Philippe Flajolet and Symbolic Computation

Transcription

1 Philippe Flajolet and Symbolic Computation Bruno Salvy 1. Combinatorics and Analysis of Algorithms 4 2. Complex Analysis 5 3. Symbolic Computation 7 While Philippe Flajolet did not publish many articles in journals or conferences devoted specifically to symbolic computation, symbolic computation played an important rôle in the way he dealt with mathematics. This is obvious in the Symbolic Method that is the first stage of Analytic Combinatorics and is briefly summarized in 1 below. It is also very important in his approach to complex analysis as we develop in 2. Finally, his taste for computation also expressed itself in the development of a few new and useful algorithms in computer algebra per se, presented in 3. This introduction aims at presenting the basic ideas in the articles of this chapter, with a few comments on the historical context and the evolutions of these ideas. The articles discussed here are: Computer algebra libraries for combinatorial structures [46]; A finite sum of products of binomial coefficients [81]; The SIGSAM challenges: symbolic asymptotics in practice [49]; On the non-holonomic character of logarithms, powers, and the nth prime function [28]; Lindelöf representations and (non-)holonomic sequences [29]; Holonomic symmetric functions [79]; Fast computation of special resultants [5]. The first one contains a longer summary of the computational aspects of the work led in the Algorithms project till

2 4 PHILIPPE FLAJOLET AND SYMBOLIC COMPUTATION 1. Combinatorics and Analysis of Algorithms The most important connection between Philippe Flajolet s work and Symbolic Computation is the automatic analysis of algorithms. The idea dates back to early work with Jean-Marc Steyaert around 1981 [71] and is described together with related works [72, 53, 54, 56] in Chapter II of this Volume. The principle is that Analytic Combinatorics (see Vol. 1) is sufficiently systematic that it can be automated, giving rise to a symbolic computation of combinatorial structures and also of complexities of algorithms operating over these structures. At this time, in the early 80 s, the classical idea of using generating functions for the enumeration of structured combinatorial objects was complemented by a new idea, namely that the complexity of algorithms could be captured by another kind of generating function, called complexity descriptor. Thus, for a combinatorial structure C and a programme P operating over objects of C (with a size function. ), two functions are related: C(z) = X c2c z c ( c!), P(z) =X c2c cost(p(c)) z c ( c!), where the factorial in the denominators is present only when different labelings of the individual nodes of the structures have to be distinguished. In the same way that a simple dictionary translates equations over combinatorial classes into equations over generating functions, another dictionary emerged, that converted program operations (sequence, loop, branching) into equations over the complexity descriptors. Next, the asymptotic behaviour could be extracted from the singular behaviour of these generating functions. The automation of this principle became more concrete with a sabbatical of Gaston Gonnet at Inria in 1984/85. At that time Gaston Gonnet was finishing the first public version of the Maple computer algebra system. Philippe Flajolet immediately saw the interest of this tool, both in itself and for its applications to the analysis of algorithms. As soon as the next year, he co-organized with Jean-Marc Steyaert and Marc Giusti a course in computer algebra with Maple at École polytechnique. Among the 20 students or so were Paul Zimmermann and myself, who got hooked and would spend the following years doing their PhDs on the automatic analysis of algorithms and the system. We refer to Paul s introduction to the next Chapter for more information on these works [71, 72, 53, 54, 56]. Rather than a closed system like, it soon became clear that users were interested in having a toolbox of computer algebra routines dealing with combinatorial structures and their generating functions. The last sparkle leading to the development of the Maple combstruct package was the new algorithm for random generation developed by Flajolet, Zimmermann and Van Cutsem in 1994 [77] and presented in Chapter 3 of this Volume. This thread of work evolved in its own direction, but the idea of analyzing algorithms operating on recursively structured data-sets was never completely forgotten. To date, the nicest approach, generalizing and leading to multivariate generating functions for the complexity estimates, is the one developed by Marni Mishna [83] using attribute grammars.

