Numerics of classical elliptic functions, elliptic integrals and modular forms

Transcription

1 1 / 49 Numerics of classical elliptic functions, elliptic integrals and modular forms Fredrik Johansson LFANT, Inria Bordeaux & Institut de Mathématiques de Bordeaux KMPB Conference: Elliptic Integrals, Elliptic Functions and Modular Forms in Quantum Field Theory DESY, Zeuthen, Germany 24 October 2017

2 2 / 49 Introduction Elliptic functions F(z + ω 1 m + ω 2 n) = F(z), m, n Z Can assume ω 1 = 1 and ω 2 = τ H = {τ C : Im(τ) > 0} Elliptic integrals R(x, P(x))dx; inverses of elliptic functions Modular forms/functions on H F( aτ+b cτ+d ) = (cτ + d)k F(τ) for ( a b c d ) PSL2 (Z) Related to elliptic functions with fixed z and varying lattice parameter ω 2 /ω 1 = τ H Jacobi theta functions (quasi-elliptic functions) Used to construct elliptic and modular functions

3 3 / 49 Numerical evaluation Lots of existing literature, software (Pari/GP, Sage, Maple, Mathematica, Matlab, Maxima, GSL, NAG,... ). This talk will mostly review standard techniques (and many techniques will be omitted). My goal: general purpose methods with Rigorous error bounds Arbitrary precision Complex variables Implementations in the C library Arb (

4 4 / 49 Why arbitrary precision? Applications: Mitigating roundoff error for lengthy calculations Surviving cancellation between exponentially large terms High order numerical differentiation, extrapolation Computing discrete data (integer coefficients) Integer relation searches (LLL/PSLQ) Heuristic equality testing Also: Can increase precision if error bounds are too pessimistic Most interesting range: digits. (Millions, billions...?)

5 Ball/interval arithmetic 5 / 49 A real number in Arb is represented by a rigorous enclosure as a high-precision midpoint and a low-precision radius: [ ± ] Complex numbers: [m 1 ± r 1 ] + [m 2 ± r 2 ]i. Key points: Error bounds are propagated automatically As cheap as arbitrary-precision floating-point To compute f (x) = k=0 N 1 k=0 rigorously, only need analysis to bound k=n Dependencies between variables may lead to inflated enclosures. Useful technique is to compute f ([m ± r]) as [f (m) ± s] where s = r sup x m r f (x).

6 6 / 49 Reliable numerical evaluation Example: sin(π ) IEEE 754 double precision result: e-16 Adaptive numerical evaluation with Arb: 164 bits: [± ] 128 bits: [ ± ] 192 bits: [ ± ] Can be used to implement reliable floating-point functions, even if you don t use interval arithmetic externally: Float input Arb function Increase precision Output midpoint yes Accurate enough? no

7 Elliptic and modular functions in Arb 7 / 49 PSL 2 (Z) transformations and argument reduction Jacobi theta functions θ 1 (z, τ),..., θ 4 (z, τ) Arbitrary z-derivatives of Jacobi theta functions Weierstrass elliptic functions (n) (z, τ), 1 (z, τ), ζ(z, τ), σ(z, τ) Modular forms and functions: j(τ), η(τ), (τ), λ(τ), G 2k (τ) Legendre complete elliptic integrals K (m), E(m), Π(n, m) Incomplete elliptic integrals F(φ, m), E(φ, m), Π(n, φ, m) Carlson incomplete elliptic integrals R F, R J, R C, R D, R G Possible future projects: The suite of Jacobi elliptic functions and integrals Asymptotic complexity improvements

8 8 / 49 An application: Hilbert class polynomials For D < 0 congruent to 0 or 1 mod 4, H D (x) = ( ( b + )) D x j Z[x] 2a (a,b,c) where (a, b, c) is taken over all the primitive reduced binary quadratic forms ax 2 + bxy + cy 2 with b 2 4ac = D. Example: H 31 = x x x Algorithms: modular, complex analytic D Degree Bits Pari/GP classpoly CM Arb s 0.8 s 0.4 s 0.14 s s 8 s 29 s 17 s s 82 s 436 s 274 s

9 Some visualizations 9 / 49 The Weierstrass zeta-function ζ( i, τ) as the lattice parameter τ varies over [ 0.25, 0.25] + [0, 0.15]i.

10 Some visualizations 10 / 49 The Weierstrass elliptic functions ζ(z, i) (left) and σ(z, i) (right) as z varies over [ π, π], [ π, π]i.

