Numerical integration in arbitrary-precision ball arithmetic
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1 / 24 Numerical integration in arbitrary-precision ball arithmetic Fredrik Johansson (LFANT, Bordeaux) Journées FastRelax, Inria Sophia Antipolis 7 June 208
2 Numerical integration in Arb 2 / 24 New code for numerical integration in Arb ( since November 207. Paper: Sage interface: sage: C = ComplexBallField(333) sage: C.integral(lambda x, _: sin(x+exp(x)), 0, 8) [ /- 5.97e-96]
3 Example: a nice and smooth function 3 / 24 ( ) I = 0 cosh 2 (0(x 0.2)) + cosh 4 (00(x 0.4)) + cosh 6 dx (000(x 0.6)) (Test problem by Cranley and Patterson, 97)
4 Example: a nice and smooth function 3 / 24 ( ) I = 0 cosh 2 (0(x 0.2)) + cosh 4 (00(x 0.4)) + cosh 6 dx (000(x 0.6)) (Test problem by Cranley and Patterson, 97) Mathematica NIntegrate:
5 Example: a nice and smooth function 3 / 24 ( ) I = 0 cosh 2 (0(x 0.2)) + cosh 4 (00(x 0.4)) + cosh 6 dx (000(x 0.6)) (Test problem by Cranley and Patterson, 97) Mathematica NIntegrate: Octave quad: , error estimate 0 9
6 Example: a nice and smooth function 3 / 24 ( ) I = 0 cosh 2 (0(x 0.2)) + cosh 4 (00(x 0.4)) + cosh 6 dx (000(x 0.6)) (Test problem by Cranley and Patterson, 97) Mathematica NIntegrate: Octave quad: , error estimate 0 9 Sage numerical integral: , error estimate 0 4
7 Example: a nice and smooth function 3 / 24 ( ) I = 0 cosh 2 (0(x 0.2)) + cosh 4 (00(x 0.4)) + cosh 6 dx (000(x 0.6)) (Test problem by Cranley and Patterson, 97) Mathematica NIntegrate: Octave quad: , error estimate 0 9 Sage numerical integral: , error estimate 0 4 SciPy quad: , error estimate 0 9
8 Example: a nice and smooth function 3 / 24 ( ) I = 0 cosh 2 (0(x 0.2)) + cosh 4 (00(x 0.4)) + cosh 6 dx (000(x 0.6)) (Test problem by Cranley and Patterson, 97) Mathematica NIntegrate: Octave quad: , error estimate 0 9 Sage numerical integral: , error estimate 0 4 SciPy quad: , error estimate 0 9 mpmath quad:
9 Example: a nice and smooth function 3 / 24 ( ) I = 0 cosh 2 (0(x 0.2)) + cosh 4 (00(x 0.4)) + cosh 6 dx (000(x 0.6)) (Test problem by Cranley and Patterson, 97) Mathematica NIntegrate: Octave quad: , error estimate 0 9 Sage numerical integral: , error estimate 0 4 SciPy quad: , error estimate 0 9 mpmath quad: Pari/GP intnum: 0.236
10 Example: a nice and smooth function ( ) I = 0 cosh 2 (0(x 0.2)) + cosh 4 (00(x 0.4)) + cosh 6 dx (000(x 0.6)) (Test problem by Cranley and Patterson, 97) Mathematica NIntegrate: Octave quad: , error estimate 0 9 Sage numerical integral: , error estimate 0 4 SciPy quad: , error estimate 0 9 mpmath quad: Pari/GP intnum: Actual value: / 24
11 4 / 24 Results with Arb 64-bit precision: [ /- 4.43e-8] # time s 333-bit precision: [ /- 3.73e-99] # 0.04 s 3333-bit precision: [ /-.39e-00] # 9 s ( s first time)
12 Another example: violent oscillation 5 / sin(x+e x )dx S. Rump (200) noticed that MATLAB s quad returned the incorrect 0.25 after second of computation Rump s INTLAB computes the correct enclosure [ , ] in about s Mahboubi, Melquiond & Sibut-Pinote (206): digit in 80 s and 4 digits in 277 s with CoqInterval
13 Another example: violent oscillation 8 0 sin(x+e x )dx S. Rump (200) noticed that MATLAB s quad returned the incorrect 0.25 after second of computation Rump s INTLAB computes the correct enclosure [ , ] in about s Mahboubi, Melquiond & Sibut-Pinote (206): digit in 80 s and 4 digits in 277 s with CoqInterval Arb (64, 333, 3333 bits): [ /- 3.94e-5] # s [ /- 5.98e-96] # 0.02 s [ /- 2.95e-999] # s (5 s first time) 5 / 24
14 6 / 24 Yet another example: a monster 8 0 (e x e x ) sin(x+e x )dx now with 2980 discontinuities! bit precision: [+/- 2.47e+4] # time 0.5 s, aborted [ /- 4.74e-4] # time 9 s 333-bit precision: [ /- 6.78e-95] # 52 s
![Brute force interval integration 7 / 24 b a f (x)dx (b a)f ([a, b]) + adaptive subdivision of [a,](/docs-images/91/107499503/images/15-0.jpg "b] Simple and general method, but need 2 O(p) evaluations of f for p-bit accuracy when used alone!")