3 2. COMPLEX ANALYSIS 5 2. Complex Analysis Symbolic Computation is also central to Philippe Flajolet s approach to complex analysis. Thanks to Cauchy s residue theorem, a large number of computations of integrals are reduced to the location of singularities and the computation of local expansions there. Those are more amenable to a symbolic computation that the computation of the integral of a function viewed as continuous over the range of integration. Singularity Analysis. The first example is the now familiar process of Singularity Analysis due to Flajolet and Odlyzko [41] (see Volume I), where it is used for the computation of the asymptotic behaviour of the coefficients of a large family of generating functions from the classical formula relating coefficient to generating function: [z n ]f(z) = 1 2 i I f(z) dz z n+1. Mellin Transform. Another important class of applications of this point of view is provided by the Mellin transform (described in detail in Volume III). Under rather mild technical conditions, an analytic function f(z) can be written as the inverse Mellin transform of its Mellin transform f? (s). This gives rise to an integral representation of the form f(z) = 1 Z f? (s)z s ds, 2 i (a) where (a) is a vertical line inside the strip of convergence of f?. Then, by shifting the contour to the left, one recovers very precisely the behaviour of f near 0, the poles 1, 2,... of the transform and their residues c 1,c 2,... corresponding to an expansion f(z) c 1 z 1 + c 2 z 2 +. In addition, shifting the contour to the right instead of the left provides a similar expansion as z! +1. Numerous examples are given in Volume III. Euler Sums [51]. A variant of this method leading to summatory formulæ is the basis of the automatic computation of many identities relating so-called Euler sums. These are sums like (1) X n 1 H n n 2 =2 (3), discovered by Euler in 1742, where H n =1+1/2+ +1/n is the nth harmonic number (below we also use the generalized harmonic number H n (k) =1+1/2 k + + 1/n k ). The idea is to synthesize this sum as the sum of residues of a simple function. In the case of this example, the function of interest is f(s) :=( ( s) + ) 2 /s 2, where is Euler s constant and is the logarithmic derivative of Euler s function. By differentiating the functional equation (s + 1) = s (s), one gets that (s + 1) =

4 6 PHILIPPE FLAJOLET AND SYMBOLIC COMPUTATION (s)+1/s, from which it follows that has poles at the nonpositive integers and only there. By induction, the residue at n 2 N is obtained as 1 (2) ( s) = s!n s n + H n + X (( 1) k H n (k+1) (k + 1))(s n) k, k 1 where it is understood that H 0 = H (k+1) 0 =0. From there, it follows that the residue of f(s) at n for a positive integer n is simply 2H n /n 2 2/n 3. Conditions on the growth of f(s) at infinity ensure that the sum of its residues over all n>0is equal to its residue at 0, namely 2 (3), which proves Eq. (1). (See [51] for details.) A small collection of expansions similar to Eq. (2) makes it possible to build a whole family of kernels like, leading to a variety of identities including a few ones like 1X (H n ) 3 (n + 1) 4 = (7) 33 (3) (4) + 2 (2) (5) 4 n=1 that had been conjectured from numerical experiments by Bailey, Borwein and Girgensohn [1]. The process can be automated completely: we wrote a program that takes as input a kernel and returns the corresponding summatory formula, with which we produced most of the formulas in [51]. Magic Duality. The kernel / sin s is particularly useful. It lets one write an alternating sum F (z) = X n>0 (n)( z) n, z! 0 as a Lindelöf integral (3) Z 1 2 i C (s)z s sin s ds provided an appropriate analytic lifting of the coefficient sequence (n) can be found. From there, shifting the contour to the right yields the asymptotic expansion F (z) X n 0 ( n)( z) n, z!1 under technical conditions or minor adjustments. This is the principle of what Philippe Flajolet liked to call Ramanujan s magic duality. It was used for instance in the analysis of quadtrees to obtain a symbolic representation of a constant that governs the subdominant term in the asymptotic behaviour [33]. Rice s Method. Yet another variant of that same idea, with kernel (z n)/ (z+ 1) leads to useful integral representations of binomial sums P n n k=0 k ( 1)n k f(k) and forms the basis of Rice s method, detailed further in Volume III.

