The subject matter of this paper is the solution of the linear differential equation y = a#t#y, y#0# = y0 , where y0 2 G, a# # #:R !gand g is a Lie algebra of the Lie group G. By building upon an earlier work of Wilhelm Magnus [16], we represent the solution as an infinite series whose terms are indexed by binary trees. This relationship between the infinite series and binary trees leads both to a convergence proof and to a constructive computational algorithm. This numerical method requires the evaluation of a large number of multivariate integrals but this can be accomplished in a tractable manner by using quadrature schemes in a novel manner and by exploiting the structure of the Lie algebra.

Many numerical algorithms involve computations in Lie algebras, like composition and splitting methods, methods involving the Baker-Campbell-Hausdorff formula and the recently developed Lie group methods for integration of differential equations on manifolds. This paper is concerned with complexity and optimization of such computations in the general case where the Lie algebra is free, i.e. no specific assumptions are made about its structure. It is shown how transformations applied to the original variables of a problem yield elements of a graded free Lie algebra whose homogeneous subspaces are of much smaller dimension than the original ungraded one. This can lead to substantial reduction of the number of commutator computations. Witts formula for counting commutators in a free Lie algebra is generalized to the case of a general grading. This provides accurate bounds on the complexity. The interplay between symbolic and numerical computations is also discussed, exemplified by the new...

. Runge--Kutta methods are formulated via coordinate independent operations on manifolds. It is shown that there is an intimate connection between Lie series and Lie groups on one hand and Butcher's celebrated theory of order conditions on the other. In Butcher's theory the elementary differentials are represented as trees. In the present formulation they appear as commutators between vector fields. This leads to a theory for the order conditions, which can be developed in a completely coordinate free manner. Even if this theory is developed in a language that is not widely used in applied mathematics, it is structurally simple. The recursion for the order conditions rests mainly on three lemmas, each with very short proofs. The techniques used in the analysis are prepared for studying RK--like methods on general Lie groups and homogeneous manifolds, but these themes are not studied in detail within the present paper. AMS subject classification: 65L06. Key words: Butcher theory, Run...

We present anoverview of intrinsic integration schemes for differential equations evolving on manifolds, paying particular attention to homogeneous spaces. Various examples of applications are introduced, Runge-Kutta methods. We argue that homogeneous spaces are the natural structures for the study and the analysis of these methods.

We de#ne the Newton iteration for solving the equation f#y# = 0, where f is a map from a Lie group to its corresponding Lie algebra. Twoversions are presented, which are formulated independently of any metric on the Lie group. Both formulations reduce to the standard method in the Euclidean case, and are related to existing algorithms on certain Riemannian manifolds. In particular, we show that, under classical assumptions on f , the proposed method converges quadratically.We illustrate the techniques by solving a #xed-point problem arising from the numerical integration of a Lie-type initial value problem via implicit Euler. # This work was in part sponsored by The Norwegian Research Council under contract no. 111038#410, through the SYNODE project. WWW: http:##www.imf.unit.no#num#synode y Email: Brynjulf.Owren@imf.unit.no, WWW: http:##www.imf.unit.no#~bryn z Email: bdw@math.la.asu.edu, WWW: http:##math.la.asu.edu#~bdw 1 1 Motivation Recently, there has been an increased intere...

We consider numerical integration methods for differentiable manifolds as proposed by Crouch and Grossman. The differential system is phrased by means of a system of frame vector fields E1 , ..., En on the manifold. The numerical approximation is obtained by composing flows of certain vector fields in the linear span of E1 , ..., En that are tangent to the differential system at various points. The methods reduce to traditional Runge-Kutta methods if the frame vector fields are chosen as the standard basis of euclidean R n . A complete theory for the order conditions involving ordered rooted trees is developed. Examples of explicit and diagonal implicit methods are presented, along with some numerical results.

We present a new class of integration methods for differential equations on manifolds, in the framework of Lie group actions. Canonical coordinates of the second kind is used for representing the Lie group locally by means of its corresponding Lie algebra. The coordinate map itself can, in many cases, be computed inexpensively, but the approach also involves the inversion of its differential, a task that can be challenging. To succeed, it is necessary to consider carefully howtochoose a basis for the Lie algebra, and the ordering of the basis is important as well. For semisimple Lie algebras, one may take advantage of the root space decomposition to provide a basis with desirable properties. The problem of ordering leads us to introduce the concept of an admissible ordered basis (AOB). The existence of an AOB is established for some of the most important Lie algebras. The computational cost analysis shows that the approachmay lead to more efficient solvers for ODEs on manifolds than those ba...

The method of Magnus series has recently been analysed by Iserles & Nørsett (1997). It approximates the solution of linear differential equations y 0 = a(t)y in the form y#t#=e ##t# , solving a nonlinear differential equation for # by means of an expansion in iterated integrals of commutators. An appealing feature of the method is that, whenever the exact solution evolves in a Lie group, so does the numerical solution. The subject matter of the present paper is practical implementation of the method of Magnus series. We commence by briefly reviewing the method and highlighting its connection with graph theory. This is followed by the derivation of error estimates, a task greatly assisted by the graph-theoretical connection. These error estimates have been incorporated into a variable-step fourth-order code. The concluding section of the paper is devoted to a number of computer experiments that highlight the promise of the proposed approach even in the absence of a Lie-group structure....

. In the context of devising geometrical integrators that retain qualitative features of the underlying solution, we present a family of numerical methods (the method of iterated commutators, [5, 13]) to integrate ordinary differential equations that evolve on matrix Lie groups. The schemes apply to the problem of finding a numerical approximation to the solution of Y 0 = A(t;Y )Y; Y (0) = Y0 ; whereby the exact solution Y evolves in a matrix Lie group G and A is a matrix function on the associated Lie algebra g. We show that the method of iterated commutators, in a linear setting, is intrinsically related to Lie's reduction method for finding the fundamental solution of the Lie-group equation Y 0 = A(t)Y . 1 Introduction In many applications of practical (and theoretical) interest, a given problem is often reduced to the solution of a differential equation or a family of differential equations evolving on some prescribed manifold. In the framework of devising structural discre...

RKMK methods and Crouch-Grossman methods are two classes of Lie group methods. The former is using flows and commutators of a Lie algebra...