11 Some visualizations 11 / 49 The function j(τ) on the complex interval [ 2, 2] + [0, 1]i. The function η(τ) on the complex interval [0, 24] + [0, 1]i.

12 Some visualizations 12 / 49 Plot of j(τ) on [ 13, ] + [0, ]i. Plot of η(τ) on [ 2, ] + [0, ]i.

13 Approaches to computing special functions 13 / 49 Numerical integration (integral representations, ODEs) Functional equations (argument reduction) Series expansions Root-finding methods (for inverse functions) Precomputed approximants (not applicable here)

14 14 / 49 Brute force: numerical integration For analytic integrands, there are good algorithms that easily permit achieving 100s or 1000s of digits of accuracy: Gaussian quadrature Clenshaw-Curtis method (Chebyshev series) Trapezoidal rule (for periodic functions) Double exponential (tanh-sinh) method Taylor series methods (also for ODEs) Pros: Simple, general, flexible approach Can deform path of integration as needed Cons: Usually slower than dedicated methods Possible convergence problems (oscillation, singularities) Error analysis may be complicated for improper integrals

15 Poisson and the trapezoidal rule (historical remark) 15 / 49 In 1827, Poisson considered the example of the perimeter of an ellipse with axis lengths 1/π and 0.6/π: I = 1 2π 2π sin 2 (θ)dθ = 2 E(0.36) = π Poisson used the trapezoidal approximation I I N = 4 N N /4 k= sin 2 (2πk/N ). With N = 16 (four points!), he computed I and proved that the error is < In fact I N I = O(3 N ). See Trefethen & Weideman, The exponentially convergent trapezoidal rule, 2014.

16 A model problem: computing exp(x) 16 / 49 Standard two-step numerical recipe for special functions: (not all functions fit this pattern, but surprisingly many do!) 1. Argument reduction exp(x) = exp(x n log(2)) 2 n 2. Series expansion exp(x) = [ exp(x/2 R ) ] 2 R exp(x) = 1 + x + x2 2! + x3 3! +... Step (1) ensures rapid convergence and good numerical stability in step (2).

17 17 / 49 Reducing complexity for p-bit precision Principles: Balance argument reduction and series order optimally Exploit special (e.g. hypergeometric) structure of series How to compute exp(x) for x 1 with an error of ? Only reduction: apply x x/2 reduction 1000 times Only series evaluation: use 170 terms (170! > ) Better: apply 1000 = 32 reductions and use 32 terms This trick reduces the arithmetic complexity from p to p 0.5 (time complexity from p 2+ε to p 1.5+ε ). With a more complex scheme, the arithmetic complexity can be reduced to O(log 2 p) (time complexity p 1+ε ).

18 Evaluating polynomials using rectangular splitting (Paterson and Stockmeyer 1973; Smith 1989) N i=0 xi in O(N ) cheap steps + O(N 1/2 ) expensive steps ( + x + x 2 + x 3 ) + ( + x + x 2 + x 3 ) x 4 + ( + x + x 2 + x 3 ) x 8 + ( + x + x 2 + x 3 ) x 12 This does not genuinely reduce the asymptotic complexity, but can be a huge improvement (100 times faster) in practice. 18 / 49

19 19 / 49 Elliptic functions Elliptic integrals Argument reduction Move to standard domain (periodicity, modular transformations) Move parameters close together (various formulas) Series expansions Theta function q-series Multivariate hypergeometric series (Appell, Lauricella... ) Special cases Modular forms & functions, theta constants Complete elliptic integrals, ordinary hypergeometric series (Gauss 2 F 1 )

20 Modular forms and functions 20 / 49 A modular form of weight k is a holomorphic function on H = {τ : τ C, Im(τ) > 0} satisfying ( ) aτ + b F = (cτ + d) k F(τ) cτ + d for any integers a, b, c, d with ad bc = 1. A modular function is meromorphic and has weight k = 0. Since F(τ) = F(τ + 1), the function has a Fourier series (or Laurent series/q-expansion) F(τ) = n= m c n e 2iπnτ = n= m c n q n, q = e 2πiτ, q < 1

21 Some useful functions and their q-expansions 21 / 49 Dedekind ( eta ) function η aτ+b cτ+d = ε(a, b, c, d) cτ + dη(τ) η(τ) = e πiτ/12 n= ( 1)n q (3n2 n)/2 The j-invariant ( ) j aτ+b cτ+d = j(τ) j(τ) = 1 q q q2 + j(τ) = 32(θ θ8 3 + θ8 4 )3 /(θ 2 θ 3 θ 4 ) 8 Theta constants (q = e πiτ ) (θ 2, θ 3, θ 4 ) = n= (q (n+1/2)2, q n2, ( 1) n q n2) Due to sparseness, we only need N = O( p) terms for p-bit accuracy (so the evaluation takes p 1.5+ε time).

