The Dynamic Dictionary of Mathematical Functions

Transcription

1 The Dynamic Dictionary of Mathematical Functions AISC July 6, 2010 Joint work with: Alexandre Benoit, Alin Bostan, Frédéric Chyzak, Alexis Darrasse, Stefan Gerhold, Marc Mezzarobba 1 / 31

2 I Motivation 2 / 31

3 Mathematical Functions exp, log, sin, cos, tan, sinh, cosh, tanh, arcsin, arccos, arctan, arcsinh, arccosh, arctanh, Bessel J ν, I ν, Y ν, K ν, Airy Ai and Bi, hypergeometric, generalized hypergeometric, classical orthogonal polynomials, Struve H ν, L ν, M ν, K ν, Lommel s µ,ν and S µ,ν, Anger J ν, Weber E ν, Whittaker W κ,µ, M κ,µ... Special functions Functions that have been met sufficiently often to deserve a name. Scientists need help with these functions 3 / 31

4 Dictionaries Motivation Demonstration DynaMoW Symbolic Computation Among the most cited documents in the scientific literature. Thousands of useful mathematical formulas, computed, compiled and edited by hand. Started between 60 and 30 years ago. 4 / 31

5 Motivation Demonstration DynaMoW Symbolic Computation Progress in the Past 30 Years Two important changes in the way we work: Symbolic Computation. Several million users. Mathematical functions implemented from these dictionaries. The Web New kinds of interaction with documents. 5 / 31

6 First Step: the NIST DLMF Navigation, Exports, Search Engine Still computed, compiled and edited by hand. 6 / 31

7 Motivation Demonstration DynaMoW Symbolic Computation Aim of the project DDMF = Mathematical Handbooks + Computer Algebra + Web 1 Computer algebra algorithms to generate the formulas; 2 Web-like interaction with the document and the computation. Building Blocks: 1 Linear differential equations as a data-structure; 2 New language for maths on the web. (compatible with browsers!). 7 / 31

8 II Demonstration 8 / 31

9 Demo Motivation Demonstration DynaMoW Symbolic Computation 9 / 31

10 III DynaMoW: Dynamic Mathematics on the Web 10 / 31

11 DynaMoW Motivation Demonstration DynaMoW Symbolic Computation Principle The document being generated by the computer algebra system is an object of the language. Consequences: 1 the structure of the document may depend on values that have been computed; 2 intermediate steps can be turned into a mathematical proof in natural language; 3 easy to write demo code. 11 / 31

12 Motivation Demonstration DynaMoW Symbolic Computation Architecture l Web Server Dynamow l Symbolic Computation Engine 12 / 31

13 DynaMoW, an OCaml Library DynaMoW = ocaml + quotations + antiquotations Symbolic result, converted to LaTeX, put into a paragraph let res = <:symb< symbolic expression >> in <:par< some text <:imath< some latex $(symb:res) >> >> Using ocaml values in symbolic computations let n = 23 and s = "foo" in <:symb< f($(int:n), $(str:s)) >> Symbolic objects cast to ocaml types let n = 23 + <:int< symbolic expression >> in... if <:bool< symbolic expression >> then... else... <:unit< f := symbolic expression >> [ChyzakDarrasse] 13 / 31

14 Example Motivation Demonstration DynaMoW Symbolic Computation The proof that a function is odd 14 / 31

15 15 / 31

16 IV Symbolic Computation 16 / 31

17 Defining a Mathematical Function by an Equation Classical: polynomials represent their roots better than radicals. Algorithms: Euclidean division and algorithm, Gröbner bases. More Recent: same for linear differential or recurrence equations. Algorithms: non-commutative analogues & gen. func. About 25% of Sloane s encyclopedia, 60% of Abramowitz & Stegun. eqn+ini. cond.=data structure 17 / 31

18 Guaranteed Numerical Evaluation 1. Inside the disk of convergence 1 Effective majorant-series analysis; 2 Efficient evaluation of truncated series; 3 Time complexity quasi-linear wrt precision. 2. Effective analytic continuation Path: [MezzarobbaSalvy2010] 18 / 31

19 Computation of Identities by Confinement Starting Point k + 1 vectors in dimension k an identity. 19 / 31

20 First Example: Contiguity of Hypergeometric Series F (a, b; c; z) := n=0 (a) n (b) n (c) n n! zn, (x) n := x(x +1) (x +n 1). { z(1 z)f + (c (a + b + 1)z)F abf = 0. S a F := F (a + 1, b; c; z) = z a F + F. S a Gauss 1812: contiguity relation. dim=2 S 2 a F, S a F, F linearly dependent. z 20 / 31

21 Second Example: Mehler s Identity by Confinement n=0 ( ) H n (x)h n (y) un exp 4u(xy u(x 2 +y 2 )) n! = 1 4u 2 1 4u 2 1 Definition of Hermite polynomials H n (t): recurrence of order 2 vector space of dimension 2 over Q(t, n); 2 Product: vector space over Q(x, y, n) generated by H n (x)h n (y) n!, H n+1(x)h n (y) n!, H n(x)h n+1 (y), H n+1(x)h n+1 (y) n! n! recurrence of order at most 4; (confinement) 3 Translate into differential equation (and solve). 21 / 31

