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数値解析セミナー
過去の記録 ～06/21｜次回の予定｜今後の予定 06/22～
| 開催情報 | 火曜日 16:30～18:00 数理科学研究科棟(駒場) 002号室 |
|---|---|
| 担当者 | 齊藤宣一、柏原崇人 |
| セミナーURL | https://sites.google.com/g.ecc.u-tokyo.ac.jp/utnas-bulletin-board/ |
2015年02月18日(水)
16:30-18:00 数理科学研究科棟(駒場) 002号室
福島登志夫 氏 (国立天文台)
Precise and fast computation of elliptic integrals and elliptic functions (日本語)
福島登志夫 氏 (国立天文台)
Precise and fast computation of elliptic integrals and elliptic functions (日本語)
[ 講演概要 ]
Summarized is the recent progress of the methods to compute (i) Legendre's normal form complete elliptic integrals of all three kinds, K(m), E(m), and Π(n|m), (ii) Legendre's normal form incomplete elliptic integrals of all three kinds, F(ϕ|m), E(ϕ|m), and Π(ϕ,n|m), (iii) Jacobian elliptic functions, sn(u|m), cn(u|m), dn(u|m), and am(u|m), (iv) the inverse functions of K(m) and E(m), mK(K) and mE(E), (v) the inverse of a general incomplete elliptic integral in Jacobi's form, G(am(u|m),n|m), with respect to u, and (vi) the partial derivatives of sn(u|m), cn(u|m), dn(u|m), E(am(u|m)|m), and Π(am(u|m),n|m) with respect to u and those of F(ϕ|m), E(ϕ|m), and Π(ϕ,n|m) with respect to ϕ. In order to avoid the information loss when n≪1 and/or m≪1, focused are the associate incomplete elliptc integrals defined as B(ϕ|m)=[E(ϕ|m)−(1−m)F(ϕ|m)]/m, D(ϕ|m)=[F(ϕ|m)−E(ϕ|m)]/m, and J(ϕ,n|m)=[Π(ϕ,n|m)−F(ϕ|m)]/n, and their complete versions, B(m)=[E(m)−(1−m)K(m)]/m, D(m)=[K(m)−E(m)]/m, and J(n|m)=[Π(n|m)−K(m)]/n. The main techniques used are (i) the piecewise approximation for single variable functions as K(m), and (ii) the combination of repeated usage of the half and double argument transformations and the truncated Maclaurin series expansions with respect to u=F(ϕ|m). The new methods are of the full double precision accuracy without any chance of cancellation against small input arguments. They run significantly faster than the existing methods: (i) 2.5 times faster than Cody's Chebyshev polynomial approximations for K(m) and E(m), (ii) 2.5 times faster than Bulirsch's cel for Π(n|m), (iii) slightly faster than Bulirsch's el1 for F(ϕ|m), (iv) 3.5 times faster than Carlson's RD for E(ϕ|m), (v) 3.5 times faster than Carlson's RC, RD, RF, and RJ for Π(ϕ,n|m), and (vi) 1.5 times faster than Bulirsch's \texttt{sncndn} for sn(u|m), cn(u|m), and dn(u|m).
Summarized is the recent progress of the methods to compute (i) Legendre's normal form complete elliptic integrals of all three kinds, K(m), E(m), and Π(n|m), (ii) Legendre's normal form incomplete elliptic integrals of all three kinds, F(ϕ|m), E(ϕ|m), and Π(ϕ,n|m), (iii) Jacobian elliptic functions, sn(u|m), cn(u|m), dn(u|m), and am(u|m), (iv) the inverse functions of K(m) and E(m), mK(K) and mE(E), (v) the inverse of a general incomplete elliptic integral in Jacobi's form, G(am(u|m),n|m), with respect to u, and (vi) the partial derivatives of sn(u|m), cn(u|m), dn(u|m), E(am(u|m)|m), and Π(am(u|m),n|m) with respect to u and those of F(ϕ|m), E(ϕ|m), and Π(ϕ,n|m) with respect to ϕ. In order to avoid the information loss when n≪1 and/or m≪1, focused are the associate incomplete elliptc integrals defined as B(ϕ|m)=[E(ϕ|m)−(1−m)F(ϕ|m)]/m, D(ϕ|m)=[F(ϕ|m)−E(ϕ|m)]/m, and J(ϕ,n|m)=[Π(ϕ,n|m)−F(ϕ|m)]/n, and their complete versions, B(m)=[E(m)−(1−m)K(m)]/m, D(m)=[K(m)−E(m)]/m, and J(n|m)=[Π(n|m)−K(m)]/n. The main techniques used are (i) the piecewise approximation for single variable functions as K(m), and (ii) the combination of repeated usage of the half and double argument transformations and the truncated Maclaurin series expansions with respect to u=F(ϕ|m). The new methods are of the full double precision accuracy without any chance of cancellation against small input arguments. They run significantly faster than the existing methods: (i) 2.5 times faster than Cody's Chebyshev polynomial approximations for K(m) and E(m), (ii) 2.5 times faster than Bulirsch's cel for Π(n|m), (iii) slightly faster than Bulirsch's el1 for F(ϕ|m), (iv) 3.5 times faster than Carlson's RD for E(ϕ|m), (v) 3.5 times faster than Carlson's RC, RD, RF, and RJ for Π(ϕ,n|m), and (vi) 1.5 times faster than Bulirsch's \texttt{sncndn} for sn(u|m), cn(u|m), and dn(u|m).