22 22 / 49 Argument reduction for modular forms PSL 2 (Z) is generated by ( ) and ( By repeated use of τ τ + 1 or τ 1/τ, we can move τ to the fundamental domain { τ H : z 1, Re(z) 1 2}. In the fundamental domain, q exp( π 3) = , which gives rapid convergence of the q-expansion. )

23 Practical considerations 23 / 49 Instead of applying F(τ + 1) = F(τ) or F( 1/τ) = τ k F(τ) step by step, build transformation matrix g = ( ) a b c d and apply to F in one step. This improves numerical stability g can usually be computed cheaply using machine floats If computing F via theta constants, apply transformation for F instead of the individual theta constants.

24 Fast computation of eta and theta function q-series 24 / 49 Consider N n=0 qn2. More generally, q P(n), P Z[x] of degree 2. Naively: 2N multiplications. Enge, Hart & J, Short addition sequences for theta functions, 2016: Optimized addition sequence for P(0), P(1),... (2 speedup) Rectangular splitting: choose splitting parameter m so that P has few distinct residues mod m (logarithmic speedup, in practice another 2 speedup) Schost & Nogneng, On the evaluation of some sparse polynomials, 2017: N 1/2+ε method (p 1.25+ε time complexity) using FFT Faster for p > in practice

25 Jacobi theta functions 25 / 49 Series expansion: θ 3 (z, τ) = q n2 w 2n, n= and similarly for θ 1, θ 2, θ 4. q = e πiτ, w = e πiz The terms eventually decay rapidly (there can be an initial hump if w is large). Error bound via geometric series. For z-derivatives, we compute the object θ(z + x, τ) C[[x]] (as a vector of coefficients) in one step. θ(z+x, τ) = θ(z, τ)+θ (z, τ)x+...+ θ(r 1) (z, τ) x r 1 +O(x r ) C[[x]] (r 1)!

26 Argument reduction for Jacobi theta functions 26 / 49 Two reductions are necessary: Move τ to τ in the fundamental domain (this operation transforms z z, introduces some prefactors, and permutes the theta functions) Reduce z modulo τ using quasiperiodicity General formulas for the transformation τ τ = aτ+b cτ+d are given in (Rademacher, 1973): z = θ n (z, τ) = exp(πir/4) A B θ S (z, τ ) z i cτ + d, A = cτ + d, ( B = exp z 2 ) πic cτ + d R, S are integers depending on n and (a, b, c, d). The argument reduction also applies to θ(z + x, τ) C[[x]].

27 Elliptic functions 27 / 49 The Weierstrass elliptic function (z, τ) = (z + 1, τ) = (z + τ, τ) (z, τ) = 1 z 2 + [ ] 1 (z + m + nτ) 2 1 (m + nτ) 2 n 2 +m 2 0 is computed via Jacobi theta functions as (z, τ) = π 2 θ2(0, 2 τ)θ3(0, 2 τ) θ2 4 (z, τ) π2 [ θ1 2 θ 4 (z, τ) 3 3 (0, τ) + θ3(0, 4 τ) ] Similarly σ(z, τ), ζ(z, τ) and (k) (z, τ) using z-derivatives of theta functions. With argument reduction for both z and τ already implemented for theta functions, reduction for is unnecessary (but can improve numerical stability).