5 3. SYMBOLIC COMPUTATION 7 3. Symbolic Computation Holonomy [81, 49]. In the early 90 s, Philippe Flajolet came back from a visit to D. E. Knuth having learned about Doron Zeilberger s work on algorithms for definite summation and integration based on the theory of holonomic functions. During the next academic year (91-92), he gave a presentation of the univariate theory at the Algorithms Seminar and convinced Kevin Compton and Dominique Gouyou-Beauchamps to give presentations of the multivariate case and of the symmetric function case, respectively. He also wrote a summary of that last talk for the proceedings [79]. In a word, holonomic series 1 in one variable are solutions of linear differential equations with polynomial coefficients, while holonomic sequences in one variable are solutions of linear recurrence equations with polynomial coefficients. Generating functions provide a bridge between both notions. These series and sequences enjoy many closure properties, since the sum or product of holonomic series or sequences is again holonomic. Many elementary and special functions fall into that class, including exp, sin, cos, their hyperbolic counterparts or their inverses (arcsin,... ), functions derived from the Laplacian like Bessel functions, Airy functions,... and the whole class of algebraic functions that contains the generating series of various classes of trees. With Paul Zimmermann, we then developed the first version of the gfun package in 1992 (the paper [84] appeared much later, having been lost by the editor!). The principle is that differential or recurrence equations can be viewed as a data-structure to represent their solutions. Operations are then performed directly on the equations. As an example, we dealt [81] with a question from the Problems and Solutions Section of the December 1992 issue of SIAM Review, asking to determine nx 2 2 1/4 1/4 =: S, m n m m=0 that arises from the calculation of the shift of the frequency of an electromagnetic TM wave-mode caused by a small metallic cylinder in a resonant cavity. Defining a n as the inner binomial, b n as its square and c n as the sum of b m b n m, we obtain automatically a recurrence for the sum in three steps. First, the series (1+z) 1/4 is defined by a linear differential equation, from which a linear recurrence for its coefficients a n is deduced. By closure, a recurrence for b n = a 2 n is then obtained, whence a differential equation for its generating series B(z). Again by closure, the square C(z) =B(z) 2 satisfies a linear differential equation from where a linear recurrence for its coefficients c n follows. It turns out that this recurrence has order 1 and an explicit solution is easily computed. S =( 1) n 1/2 n 3 1. This is the terminology used by Philippe Flajolet and part of the combinatorial community. These series are sometimes called D-finite, which avoids a blurring effect on the notion of holonomy in several variables. This introduction sticks to Philippe s vocabulary.

6 8 PHILIPPE FLAJOLET AND SYMBOLIC COMPUTATION Another example was provided by a challenge from SIGSAM, asking for the value of the coefficient [x 6N ](1 + x) 4N (1 + x + x 2 ) 2N (1 + x + x 2 + x 3 + x 4 ) N for the specific value N = 500. For large N, expanding this polynomial of degree 12N is not possible. However, the three polynomial factors satisfy linear differential equations of order 1 (obtained, e.g., by taking the logarithmic derivative), hence by closure their product satisfies a linear differential equation too (note that here N can be kept as a parameter, if desired). This differential equation can then be translated into a linear recurrence for the coefficients and then the coefficient be computed efficiently both in time and space [49]. Today (2013), with a recent version of gfun, this computation takes about half a second, while values up to N = 10, 000 are obtained in half a minute (this is an integer with 28,571 decimal digits). The theory generalizes to multivariate series or sequences. A series S(z 1,...,z n ) is called D-finite when its partial derivatives at all orders generate a finite-dimensional vector space over the rational functions in z 1,...,z n. Besides the previous closure properties (sum, product) that are easily seen to hold, a new one occurs: the diagonal of a D-finite series is D-finite [82]. Consequently, other operations like the Hadamard product, or from there taking the odd or even part of a series, also preserve D-finiteness. Algorithms for integration and summation have been discovered in this context by Doron Zeilberger and later developed in the Algorithms Project by Frédéric Chyzak [8, 6], together with implementations. More recently, a lot of interesting activity in this area comes from RISC Linz (Manuel Kauers, Christoph Koutschan, Peter Paule). The generalization to symmetric functions is due to Ira Gessel [78] and has been briefly summarized by Philippe Flajolet [79]. Symmetric functions are series in an infinite number of variables x 1,x 2,... subject to a certain invariance under renumbering of the variables. Many bases of symmetric functions are known and in particular the power symmetric functions, with p n defined as the sum of the nth power of all variables. Such a series is called D-finite by Gessel when each specialization of all but a finite number of the p i s to 0 makes it D-finite in the previous sense. Again, this definition is motivated by a new closure property, this time under a specific scalar product. From there, Gessel was able to prove the D-finiteness of several combinatorial counting functions, like k-regular graphs or standard tableaux of bounded height. Some of this theory was later made algorithmic in the Algorithms Project [7] Non-holonomy [28, 29]. The algorithmic richness of holonomic series and sequences makes it desirable to detect when a sequence or series is D-finite. Conversely, it is useful to be able to prove that sequences like (4) log n, p 1 n, p n,,n n,e pn,e 1/n, (n p 2) H n do not satisfy a linear recurrence with polynomial coefficients (here p n is the nth prime number). Here also, a nice interplay between symbolic computation and complex analysis was developed by Philippe Flajolet and co-authors [28, 29]. The basic idea is that solutions of linear differential equations with polynomial coefficients are