22 I. Definition O R 1 := {H(n C 2 ) =(K2 n K 2 ) H(n ) C 2 H(n C 1 ) x, H(0 ) = 1, H(1 ) = 2 x } : O R 2 := subs0h = H 2, x = y, R 1 1; II. Product O R 3 := gfun :- poltorec0 H(n )$ H 2 (n )$ v (n ), 2 R 1, R 2, {v (n C 1 ) $ (n C 1 ) = v (n ), v (1 ) = 1 } 3, 2 H(n ), H 2 (n ), v (n ) 3, c (n ) 1; O gfun :- rectodiffeq0 R 3, c (n ), f (u ) 1; u 3 K 16 u 2 y x K 4 u C 8 u y 2 C 8 u x 2 K 4 x y1 f (u ) C 0 16 u 4 K 8 u 2 C 1 1 Z d \ du f (u ) ] _, f (0 ) = 1 5 O dsolve(%, f (u ) ); O R 2 := 4 H 2 (0 ) = 1, H 2 (n C 2 ) = (K2 n K 2 ) H 2 (n ) C 2 H 2 (n C 1 ) y, H 2 (1 ) = 2 y5 R 3 := 4c (0 ) = 1, c (1 ) = 4 x y, c (2 ) = 8 x 2 y 2 C 2 K 4 y 2 K 4 x 2, c (3 ) = 32 3 x3 y 3 C 24 x y K 16 x y 3 K 16 x 3 y, (16 n C 16 ) c (n ) K 16 x y c (n C 1 ) C 0 K8 n K 20 C 8 y 2 C 8 x 2 1 c (n C 2 ) K 4 x c (n C 3 ) y C (n C 4 ) c (n C 4 )5 III. Differential Equation Z [ K4 x y u C y 2 C x 2 ] ^ \ (2 u K 1 ) (2 u C 1 ) _ I e f (u ) = e 0 Ky 2 K x u C 1 2 u K 1 22 / 31

23 Data-Structure 1 Univariate case: linear differential/recurrence equation + ini. conds; 2 Multivariate case: operators (a Gröbner basis of them) + ini. conds. D-finiteness finite dimension 23 / 31

24 Confinement: Beyond Finite Dimension + Use the Hilbert dimension to drive the computation by increasing degrees. 24 / 31

25 Diff. under + Integration by Parts Algorithm? 1 Ex.: cos zt dt = π 0 1 t 2 2 J 0(z), }{{} (zj zj 0 = 0, J 0 (0) = 1). f (z,t) Proof: initial conditions and z 2 z 2 f (z, t) + z f (z, t) + zf (z, t) = t ( t 2 1 t f (z, t) z ) Creative Telescoping [Zeilberger] Input: (a basis of) linear operators that annihilate f ; Output: A free of t, / t, certificate B, such that A(f ) = B(f ). t Algorithm: sometimes. (Why would they exist?) + variants for multiple sum/int. 25 / 31

26 Algorithms Motivation Demonstration DynaMoW Symbolic Computation z1 1 Hypergeometric case [Zeilberger 1990]; 2 D-finite case: [Chyzak 2000]; 3 Non-D-finite: t z2 Algorithm [ChyzakKauersSalvy2009] for s = 0, 1, 2,..., until dim H J bound: 1 reduce A t B with A := η α (z) z α, B := φ β (z, t) β, α s β M s(i) for undetermined rational η α (z), φ β (z, t). 2 extract coeffs of M s+1 (I) to form a linear system of first order w.r.t. t 3 solve and set J to the ideal of the A s return the pairs (A, B). 26 / 31

27 Typical Examples n k= πi n=0 ( ) H n (x)h n (y) un exp 4u(xy u(x 2 +y 2 )) n! = 1 4u 2 1 4u 2 q k2 (q; q) k (q; q) n k = n k= n ( 1) k q (5k2 k)/2 (q; q) n k (q; q) n+k xj 1 (ax)i 1 (ax)y 0 (x)k 0 (x) dx = ln(1 a4 ) 2πa 2 (1 + 2xy + 4y 2 ) exp ( 4x 2 y 2 1+4y 2 ) y n+1 (1 + 4y 2 ) 3 2 dy = H n(x) n/2! [Mehler1866] [Andrews1974] [GlasserMontaldi1994] [Doetsch1930] 27 / 31

28 Examples of Non-D-Finite Identities 0 0 n k=0 ( ) n i(k + i) k 1 (n k + j) n k = (n + i + j) n k n ( ) { } n k ( 1) m k n + 1 n k! = m k k + 1 m k=0 m k=0 ( ) m B n+k = ( 1) m+n k n k=0 ( ) n B m+k k x k 1 ζ(n, α + βx) dx = β k B(k, n k)ζ(n k, α) x α 1 Li n ( xy) dx = π( α)n y α 0 sin(απ) x s 1 πy s Γ(s) exp(xy)γ(a, xy) dx = sin((a + s)π) Γ(1 a) [Abel1826] [Frobenius1910] [Gessel2003] 28 / 31

29 Special Cases with Specific Algorithms 1 Recurrence for the Taylor coefficients: 1 f (z) dz 2πi z n+1 ; 2 Recurrence for the Chebyshev coefficients: [BenoitSalvy2009] 2 1 f (t)t n (t) π dt; 1 1 t 2 3 Laplace transform: + 0 e zt f (t) dt. 29 / 31

30 Future Work Motivation Demonstration DynaMoW Symbolic Computation More expansions on bases of functions; more integral transforms; families of functions or functions with parameters; automatic generation of numerical code; information on the zeros of functions; user-defined functions. Summary: you want to bookmark / 31

31 THE END 31 / 31