28 Some timings For d decimal digits (z = 5 + 7i, τ = 7 + i/ 11): Function d = 10 d = 10 2 d = 10 3 d = 10 4 d = 10 5 exp(z) log(z) η(τ) j(τ) (θ i (0, τ)) 4 i= (θ i (z, τ)) 4 i= (z, τ) (, ) ζ(z, τ) σ(z, τ) / 49

29 Elliptic integrals 29 / 49 Any elliptic integral R(x, P(x))dx can be written in terms of a small basis set. The Legendre forms are used by tradition. Complete elliptic integrals: K (m) = π/2 dt 0 = 1 1 m sin 2 t 0 E(m) = π/2 0 Π(n, m) = π/2 0 1 m sin 2 t dt = 1 0 (1 n sin 2 t) dt ( 1 t 2 )( 1 mt 2 ) dt = 1 1 m sin 2 t 0 Incomplete integrals: F(φ, m) = φ dt 0 = sin φ 1 m sin 2 t 0 E(φ, m) = φ 0 Π(n, φ, m) = φ 0 1 m sin 2 t dt = sin φ 0 (1 n sin 2 t) 1 mt 2 1 t 2 dt (1 nt 2 ) dt ( 1 t 2 )( 1 mt 2 ) dt = sin φ 1 m sin 2 t 0 1 mt 2 1 t 2 dt (1 nt 2 ) dt 1 t 2 1 mt 2 dt 1 t 2 1 mt 2

30 30 / 49 Complete elliptic integrals and 2 F 1 The Gauss hypergeometric function is defined for z < 1 by 2F 1 (a, b, c, z) = k=0 (a) k (b) k (c) k z k k!, (x) k = x(x + 1) (x + k 1) and elsewhere by analytic continuation. The 2 F 1 function can be computed efficiently for any z C. K (m) = 1 2 π 2F 1 ( 1 2, 1 2, 1, m) E(m) = 1 2 π 2F 1 ( 1 2, 1 2, 1, m) This works, but it s not the best way!

31 Complete elliptic integrals and the AGM 31 / 49 The AGM of x, y is the common limit of the sequences a n+1 = a n + b n, b n+1 = a n b n 2 with a 0 = x, b 0 = y. As a functional equation: ( x + y M(x, y) = M 2, ) xy Each step doubles the number of digits in M(x, y) x y convergence in O(log p) operations (p 1+ε time complexity). K (m) = π 2M(1, 1 m), E(m) = (1 m)(2mk (m) + K (m))

32 Numerical aspects of the AGM 32 / 49 Argument reduction vs series expansion: O(1) terms only. Slightly better than reducing all the way to a n b n < 2 p : π 4K (z 2 ) = 1 2 z2 8 5z z z O(z10 ) Complex variables: simplify to M(z) = M(1, z) using M(x, y) = xm(1, y/x). Some case distinctions for correct square root branches in AGM iteration. Derivatives: can use finite (central) difference for M (z) (better method possible using elliptic integrals), higher derivatives using recurrence relations.

33 Incomplete elliptic integrals Incomplete elliptic integrals are multivariate hypergeometric functions. In terms of the Appell F 1 function F 1 (a, b 1, b 2 ; c; x, y) = where x, y < 1, we have F(z, m) = Problems: z 0 m,n=0 (a) m+n (b 1 ) m (b 2 ) n (c) m+n m! n! x m y n dt 1 m sin 2 t = sin(z) F 1( 1 2, 1 2, 1 2, 3 2 ; sin2 z, m sin 2 z) How to reduce arguments so that x, y 1? How to perform analytic continuation and obtain consistent branch cuts for complex variables? 33 / 49

34 Branch cuts of Legendre incomplete elliptic integrals 34 / 49

35 Branch cuts of F(z, m) with respect to z / 49

36 Branch cuts of F(z, m) with respect to z (continued) 36 / 49

37 Branch cuts of F(z, m) with respect to z (continued) 37 / 49

38 Branch cuts of F(z, m) with respect to z (continued) 38 / 49

39 Branch cuts of F(z, m) with respect to z (continued) 39 / 49

40 Branch cuts of F(z, m) with respect to m 40 / 49 Conclusion: the Legendre forms are not nice as building blocks.

41 Carlson s symmetric forms 41 / 49 In the 1960s, Bille C. Carlson suggested an alternative basis set for incomplete elliptic integrals: R F (x, y, z) = 1 2 R J (x, y, z, p) = 3 2 Advantages: 0 0 dt (t + x)(t + y)(t + z) dt (t + p) (t + x)(t + y)(t + z) R C (x, y) = R F (x, y, y), R D (x, y, z) = R J (x, y, z, z) Symmetry unifies and simplifies transformation laws Symmetry greatly simplifies series expansions The functions have nice complex branch structure Simple universal algorithm for computation

42 Evaluation of Legendre forms 42 / 49 For π 2 Re(z) π 2 : F(z, m) = sin(z) R F (cos 2 (z), 1 m sin 2 (z), 1) Elsewhere, use quasiperiodic extension: F(z + kπ, m) = 2kK (m) + F(z, m), k Z Similarly for E(z, m) and Π(n, z, m). Slight complication to handle (complex) intervals straddling the lines Re(z) = (n )π. Useful for implementations: variants with z πz.