7 3. SYMBOLIC COMPUTATION 9 very much constrained. First, they can only have finitely many singularities (thus tan(z) cannot satisfy a linear differential equation, even of huge order; similarly, the sequence of numbers of integer partitions cannot satisfy a linear recurrence). Next, a classification of their possible asymptotic behaviours has been known since the end of the 19th century. It is a combination of terms of the form exp(p (Z 1/r ))Z log k Z, where P is a polynomial, r is a positive integer, is an algebraic integer, k is a nonnegative integer and different points can be considered by setting Z =(z z 0 ). So for instance, Lambert s W function, which is related to the generating function of Cayley trees and is defined by W (z)exp(w(z)) = z cannot satisfy a linear differential equation because of a term in log log z in its asymptotic expansion. As a consequence, the sequence n n of number of labelled trees cannot satisfy a linear recurrence. Finally, starting from an explicit sequence like those in Eq. (4) and using a Lindelöf representation from Eq. (3), reduces the matter to a study of the asymptotic behaviour of this integral. This can be achieved with more or less technicalities in a large number of cases, see [28, 29] for details Fast resultant computation [5]. Philippe Flajolet, together with Cyril Banderier [3], studied lattice paths starting at the origin and such that each step increases the abscissa by 1, while the ordinate is increased (or decreased) by an amount belonging to a given fixed finite set S Z. They considered variants depending on whether the path is completely unconstrained, or has to stay in the positive half quadrant. In both cases, they distinguished those paths ending at height 0 from the others. They give explicit forms of all four generating functions, making explicit their algebraic character (See also Volume V, Chapter 4). To the set S is associated a characteristic polynomial P (u) = X b2s u b =: dx k= c p k u k, (c, d in N). It is straightforward that the unconstrained walks with z marking size and u marking final height are enumerated by the rational generating function 1/(1 zp(u)). More interestingly, the excursions (walks staying in N N and ending at height 0) are enumerated by the algebraic generating function E(z) = ( 1)c 1 cy u j (z), p c z where the series u j (z), j =1,...,c are the formal power series solutions of 1 zp(u) =0(thus branches tending to 1 at 0 are excluded). The other two generating series are similarly expressed in terms of the u j s. Computationally, the next step is to compute a bivariate polynomial that vanishes at E(z), given P (u). This is a standard question in elimination theory that is classically rephrased in terms of resultants. In terms of complexity, the resultant of two univariate polynomials of degree n can be computed in a quasi-optimal number O(n log 2 n) of arithmetic operations. The main use of resultants for elimination is, an in this application, in the case of bivariate polynomials. The resultant is then a j=1

8 10 PHILIPPE FLAJOLET AND SYMBOLIC COMPUTATION polynomial of degree n 2. It can be computed by evaluating both polynomials at n 2 +1 points, computing their univariate resultants efficiently and reconstructing the result by interpolation, but this requires O(n 3 log 2 n) operations, which is far from optimal when the input and output have size O(n 2 ). In this generality, we do not know of any better algorithm. However, for the computation of E(z), Banderier and Flajolet designed a faster algorithm (in O((c + d) log(c + d)) operations). Their basic idea was then extended to an important class of resultants [5]. The starting point is to represent a polynomial by its Newton sums: if P (z) = Q i (1 iz), then the generating series of its Newton sums is given by the logarithmic derivative: S(z) = X ( X n>0 i n i )z n = zp 0 (z) P (z). Using Newton iteration, this expansion can be computed at precision N in O(N log N) operations. Conversely, taking S(z) at precision N = n + 1is sufficient to reconstruct P (z) =exp R ( S/z), again in O(N log N) operations using fast algorithms of computer algebra. Given another polynomial Q(z) = Q j (1 jz) also in this representation, one can then efficiently compute the polynomials Y Y (1 i j z) and (1 ( i + j )z) i,j from their Newton sums, obtained via X X i n n j = X ( i j ) n, exp( X i j i,j i i,j i z)exp( X j jz) =exp( X i,j ( i + j )z). That way, resultants in those important classes can all be computed in O(n 2 log n) operations (see [5] for details).