Power Series Composition and Change of Basis Alin Bostan Bruno Salvy Eric´ Schost Algorithms Project Algorithms Project ORCCA and CSD INRIA Rocquencourt INRIA Rocquencourt University of Western Ontario France France London, ON, Canada [email protected] [email protected] [email protected] ABSTRACT less than n can be multiplied in M(n) operations in K. We Efficient algorithms are known for many operations on trun- impose the usual super-linearity conditions of [17, Chap. 8]. cated power series (multiplication, powering, exponential, Using Fast Fourier Transform algorithms, M(n) can be taken . ). Composition is a more complex task. We isolate a in O(n log(n)) over fields with suitable roots of unity, and large class of power series for which composition can be O(n log(n) log log(n)) in general [31, 14]. performed efficiently. We deduce fast algorithms for con- If g(0) = 0,p the best known algorithm, due to Brent and verting polynomials between various bases, including Euler, Kung, uses O( n log n M(n)) operations in K [11]; in small Bernoulli, Fibonacci, and the orthogonal Laguerre, Hermite, characteristic, a quasi-linear algorithm is known [5]. There Jacobi, Krawtchouk, Meixner and Meixner-Pollaczek. are however special cases of power series g with faster algo- rithms: evaluation at g = λx takes linear time; evaluation Categories and Subject Descriptors: at g = xk requires no arithmetic operation. A non-trivial I.1.2 [Computing Methodologies]: Symbolic and Alge- example is g = x + a, which takes time O(M(n)) when braic Manipulation { Algebraic Algorithms the base field has characteristic zero or large enough [1]. General Terms: Algorithms, Theory Brent and Kung [11] also showed how to obtain a cost in Keywords: Fast algorithms, transposed algorithms, basis O(M(n) log(n)) when g is a polynomial; this was extended conversion, orthogonal polynomials. by van der Hoeven [22] to the case where g is algebraic over K(x). In x2, we prove that evaluation at g = exp(x) − 1 and at g = log(1 + x) can also be performed in O(M(n) log(n)) 1. INTRODUCTION operations over fields of characteristic zero or larger than n. Through the Fast Fourier Transform, fast polynomial mul- Using associativity of composition and the linearity of tiplication has been the key to devising efficient algorithms the map Evalm;n, we show in x2 how to use these spe- for polynomials and power series. Using techniques such cial cases as building blocks, to obtain fast evaluation al- as Newton iteration or divide-and-conquer, many problems gorithms for a large class of power series. This idea was have received satisfactory solutions: polynomial evaluation first used by Pan [28], who applied it to functions of the and interpolation, power series exponentiation, logarithm, form (ax + b)=(cx + d). Our extensionsp cover further exam- . can be performed in quasi-linear time. ples such as 2x=(1 + x)2 or (1 − 1 − x2)=x, for which we In this article, we discuss two questions for which such fast improve the previously known costs. algorithms are not known: power series composition and Bivariate problems. Our results on the cost of evalu- change of basis for polynomials. We isolate special cases, ation (and of the transposed operation) are applied in x3 including most common families of orthogonal polynomials, to special cases of a more general composition, reminiscent for which our algorithms reach quasi-optimal complexity. of umbral operations [30]. Given a bivariate power series Composition. Given a power series g with coefficients in P j F = j≥0 ξj (x)t , we consider the linear map a field K, we first consider the map of evaluation at g X n Evaln(:; F; t):(a0; : : : ; an−1) 7! ξj (x)aj mod x : Eval (:; g): A 2 [x] 7! A(g) mod xn 2 [x] : m;n K m K n j<n Here, K[x]m is the m-dimensional K-vector space of polyno- For instance, with mials of degree less than m. We note Evaln for Evaln;n. 1 X j j To study this problem, as usual, we denote by M a mul- F = = g(x) t ; 1 − tg(x) tiplication time function, such that polynomials of degree j≥0 this is the map Evaln(:; g) seen before. For general F, the conversion takes quadratic time (one needs n2 coefficients for F). Hence, better algorithms can only been found for Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are structured cases; in x3, we isolate a large family of bivari- not made or distributed for profit or commercial advantage and that copies ate series F for which we can provide such fast algorithms. bear this notice and the full citation on the first page. To copy otherwise, to This approach follows Frumkin's [16], which was specific to republish, to post on servers or to redistribute to lists, requires prior specific Legendre polynomials. permission and/or a fee. ISSAC’08, July 20–23, 2008, Hagenberg, Austria. Change of basis. Our framework captures in particular Copyright 2008 ACM 978-1-59593-904-3/08/07 ...