兾傪媮傔傞岞幃

丂丂丂丂丂丂丂丂2002/12/20 擔棫惢嶌強 & 憗堫揷戝妛丂屻 曐斖 ( Ushiro Yasunori )

1. 奣梫

丂寁嶼婡偵傛傞兾偺寁嶼偼1949擭Arctan岞幃偱奐巒偝傟丄1983擭埲崀偼庡偵憡壛憡忔朄
偵側傝丄Arctan岞幃偼巊梡偝傟側偔側偨丅偦偺棟桼偼丄n寘偺兾偺寁嶼偵梫偡傞墘嶼検偑
Arctan岞幃偺O(n2)偵懳偟憡壛憡忔朄偼O(n(log2n)2)偲 戝暆偵嶍尭偱偒傞偨傔偱偁傞丅
丂偟偐偟丄DRM朄 偺敪尒偵傛傝arctan岞幃摍偺桳棙悢媺悢傗楢暘悢傕憡壛憡忔朄偲傎傏
摨摍偺墘嶼検偱寁嶼壜擻側偙偲偑敾柧偟偨丅
丂岞幃偵懳墳偡傞FORTRAN偲C偺兾寁嶼僾儘僌儔儉傪弴偵宖嵹丅杮僾儘僌儔儉偼撪梕傪棟夝
偟偰傕傜偆偙偲傪栚揑偲偟丄寁嶼帪娫媦傃巊梡儊儌儕検偺岠棪壔偼栚揑偲偟偰偄側偄丅

2. 媺悢岞幃

2.1 Arctan岞幃

丂arctan(1/p) = 1/p - 1/(3p3) + 1/(5p5) - 1/(7p7) + 乧乧 側傞揥奐幃偱寁嶼丅
丂懡悢偁傞偑壓婰偵桳柤側岞幃媦傃擔杮恖偺敪尒偟偨岞幃傪帵偡丅
丂(1) 儅僠儞偺岞幃丂丗丂丂丂丂兾/4 = 4arctan(1/5) - arctan(1/239)
丂(2) 僈僂僗偺岞幃丂丗丂丂丂丂兾/4 = 12arctan(1/18) + 8arctan(1/57) - 5arctan(1/239)
丂(3) 肛莨掭辑猫倥偺岞幃丗兾/4 = 8arctan(1/10) - arctan(1/239) - 4arctan(1/515)
丂(4) 辑觅习偺岞幃丂丗丂丂丂兾/4 = 44arctan(1/57) + 7arctan(1/239) - 12arctan(1/682)
丂丂丂丂丂丂丂丂丂丂丂丂丂丂丂丂丂丂丂 + 24arctan(1/12943)
丂(5) 崅栰婌媣梇偺岞幃丗 兾/4 = 12arctan(1/49) + 32arctan(1/57) - 5arctan(1/239)
丂丂丂丂丂丂丂丂丂丂丂丂丂丂丂丂丂丂丂 + 12arctan(1/110443)

2.2 儔儅僰僕儍儞宆岞幃

丂(1) 僠儏僪僲僼僗僉乕偺岞幃
丂丂丂1/兾 = 12/A丒儼n = 0[儺nk=1 (2k-1)(6k-5)(6k-1)/(9k3B3)](-1)n(Cn+D)
丂丂丂A = 6403203/2丂,丂B = 106720丂,丂C = 545140134丂,丂D = 13591409
丂(2) 儔儅僰僕儍儞偺岞幃
丂丂丂1/兾 = 23/2/9801丒儼n = 0[儺nk=1 (2k-1)(4k-3)(4k-1)/(2k31984)](26390n+1103)
丂(3) 儔儅僰僕儍儞偺柍暯曽崻岞幃
丂丂丂1/兾 = 1/3528丒儼n=0[儺nk=1 (2k-1)(4k-3)(4k-1)/(2k335282)](-1)n(21460n+1123)

3. 嶼弍婔壗暯嬒(AGM, Arithmetic Geometric Mean)

3.1 僈僂僗丒儖僕儍儞僪儖偺岞幃

丂弶婜抣丗丂a0=1,b0=2-1/2,t0=1/4 偲偟n寘寁嶼偱偼p=log2n傑偱壓婰傪k=1,2,乧乧,p偲斀暅寁嶼
丂丂丂ak = (ak-1+bk-1)/2丂, 丂bk = (ak-1bk-1)1/2丂, 丂tk = tk-1-2k-1(ak-1-ak)2

丂偡傋偰偺寁嶼偼n寘(+僈乕僪寘)偱峴偄丄n寘偺兾偼壓婰偱媮傑傞丅
丂丂丂兾 = (ap+bp)2/(4tp)

3.2 儃乕儖僂僄僀儞偺4師偺岞幃

丂弶婜抣丗丂a0=1,b0=2-1/4,t0=(5-81/2)/8 偲偟p=(log2n)/2傑偱壓婰傪k=1,2,乧乧,p偲斀暅寁嶼
丂丂丂ak = (ak-1+bk-1)/2丂, 丂bk = (ak-1bk-1(ak-12+ bk-12)/2)1/4
丂丂丂tk = tk-1+4k{ak4 -[(ak2+bk2)/2]2}

丂偡傋偰偺寁嶼偼n寘(+僈乕僪寘)偱峴偄丄n寘偺兾偼壓婰偱媮傑傞丅
丂丂丂兾 = ap+14/(1-2tp)

4. 楢暘悢岞幃

4.1 儭娭悢宯岞幃

丂n寘寁嶼偱n偲摨掱搙偺惍悢k傪梡偄偰壓婰偺傛偆偵寁嶼偡傞丅
丂丂丂兾 = { (2񑺶c乧(2k)) / (1񑧉c乧(2k-1)) }2 / k F(v)

丂丂丂丂丂F(v) = 1 - 1 / [ 1 + v + (13) / { v + (35) / ( v + (57) / ( v + 乧乧 )) } ]
丂丂丂丂丂丂丂,丂v = 8k