Precise & Fast Computation of Elliptic Functions & Elliptic ... - arith 22

Precise & Fast Computation of Elliptic Functions & Elliptic Integrals aka

How Sushi Mania became Samurai Swordsmith Toshio Fukushima (Nat’l Astron. Obs. Japan) ResearchGate Fukushima Click

Sushi

Samurai Sword

3

Sushi


Asteroid Itokawa, Eros, Ida, Kleopatra, …

4

Why Elliptic Function/Integral?


Samurai Sword for Astronomer






Elliptic F. = Extension of Trig./Hyp. F






Elliptic Function: 3D Rotational Dynamics Elliptic Integral: Gravity Field Evaluation sn(u|0) = sin u, cn(u|0) = cos u, dn(u|0) = 1 sn(u|1) = tanh u, cn(u|1) = dn(u|1) = 1/cosh u

Elliptic I. = Ext. of Inverse Trig./Hyp. F


F(phi|0) = phi, F(phi|1) = atanh(sin phi) 5

General Elliptic Integral


Pdx Definition: G ≡ ∫ Q
P: Polyn.





Q: Degree 3 or 4 Polyn.

Reduction Th.: Legendre (1825-1832)

G (ϕ , n | m ) = aF (ϕ | m ) + bE (ϕ | m ) + cΠ (ϕ , n | m ) + R (ϕ , n | m )




F, E, Pi: Legendre’s 1st, 2nd, & 3rd Kind Elliptic I R: General Rational F phi: Amplitude, m: Parameter, n: Characteristic 6

st 1 ,

nd 2 ,

rd 3

& Kind Elliptic I st
1 nd
2

F (ϕ | m ) ≡ ∫

ϕ

0

dθ ∆ (θ | m )

ϕ

E (ϕ | m ) ≡ ∫ ∆ (θ | m ) dθ 0

rd
3




ϕ

dθ

0

(1 − n sin θ ) ∆ (θ | m )

Π (ϕ , n | m ) ≡ ∫

Jacobi’s Delta F

2

∆ (ϕ | m ) ≡ 1 − m sin 2 θ 7

Associate Elliptic I B (ϕ | m ) ≡ ∫

ϕ

0

E (ϕ | m ) − (1 − m ) F (ϕ | m ) = m 1 − m sin 2 θ 2

cos θ dθ

D (ϕ | m ) ≡ ∫

ϕ

0

J (ϕ , n | m ) ≡ ∫

ϕ

0

F (ϕ | m ) − E (ϕ | m ) = 2 m 1 − m sin θ sin 2 θ dθ

Π (ϕ , n | m ) − F (ϕ | m ) = n 1 − m sin 2 θ

sin 2 θ dθ 2 θ) − 1 n sin (

8

Jacobian Elliptic F




Gauss and Abel Trio: sn, cn, dn Amplitude F = Inv. Incompl. EI: 1st Kind

ϕ = am(u | m) ↔ u = F (ϕ | m )


Jacobian EF = Trig. F of Amplitude

sn ( u | m ) = sin ϕ , cn ( u | m ) = cos ϕ , dn ( u | m ) = ∆ (ϕ | m ) 9

10

Existing Methods: I



Complete EI of 1st, 2nd Kind: K(m), E(m) Cody’s Chebyshev Approx.


Genius of Hastings (1955)

K ( m ) = K1 ( m ) ln (1 − m ) + K 0 ( m )


Cody and Wait (1980)


“Software Manual for the Elementary Functions” 11

Existing Methods: II


Complete EI





Incomplete EI







Bulirsch’s cel: Precise but not-so-Fast. Bulirsch’s el1: Fast and Precise Bulirsch’s el2, el3: Fast but Imprecise Carlson’s RF, RD, RJ : Precise but Slow

Jacobian EF


Bulirsch’s sncndn: Precise but not-so-Fast. 12

Bulirsch and Carlson


R.Z. Bulirsch


Elliptic I/F, GBS Method




B.C. Carlson


Symmetric Elliptic I

13

Necessity is Mother of Invention 14

New Methods: I





Single Variable F: K(m), E(m), … “Divide & Rule”: Piecewise Minimax Appr. Fukushima (2009b, 2011a, 2013c, 2015) Estrin’s Scheme Most Efficient Parallel Algorithm
2.0-3.5 times faster than Horner if using AVX2 @ Intel Haswell CPU


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New Methods: II



Addition Th. of EF: Lagrange Double Argument Transf. 2 scd c2 − s2d 2 d 2 − ms 2 c 2 sn 2u = , cn 2u = , dn 2u = 4 4 1 − ms 1 − ms 1 − ms 4




scd2 (Fukushima, NM, 2013a)




1. Reduce u to sufficiently small u’=u/2N 2. Evaluate Truncated Maclaurin Series w.r.t. u’ 3. Apply Double Argument Transf. N times 16

Application to st 1 Kind Incomplete EI




y

Half Arg. Formula y ' = 1 + 1 − 1 + 1 − y my ( )( ) 2 y=sn (u|m) elf (Fukushima, NM, 2010a)





1. 2. 3. 4.

Compute y=sin2(phi) Apply Half Arg. Formula N times to Reduce y Evaluate F’ by Maclaurin Series w.r.t. y’ F(phi|m) = 2N F’ 17

Application to nd rd 2 & 3 Kind EI


Double Arg. Formulas of D and J

D = 2 D '+ t , J = 2 J '+ T ( t , h )


t = s / c, h = n (1 − n )( n − m )

elbdj (Fukushima, JCAM, 2011b,2012a)





1. 2. 3. 4.

Compute y=sin2(phi) Apply Half Arg. Formula N times to Reduce y Evaluate D’ & J’ by Maclaurin Series w.r.t. y’ Apply Double Arg. Formula N times for D and J 18

Universal ArcTangent F


Definition

 tan −1 ht / h ( h > 0)   T (t, h ) ≡  t ( h = 0)  −1 tanh −ht / −h ( h < 0 ) 

(

(




)

)

uatan (Fukushima, JCAM, 2012a, Appendix)

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New Method for J(n|m)



Add. Th. w.r.t. n: Jacobi Double Charact. Transf. 2 ( c + d ) J − yK J'= c (1 + c )(1 + d )




y ≡ n/m c = 1 − y , d = 1 − my

celj (Fukushima, JCAM, 2013d) 20

CPU Time Comparison


nanosec @ PC with Intel Core i7-2675QM

Elliptic Integral/Function K(m)

Bulirsch Carlson Cody new 93

330

B(m), D(m)

186

922

29

B(m), D(m), J(n|m)

383

1871

159

F(phi|m)

178

313

173

712

213

1426

464

B(phi|m), D(phi|m) B(phi|m), D(phi|m), J(phi,n|m) sn(u|m), cn(u|m), dn(u|m)

165

62

24

116 21

Conclusion



Single Var. F: Piecewise Minimax Approx. Multi-Var. F: Half/Double Arg. Transf.








Forward Problem: All Elliptic I/F Inverse Problem: General Elliptic I Derivatives: Recursive Computation

1.4-32 times Speed-Up except F(phi|m) Fortran 90 Software @ ResearchGate 22