$5.00. the generating series of many classical polynomial families, 269 for which it yields at once conversion algorithms between tween particular families, such as Chebyshev, Legendre and the monomial and polynomial bases, in both directions. B´ezier[27, 4], with however a quadratic cost. Floating-point Thus, we obtain in x4 change of basis algorithms of cost algorithms are known as well, of cost O(n) for conversion only O(M(n)) for all of Jacobi, Laguerre and Hermite or- from Legendre to Chebyshev bases [2] and O(n log(n)) for thogonal polynomials, as well as Euler, Bernoulli, and Mott conversions between Gegenbauer bases [25], but the results polynomials (see Table 3). These algorithms are derived in are approximate. Approximate conversions for the Hermite a uniform manner from our composition algorithms; they basis are discussed in [26], with cost O(M(n) log(n)). improve upon the existing results, of cost O(M(n) log(n)) or Note on the base field. For the sake of simplicity, in all 2 O(M(n) log (n)) at best (see below for historical comments). that follows, the base field is supposed to have character- We also obtain O(M(n) log(n)) conversion algorithms for a istic 0. All results actually hold more generally, for fields large class of Sheffer sequences [30, Chap. 2], including actu- whose characteristic is sufficiently large with respect to the arial polynomials, Poisson-Charlier polynomials and Meix- target precision of the computation. However, completely ner polynomials (see Table 4). explicit estimates would make our statements cumbersome. Transposition. A key aspect of our results is their heavy use of transposed algorithms. Introduced under this name 2. COMPOSITION by Kaltofen and Shoup, the transposition principle is an Associativity of composition can be read both ways: in algorithmic theorem with the following content: given an the identity A(f ◦ g) = A(f) ◦ g, f is either composed on algorithm that performs an r ×s matrix-vector product b 7! the left of g or on the right of A. In this section, we discuss Mb, one can deduce an algorithm with the same complexity, the consequences of this remark. We first isolate a class of up to O(r +s) operations, and that performs the transposed t operators f for which both left and right composition can matrix-vector product c 7! M c. In other words, this relates be computed fast. Most results are known; we introduce the cost of computing a K-linear map f : V ! W to that t ∗ ∗ two new ones, regarding exponentials and logarithms. Us- of computing the transposed map f : W ! V . ing these as building blocks, we then define composition se- For the transposition principle to apply, some restrictions quences, which enable us to obtain more complex functions must be imposed on the computational model: we require by iterated compositions. We finally discuss the cost of the that only linear operations in the coefficients of b are per- map Evaln and of its inverse for such functions, showing how formed (all our algorithms satisfy this assumption). See [12] to reduce it to O(M(n)) or O(M(n) log(n)). for a precise statement, Kaltofen's \open problem" [23] for further comments and [7] for a systematic review of some 2.1 Basic Subroutines classical algorithms from this viewpoint. We now describe a few basic subroutines that are the To make the design of transposed algorithms transparent, building blocks in the rest of this article. we choose as much as possible to describe our algorithms in a \functional" manner. Most of our questions boil down Left operations on power series. In Table 1, we list basic composition operators, defined on various subsets of to computing linear maps K[x]m ! K[x]n; expressing algo- rithms as a factorization of these maps into simpler ones K[[x]]. Explicitly, any such operator o is defined on a domain makes their transposition straightforward. In particular, dom(o), given in the third column. Its action on a power series g 2 dom(o) is given in the second column, and the this leads us to systematically indicate the dimensions of n the source (and often target) space as a subscript. cost of computing o(g) mod x is given in the last column. Previous work. The question of efficient change of ba- Operator Action Domain Cost sis has naturally attracted a lot of attention, so that fast Aa (add) a + g K[[x]] 1 algorithms are already known in many cases.