 O(N−4 ) O(N−5 )

1e-016 1e-014 1e-012 1e-010 1e-008 1e-006

10 100

Err46

N

Err46 O(N−4 ) O(N−5 )

1e-016 1e-014 1e-012 1e-010 1e-008 1e-006

10 100

(1)Err45 (2)Err46

Err47

N

Err47 O(N−4 ) O(N−5 )

1e-016 1e-014 1e-012 1e-010 1e-008 1e-006

10 100

Err48

N

Err48 O(N−4 ) O(N−5 )

1e-016 1e-014 1e-012 1e-010 1e-008 1e-006

10 100

(3)Err47 (4)Err48

Fig. 11. Behaviors of errors for f4(x) (M = 62).

Err47

N

Err47 O(N−4 ) O(N−5 )

1e-016 1e-014 1e-012 1e-010 1e-008 1e-006

10 100

Err48

N

Err48 O(N−4 ) O(N−5 )

1e-016 1e-014 1e-012 1e-010 1e-008 1e-006

10 100

(1)Err47 (2)Err48

Err49

N

Err49 O(N−4 ) O(N−5 )

1e-016 1e-014 1e-012 1e-010 1e-008 1e-006

10 100

Err50

N

Err50 O(N−4 ) O(N−5 )

1e-016 1e-014 1e-012 1e-010 1e-008 1e-006

10 100

(3)Err49 (4)Err50

Fig. 12. Behaviors of errors for f4(x) (M = 64).