15 Brute force interval integration 7 / 24 b a f (x)dx (b a)f ([a, b]) + adaptive subdivision of [a, b] Simple and general method, but need 2 O(p) evaluations of f for p-bit accuracy when used alone!
16 8 / 24 Efficient integration for analytic f We can achieve p-bit accuracy with n = O(p) work using: Taylor series truncated to order n Quadrature rule with n evaluation points Gauss-Legendre quadrature fast generation of quadrature rules due to F.J. and M. Mezzarobba (208) Error bounds: Using derivatives f (n) on [a, b] Using f on a complex domain around [a, b]
17 Error bounds for Gauss-Legendre quadrature 9 / 24 If f is analytic with f (z) M on an ellipse E with foci, and semi-axes X, Y with ρ = X + Y >, then n f (x)dx w k f (x k ) M ρ 2n C ρ k= X =.25, Y = 0.75, ρ = 2.00 X = 2.00, Y =.73, ρ = 3.73 Fast convergence when no singularities are close to [a, b], but should be combined with subdivision otherwise!
18 Adaptive integration algorithm 0 / 24. Compute (b a)f ([a, b]). If the error is ε, done! 2. Compute f (z) and check analyticity of f on some ellipse E around [a, b]. If the error of Gauss-Legendre quadrature is ε, compute it done! 3. Split at m = (a + b)/2 and integrate on [a, m], [m, b] recursively. Knut Petras (Self-validating integration and approximation of piecewise analytic functions, 2002) pointed out that this guarantees rapid convergence for a large class of functions.
19 Adaptive subdivision performed by Arb / 24 ( ) I = 0 cosh 2 (0(x 0.2)) + cosh 4 (00(x 0.4)) + cosh 6 dx (000(x 0.6)) 49 terminal subintervals (smallest width 2 2 )
20 Adaptive subdivision, complex view 2 / 24 Blue ellipses used for error bounds on the subintervals Red dots: poles of the integrand
21 3 / 24 Benchmark: smooth integrands p Pari/GP mpmath Arb Sub Eval Pari/GP mpmath Arb Sub Eval I 0 = 0 /( + x2 )dx I = 3 0 k= sech2k (0 k (x 0.2k)) dx I 2 = π 0 x sin(x)/(+cos2 (x))dx I 3 = W 0 (x)dx I 4 = 00 0 sin(x)dx I 5 = 8 0 sin(x + ex ) dx I 6 = ( 250 ) e x erf x dx I 7 = +000i Γ(x)dx
22 4 / 24 Endpoint singularities and infinite intervals Convergence requires a, b, f <. Can use manual truncation, e.g. 0 f (x)dx N ε f (x)dx otherwise. p Pari/GP mpmath Arb Sub Eval Pari/GP mpmath Arb Sub Eval E 0 = 0 x 2 dx E = 0 /( + x 2 ) dx M M E 2 = 0 log(x)/( + x)dx E 3 = 0 sech(x) dx M E 4 = 0 e x2 +ix dx E 5 = 0 e x Ai( x) dx M
23 Branch cuts 5 / 24 sage: F = lambda z, _: z.sqrt() sage: F2 = lambda z, a: z.sqrt(analytic=a) # WRONG! # correct 2 z dz sage: CBF.integral(F,, 2) [ /- 7.96e-6] sage: CBF.integral(F2,, 2) [ /- 3.73e-5] # WRONG! # correct +i i z dz sage: CBF.integral(F2, -+CBF(I), --CBF(I)) [+/-.4e-4] + [ /- 5.8e-4]*I
24 Piecewise and discontinuous functions 6 / 24 Functions like x and x on R can be extended to piecewise holomorphic functions on C. f (x) = x f (x + yi) = { (x + yi) 2 x + yi x > 0 = (x + yi) x < 0 (discontinuous at x = 0) f (x) = x f (x + yi) = x (discontinuous at x Z) Note: this trick does not work for b a f (z) dz where f is a complex function. However, if we have a decomposition f (z) = g(z) + h(z)i, we can use f (z) = g(z) 2 + h(z) 2.
