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, …

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Why Elliptic Function/Integral?

� Samurai Sword for Astronomer� Elliptic Function: 3D Rotational Dynamics

� Elliptic Integral: Gravity Field Evaluation

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

GeneralElliptic Integral

� Definition:� P: Polyn.

� Q: Degree 3 or 4 Polyn.

� Reduction Th.: Legendre (1825-1832)

� F, E, Pi: Legendre’s 1st, 2nd, & 3rd Kind Elliptic I

� R: General Rational F

� phi: Amplitude, m: Parameter, n: Characteristic

PdxG

Q≡ ∫

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( ) ( ) ( ) ( ) ( ), | | | , | , |G n m aF m bE m c n m R n mϕ ϕ ϕ ϕ ϕ= + + Π +

1st, 2nd, & 3rd

Kind Elliptic I

� 1st

� 2nd

� 3rd

� Jacobi’s Delta F ( ) 2| 1 sinm mϕ θ∆ ≡ −

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( ) ( )0

| |E m m dϕ

ϕ θ θ≡ ∆∫

( )( ) ( )20

, |1 sin |

dn m

n m

ϕ θϕ

θ θΠ ≡

− ∆∫

( )( )0

||

dF m

m

ϕ θϕ

θ≡

∆∫

Associate Elliptic I

( )( ) ( ) ( )2

20

| 1 |cos|

1 sin

E m m F mdB m

mm

ϕ ϕ ϕθ θϕ

θ

− −≡ =

−∫

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( )( ) ( )2

20

| |sin|

1 sin

F m E mdD m

mm

ϕ ϕ ϕθ θϕ

θ

−≡ =

−∫

( )( )

( ) ( )2

2 20

, | |sin, |

1 sin 1 sin

n m F mdJ n m

nn m

ϕ ϕ ϕθ θϕ

θ θ

Π −≡ =

− −∫

Jacobian Elliptic F

� Gauss and Abel

� Trio: sn, cn, dn

� Amplitude F = Inv. Incompl. EI: 1st Kind

� Jacobian EF = Trig. F of Amplitude

( )am( | ) |u m u F mϕ ϕ= ↔ =

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

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Existing Methods: I

� Complete EI of 1st, 2nd Kind: K(m), E(m)

� Cody’s Chebyshev Approx.� Genius of Hastings (1955)

� Cody and Wait (1980)� “Software Manual for the

Elementary Functions”

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

Existing Methods: II

� Complete EI� Bulirsch’s cel: Precise but not-so-Fast.

� Incomplete EI� 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.

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Bulirsch and Carlson� R.Z. Bulirsch

� Elliptic I/F, GBS Method

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� B.C. Carlson� Symmetric Elliptic I

Necessity is Mother of Invention

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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.

� 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

2 2 2 2 2 2

4 4 4

2sn 2 , cn 2 , dn 2

1 1 1

scd c s d d ms cu u u

ms ms ms

− −= = =

− − −

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Application to1st Kind Incomplete EI

� Half Arg. Formula

� y=sn2(u|m)

� elf (Fukushima, NM, 2010a)� 1. Compute y=sin2(phi)

� 2. Apply Half Arg. Formula N times to Reduce y

� 3. Evaluate F’ by Maclaurin Series w.r.t. y’

� 4. F(phi|m) = 2N F’

( )( )'

1 1 1 1

yy

y my=

+ − + −

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Application to2nd & 3rd Kind EI

� Double Arg. Formulas of D and J

� elbdj (Fukushima, JCAM, 2011b,2012a)� 1. Compute y=sin2(phi)

� 2. Apply Half Arg. Formula N times to Reduce y

� 3. Evaluate D’ & J’ by Maclaurin Series w.r.t. y’

� 4. Apply Double Arg. Formula N times for D and J

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

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( )( )/ , 1t s c h n n n m= = − −

Universal ArcTangent F

� Definition

� uatan (Fukushima, JCAM, 2012a, Appendix)

( )

( ) ( )

( )

( ) ( )

1

1

tan / 0

, 0

tanh / 0

ht h h

T t h t h

ht h h

−

−

>

≡ =

− − <

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New Methodfor J(n|m)

� Add. Th. w.r.t. n: Jacobi

� Double Charact. Transf.

� celj (Fukushima, JCAM, 2013d)

( )

( )( )

2'

1 1

c d J yKJ

c c d

+ −=

+ +

20

/y n m≡

1 , 1c y d my= − = −

CPU Time Comparison� nanosec @ PC with Intel Core i7-2675QM

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Elliptic Integral/Function Bulirsch Carlson Cody new

K(m) 93 330 62 24

B(m), D(m) 186 922 29

B(m), D(m), J(n|m) 383 1871 159

F(phi|m) 178 313 173

B(phi|m), D(phi|m) 712 213

B(phi|m), D(phi|m), J(phi,n|m) 1426 464

sn(u|m), cn(u|m), dn(u|m) 165 116

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 @ ResearchGate22