43 Symmetric argument reduction 43 / 49 We have the functional equation ( x + λ R F (x, y, z) = R F 4, y + λ 4, z + λ ) 4 where λ = x y + y z + z x. Each application reduces the distance between x, y, z by a factor 1/4. Algorithm: apply reduction until the distance is ε, then use an order-n series expansion with error term O(ε N ). For p-bit accuracy, need p/(2n ) argument reduction steps. (A similar functional equation exists for R J (x, y, z, p).)

44 Series expansion when arguments are close ( R F (x, y, z) = R 1 1/2 2, 1 2, 1 2, x, y, z) ( R J (x, y, z, p) = R 1 3/2 2, 1 2, 1 2, 1 2, 1 2, x, y, z, p, p) Carlson s R is a multivariate hypergeometric series: R a (b; z) = = M=0 M=0 (a) M ( n j=1 b T M (b 1,..., b n ; 1 z 1,..., 1 z n ) j) M (a) M ( n j=1 b T M j) M z a n ( b 1,..., b n 1 ; 1 z 1 z n,..., 1 z n 1 z n ), T M (b 1,..., b n, w 1,..., w n ) = m m n=m n j=1 (b j ) mj (m j )! wm j j Note that T M Const p(m) max( w 1,..., w n ) M, so we can easily bound the tail by a geometric series. 44 / 49

45 45 / 49 A clever idea by Carlson: symmetric polynomials Using elementary symmetric polynomials E s (w 1,..., w n ), T M ( 1 2, w) = m 1 +2m nm n=m ( 1) M+ j m j ( 1 2) j m j n j=1 E m j j (w) (m j )! We can expand R around the mean of the arguments, taking w j = 1 z j /A where A = 1 n n j=1 z j. Then E 1 = 0, and most of the terms disappear! Carlson suggested expanding to M < N = 8: A 1/2 R F (x, y, z) = 1 E E E E 2E E E E 2 2 E O(ε8 ) Need p/16 argument reduction steps for p-bit accuracy.

46 46 / 49 Rectangular splitting for the R series The exponents of E m 2 2 E m 3 3 appearing in the series for R F are the lattice points m 2, m 3 Z 0 with 2m 2 + 3m 3 < N. m 3 (terms appearing for N = 10) m 2 Compute powers of E 2, use Horner s rule with respect to E 3. Clear denominators so that all coefficients are small integers. O(N 2 ) cheap steps + O(N ) expensive steps For R J, compute powers of E 2, E 3, use Horner for E 4, E 5.

47 47 / 49 Balancing series evaluation and argument reduction Consider R F : p = wanted precision in bits O(ε N ) = error due to truncating the series expansion O(N 2 ) = number of terms in series O(p/N ) = number of argument reduction steps for ε N = 2 p Overall cost O(N 2 + p/n ) is minimized by N p 0.333, giving p arithmetic complexity (p time complexity). Empirically, N 2p 0.4 is optimal (due to rectangular splitting). Speedup over N = 8 at d digits precision: d = 10 d = 10 2 d = 10 3 d = 10 4 d =

48 48 / 49 Some timings We include K (m) (computed by AGM), F(z, m) (computed by R F ) and the inverse Weierstrass elliptic function: 1 (z, τ) = 1 2 z dt (t e1 )(t e 2 )(t e 3 ) = R F (z e 1, z e 2, z e 3 ) Function d = 10 d = 10 2 d = 10 3 d = 10 4 d = 10 5 exp(z) log(z) η(τ) K (m) F(z, m) (z, τ) (z, τ)

49 49 / 49 Quadratic transformations It is possible to construct AGM-like methods (converging in O(log p) steps) for general elliptic integrals and functions. Problems: The overhead may be slightly higher at low precision Correct treatment of complex variables is not obvious Unfortunately, I have not had time to study this topic. However, see the following papers: The elliptic logarithm ( 1 ): John E. Cremona and Thotsaphon Thongjunthug, The complex AGM, periods of elliptic curves over and complex elliptic logarithms, Elliptic and theta functions: Hugo Labrande, Computing Jacobi s θ in quasi-linear time, 2015.