25 7 / 24 Integrals with discontinuities in f or f In D 0, p(x) = x 4 + 0x 3 + 9x 2 6x 6 In D 3, u(x) = (x x 2 ), v(x) = max(sin(x), cos(x)) p Time Sub Eval Time Sub Eval D 0 = 0 p(x) ex dx D = 00 0 x dx D 2 = +i i x dx D3 = 0 0 u(x)v(x)dx K M High accuracy with mpmath or Pari/GP is not possible without manually splitting at the singular points.
26 8 / 24 Options The user specifies: Working precision p Absolute and relative tolerances ε abs and ε rel Configurable work limits: Maximum quadrature degree (default: O(p)) Number of calls to the integrand (default: O(p 2 )) Number of queued subintervals (default: O(p)) Use stack (default) or global priority queue for the list of subintervals generated by bisection
27 Applications Special functions: Γ(s, z) = z t s e t dt (Inverse) Laplace/Fourier/Mellin transforms Taylor/Laurent/Fourier coefficients Counting zeros and poles: N P = f (z) 2πi C f (z) dz Acceleration of series (Euler-Maclaurin summation...) 9 / 24
28 20 / 24 Example: Laurent series of elliptic functions (z; τ) = n= 2 a n (τ)z n, a n = (z) dz 2πi γ zn+ Fix τ = i (z) has poles at z = M + Ni Pick γ = square of width centered on z = 0. One segment (n = 00): (M, N Z).
29 2 / 24 Example: Laurent series of elliptic functions Time per integral (n 00): 64 bits: 0.05 seconds 333 bits: 0.8 seconds 3333 bits: 20 seconds Results with 333-bit precision: a[-2] = [ /- 3.57e-98] + [+/-.89e-98]*I a[-] = [+/- 4.e-98] + [+/- 2.57e-98]*I a[0] = [+/-.02e-97] + [+/- 5.39e-98]*I a[] = [+/-.4e-97] + [+/-.35e-97]*I a[2] = [ /- 4.44e-97] + [+/- 2.48e-97]*I a[3] = [+/- 4.47e-97] + [+/- 4.60e-97]*I... a[94] = [ /- 9.24e-70] + [+/- 8.27e-70]*I a[95] = [+/-.37e-69] + [+/-.37e-69]*I a[96] = [+/- 2.93e-69] + [+/- 2.9e-69]*I a[97] = [+/- 5.8e-69] + [+/- 5.82e-69]*I a[98] = [ /- 2.90e-68] + [+/-.7e-68]*I a[99] = [+/- 2.32e-68] + [+/- 2.32e-68]*I a[00] = [+/- 4.95e-68] + [+/- 4.95e-68]*I
30 Example: zeros of the Riemann zeta function 22 / 24 Number of zeros of ζ(s) on R = [0, ] + [0, T ]i: T p Time Eval Sub N (T ) [ /-.94e-6] [ /- 7.78e-6] [ /- 4.25e-5] [ /- 3.0e-4] [ /- 4.06e-3] [ /- 5.53e-5] [ /- 8.09e-5] [ /-.95e-4]
31 23 / 24 Example: ζ(s) -integrals (from Harald Helfgott) i 4 +8i F 9 (s + 2 )F 9(s + ) s 2 ds, F N (s) = ζ(s) p N ( p s ) We compute Taylor models f = g + hi + ε on subsegments [a, a + 0.5], and integrate g 2 + h 2.
32 Example: Stieltjes constants 24 / 24 Joint work with I. Blagouchine (arxiv.org/abs/ ) ζ(s, v) = s + π γ n (v) = 2(n + ) n=0 ( ) n γ n (v)(s ) n n! ( ( log v 2 + ix)) n+ cosh 2 dx (πx) Some pen-and-paper analysis needed for large n: Contour is deformed to go near the saddle point Tight enclosures near the saddle point γ 0 00() e e =
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