DiMan an object oriented MATLAB toolbox for solving dierential equations on manifolds Kenth Eng Department of Informatics, University of Bergen N-5020 Bergen, Norway Arne Marthinseny Department of Mathematical Sciences, NTNU N-7034 Trondheim, Norway Hans Z. Munthe-Kaasz Department of Informatics, University of Bergen N-5020 Bergen, Norway ISSN 0333-3590 REPORT NO 164 February 1999 Department of Informatics, University of Bergen, Norway Abstract We describe an object oriented MATLAB toolbox for solving dierential equations on manifolds. The software reects recent development within the area of geometric integration. Through the use of elements from dierential geometry, in particular Lie groups and homogeneous spaces, coordinate free formulations of numerical integrators are developed. The strict mathematical denitions and results are well suited for implementation in an object oriented language, and, due to its simplicity, the authors have chosen MATLAB as the working environment. The basic ideas of DiMan are presented, along with particular examples that illustrate the working of and the theory behind the software package. AMS Subject Classication: 65L06, 34A50 Key Words: geometric integration, numerical integration of ordinary dierential equations on manifolds, numerical analysis, Lie groups, Lie algebras, homogeneous spaces, object oriented programming, MATLAB, free Lie algebras Email: [email protected], WWW: http://www.ii.uib.no/~kenth/ y Email: [email protected], WWW: http://www.math.ntnu.no/~arnema/ z Email: [email protected], WWW: http://www.ii.uib.no/~hans/ 1 Contents 1 Introduction 2 Ordinary dierential equations on manifolds 2 3 3 Object orientation 6 2.1 Intuitive concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Object orientation and mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Object orientation in MATLAB and DiMan . . . . . . . . . . . . . . . . . . . . . 4 Objects in DiMan 4.1 4.2 4.3 4.4 The domain object . . . The eld object . . . . . The time stepper object The ow object . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 4 5 6 7 8 8 9 9 9 5 The structure of DiMan 10 A How to solve ODEs in DiMan 13 B An example using the free Lie algebra C The numerical time steppers in DiMan 19 21 5.1 Functor classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 5.2 Free Lie algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 A.1 How to get started . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 A.2 How to solve dierential equations in DiMan { A 5-step procedure . . . . . . . . 14 A.3 A detailed example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 1 Introduction DiMan is an object oriented MATLAB [20] toolbox designed to solve dierential equations evolv- ing on manifolds. The current version of the toolbox addresses primarily the solution of ordinary dierential equations. The solution techniques implemented fall into the category of geometric integrators { a very active area of research during the last few years. The essence of geometric integration is to construct numerical methods that respect underlying constraints, for instance the conguration space of a mechanical problem, and to render correctly geometric structures and invariants important to the underlying continuous problem. The main motivations behind the DiMan project are: To create a uniform environment where new geometric integration methods can be developed and compared. To provide researchers outside the eld of geometric integration with a tool where they can become familiar with these methods and apply them to new problems. To serve as a tool for investigating the role of abstractions and coordinate free formulations in numerical software. DiMan is based on high level abstractions aimed at modeling continuous mathematical structures, in a manner that is, to a large extent, independent of particular representations. This is a novel approach to numerical computing which cannot 2 be investigated on a purely theoretical level. The development of practical software is an essential part of this work. DiMan is developed by support through the SYNODE project and the project Coordinate Free Methods in Numerics, both projects partly sponsored by the Norwegian research council NFR. Information about the SYNODE project, papers and other links are found at http://www.math.ntnu.no/num/synode/ DiMan can be down-loaded from the DiMan home-page at URL: http://www.math.ntnu.no/num/diffman/ 2 Ordinary dierential equations on manifolds DiMan is based on recent developments within the area of generalized ordinary dierential equa- tion solvers (Lie group integrators). The framework of these solvers is based on concepts from dierential geometry. There exist a large number of excellent texts that cover introductory dierential geometry; among them are [1, 2, 18, 32, 33]. We will in this section briey review some of this material, rst in an intuitive informal manner, then in a more precise language, and nally we will illustrate the theory by some examples. 