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On Enumeration Problems in Lie–Butcher Theory H Future Generation Computer Systems 19 (2003) 1197–1205 On enumeration problems in Lie–Butcher theory H. Munthe-Kaas∗, S. Krogstad Department of Computer Science, University of Bergen, N-5020 Bergen, Norway Abstract The algebraic structure underlying non-commutative Lie–Butcher series is the free Lie algebra over ordered trees. In this paper we present a characterization of this algebra in terms of balanced Lyndon words over a binary alphabet. This yields a systematic manner of enumerating terms in non-commutative Lie–Butcher series. © 2003 Published by Elsevier Science B.V. Keywords: Butcher theory; Trees; Enumeration; Lie group integrators; Runge–Kutta methods 1. Summary 2. Introduction to Lie–Butcher theory Let g be the free Lie algebra over ordered trees, Lie series, dating back to Sophus Lie (1842–1899), with a grading induced by the number of nodes in the is a version of Taylor series adapted to general man- trees. In the first part of this paper we prove that gn, the ifolds. Butcher series, invented by Butcher [4,5] homogeneous component of degree n, has dimension: and developed further in [8] (see also [9])isan adaption of Taylor series to the study of order con- n d (g ) = 1 µ 2 . ditions of Runge–Kutta methods. In recent papers dim n n d d 2 d|n [6,10,12,14,15,17,19], efforts have been made to ex- tend the theory of Runge–Kutta methods from Rn For n = 1, 2, 3,... these numbers are 1, 1, 3, 8, 25, to Lie groups and homogeneous spaces. Some very 75, 245, 800, 2700, 9225, 32 065,... This sequence interesting recent developments establish the con- counts the number of order conditions in the nection between Butcher theory and renormalization Crouch–Grossman–Owren–Marthinsen methods for theory in physics [3]. In the light of this development integrating differential equations on manifolds. It does it seems important to further investigate the algebraic also count the number of balanced Lyndon words structure of non-commutative Lie–Butcher series on over a binary alphabet. manifolds. In the last part of the paper we establish explicitly Without going into details we will in this section the 1–1 correspondence between balanced binary Lyn- briefly review some basic results, yielding the con- don words and Hall basis elements for the free Lie nection between Lie–Butcher theory and the free Lie algebra of ordered trees. algebra over ordered trees. Definition 1. The set of ordered trees (OT) is defined ∗ Corresponding author. by the following rules: E-mail addresses: [email protected] (H. Munthe-Kaas), [email protected] (S. Krogstad). (1) The one node tree is an ordered tree, •∈OT. 0167-739X/$ – see front matter © 2003 Published by Elsevier Science B.V. doi:10.1016/S0167-739X(03)00045-1 1198 H. Munthe-Kaas, S. Krogstad / Future Generation Computer Systems 19 (2003) 1197–1205 (2) If τ1,τ2,... ,τk ∈ OT, then the ordered k-tuple The resulting linear space of all 0th and higher order τ = (τ1,τ2,... ,τk) ∈ OT. We call τi the right invariant derivations is denoted G, the universal branches of τ. enveloping algebra of g.Weletexp:g → G denote the formal exponential series: We let |τ| denote the degree of the tree (number of ∞ j nodes): V exp(V) = . j! j=0 |•|=1, |(τ1,... ,τk)|=|τ1|+···+|τk|+1. (1) Given a point p ∈ M, the Lie series development of a function φ ∈ FV(M) is given as The number of ordered trees of degree n + 1isgiven by the Catalan numbers: φ(exp(tV) · p) = exp(tV)[φ](p). ( n) n C(n) = 2 ! . In the commutative case where G = M = R , acting n!(n + 1)! upon itself by translations, this is just Taylor series. The number of ordered trees with 1, 2, 3,... nodes is In the study of numerical methods for integrating 1, 1, 2, 5, 14, 42, 132,... (2), it is very useful to characterize a curve z(t) ∈ M, We will briefly introduce Lie–Butcher series, related z(0) = p, in terms of elementary differentials of f . to Runge–Kutta methods on manifolds. Let M be a homogeneous space acted upon transitively from left Definition 2. Given a point p ∈ M and a smooth by a Lie group G, p → g · p, where g ∈ G, p ∈ f : M → g.Toeveryτ ∈ OT there corresponds an M. Let g be the Lie algebra of G and exp : g → G elementary differential Fp(τ) ∈ g, defined as follows: the exponential map. For V ∈ g and p ∈ M,welet F (•) = f(p), V · p ∈ TpM denote the tangent: p F ((τ ,τ ,... ,τ )) ∂ p 1 2 k V · p = exp(sV) · p. ∂s = (Fp(τ1) · Fp(τ2) ···Fp(τk))[f ](p). s=0 A differential equation on M can always be written There are various ways of characterizing a curve in the form: z(t) ∈ M, z(0) = p. y (t) = f(y(t)) · y(t) (2) • The approach in [16] is based on a time dependent a → R for some function f : M → g. infinitesimal generator. Let a function :OT ν(t,a,p)∈ g Let FV(M) = (M → V ) denote all smooth func- define a curve as 1 tions from M to some vector space V. Any element t|τ|−1α(τ) V ∈ g induces a (right invariant) derivation V [·]: ν(t,a,p)= a(τ)Fp(τ). (|τ|−1)! FV(M) → FV(M) as τ∈OT ∂ We characterize z(t) as the solution of a differential V φ (p) = φ( ( ) · p) [ ] ∂s exp sV for any equation: s=0 φ ∈ FV(M). z (t) = ν(t,a,p)· z(t), z(0) = p. (3) Similarly, higher order derivations are defined by iter- We define α :OT→ R to be the function satisfying ating this definition: the analytical solution to (2) for a(τ) = 1 for all τ α(•) = τ = (V · W)[φ] = V [W[φ]]. .In[19] it is shown that 1, and for (τ1,τ2,... ,τk) Linear combinations of derivations, and the identity k i (0th order derivation) are defined in the obvious way. j= |τj|−1 α(τ) = 1 α(τi) : |τi|−1 1 In this paper we consider only V = R or V = g. i=1 H. Munthe-Kaas, S. Krogstad / Future Generation Computer Systems 19 (2003) 1197–1205 1199 • A pullback series is the basis for the analysis of From this we see that the analytical solution y(t) is Crouch–Grossman [6] and Owren–Marthinsen [19]. again characterized by b(τ) = 1. Let a function b :OT→ R define a higher order • The development of z(t) is given as a curve σ(t) ∈ differential operator µ(t, b, p) ∈ G as g such that z(t) = exp(σ(t)) · p. The computation of σ(t) from the infinitesimal generator ν(t,a,p)is t|τ|−1α(τ) µ(t, b, p) = b(τ) accomplished by Magnus series [10,13,21]. In this (|τ|−1)! computation, also commutators of elementary dif- τ=(τ1,τ2,... ,τk)∈OT ferentials appears. If a curve z(t) can be represented × (Fp(τ ) · Fp(τ ) ···Fp(τk)). 1 2 as (3), then there exists a function c :fla(OT) → R The curve z(t) is characterized by the requirement such that that t|h|α(h) σ(t) = c(h)Fp(h), φ(z(t)) = µ(t, b, p) φ (p) φ ∈ F (M). (|h|)! [ ] for any R h∈H(fla(OT)) In a formal sense, the pullback series is the exponen- where H(fla(OT)) denotes a Hall basis for the free tial of the infinitesimal generator. It can be shown Lie algebra of ordered trees, to be defined below. that for a given z(t) we have (We do not discuss the extension of α from OT to ∂ H(fla(OT)) here.) Each h ∈ H(fla(OT)) is a (for- µ( ,b,p)= ν(s,a,p)+ , F (h) 1 exp ∂s mal) commutator of trees, p denotes the cor- s=0 responding commutator of elementary differentials where ν(s,a,p) is defined in (3). This yields and |h| denotes the sum of the degree of all trees in the commutator. It should be noted that in the com- b((τ ,τ ,... ,τ )) = a(τ )a(τ ) ···a(τ ). 1 2 k 1 2 k mutative theory all non-trivial commutators vanish Fig. 1. Basis elements for the free Lie algebra of ordered trees up to order 5, and their corresponding binary balanced Lyndon words. 1200 H. Munthe-Kaas, S. Krogstad / Future Generation Computer Systems 19 (2003) 1197–1205 and σ(t) reduces to the classical B-series as defined In the case of fla(OT), there are C(i−1) generators in [9]. of grade i, thus As we see from this review, the free Lie algebra of ∞ j+1 ordered trees plays a central role in the Lie–Butcher p(x) = 1 − C(j)x . theory. Our goal in this paper is to present a basis for j=0 this algebra (see Fig. 1 for a graphical presentation of λ−d the lowest order elements). The sum j j should be interpreted as the trace of the nth power of the inverse of the companion matrix of p(x), in our case given as 3. The free Lie algebra of ordered trees C(0)C(1)C(2) ··· 1 Definition 3. A Lie algebra g is free over the set I if: M = 1 . 1 (i) For every i ∈ I there corresponds an element Xi ∈ . .. g. (ii) For any Lie algebra h and any function i → Yi ∈ By recursion one finds that the non-zero diagonal el- h, there exists a unique Lie algebra homomor- d ements of M are given as phism π : g → h satisfying π(Xi) = Yi for all d i ∈ I.
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