2.1 Intuitive concepts The basic objects involved in the current Lie group integrators in DiMan are: The domain where the equation evolves is a manifold, M, which should be thought of as a linear or non-linear space looking locally like Rd. Globally it might be very dierent (e.g. a sphere looks locally but not globally like R2). We assume the existence of a set of operators G generating smooth motions on M. The operators in G can be composed and inverted. G is called a Lie group, and the set of motions is called a group action. The group action is used to advance the numerical solution on M. The action should be `easy' to compute, should be able to generate movements in the direction the dierential equation evolves, and should somehow capture some important underlying feature of the dierential equations. If the action e.g. preserves the angular momentum of a mechanical system, then also the numerical methods based on this action will preserve this property. One might compare an action in the integration of dierential equations with a preconditioner in an iterative solution of a linear equation system. Both the action and the preconditioner should be easy to compute and they should somehow capture some essential features of the system one wants to solve. The Lie group G is associated with a linear space g, the Lie algebra of the group. The algebra serves two dierent roles: { It represents locally all the tangents to G at any point. By dierentiating the action, we obtain also a representation of tangents to M at any point. This gives us a canonical representation of any dierential equation on M. { It can also be used as a at space to represent a local region on M where the dierential equation evolves. Several of the new algorithms are based on (locally) `pulling' the equation back from M to g, solving it on g, and pushing this local solution onto M to advance the solution there. 3 In the pull back { push forward process described above, one can make many choices that aect the numerical solution process, but not the canonical representation of the dierential equation on M. Some choices are: { Numerical method on g. Runge{Kutta [23] methods and multi-step methods [9] might be used. { A smooth local mapping from g to G is composed with the action of G on M to pull the equation from M to g. A simple choice is the exponential mapping [23]. Other choices are products of exponentials [29] and for quadratic groups Cayley transforms [15, 5, 19]. To obtain time-symmetric integrators one may use versions of these coordinates centered at some (time symmetric) mid-point [8]. To sum up, this theory allow us to make a lot of choices with respect to representation of the dierential equations and numerical solution techniques. A major goal of DiMan is to provide a toolbox where various domains can be easily created and where one can juggle around with all possible combinations of numerical solution techniques. 2.2 Denitions We will dene the concepts above more precisely. A manifold is a topological space M equipped with continuous local coordinate charts i : Ui M ! Rd such that all the overlap charts ij : Rd ! Rd are dieomorphisms. The overlap charts (transition functions) ij are dened as j ,i 1j (U \U ) , where ,i 1j (U \U ) means the restriction of ,i 1 to the set i (Ui \ Uj ). If p is a point Sin M, we denote by TMjp the tangent space of M at p. The tangent bundle of M is TM = p2M TMjp . A vector eld, X, on a manifold is a section of its tangent bundle, i.e. to each point p 2 M it associates a vector X(p) 2 TMjp . In DiMan we use the term domain interchangeably with the term dierentiable manifold. A Lie group is a smooth manifold equipped with a group structure such that group multiplication and group element inversion are smooth maps. Every group has an identity element, which we will denote by e. An example of a well known Lie group is Euclidean space, Rn, equipped with the group operation + (vector addition). The inverse element of u 2 Rn is ,u and the identity element is e = 0. In this case the group operation is both associative and commutative, but in general it is not commutative. A Lie algebra is a vector space, g, equipped with a bilinear, skew-symmetric form, [; ] : g g ! g, satisfying the Jacobi identity, i i j i i j [u; [v; w]]+ [w; [u; v]]+ [v; [w; u]] = 0; u; v; w 2 g: We call [; ] the Lie bracket on g. When the vector space g is Rnn, a Lie algebra is formed by taking the usual matrix commutator as the Lie bracket. The Lie algebra of a Lie group, G, can be dened as the tangent space at the identity, g = TGje, equipped with a Lie bracket @ 2 [u; v] = @[email protected] t=s=0 g(t)h(s)g,1 (t); where g(t); h(s) 2 G are two curves such that g(0) = h(0) = e, g0 (0) = u, and h0(0) = v. The general linear group, denoted by GL(n), is the Lie group that consists of non-singular n n matrices. The Lie algebra of this group, gl(n), is the vector space Rnn equipped with the matrix commutator as Lie bracket. A (left) action of a Lie group G on a manifold M is a smooth mapping : G M ! M such that (e; m) = m for all m 2 M and (g; (h; m)) = (gh; m) for all g; h 2 G and m 2 M. Any 4 2 g specify a tangent m 2 TMjm at any point m 2 M via d (g(t); m); m = dt t=0 where g(t) 2 G is a curve such that g(0) = e, g0 (0) = . One may write g(t) = exp(t), where exp : g ! G is the exponential map [23], or g(t) = (t) for any smooth function : g ! G such that (0) = e, 0(0) = I. This gives an identication 7! m 2 TMjm , which depends on the choice of action , but not on the particular choice of . In DiMan it is assumed that the dierential equation to be solved is presented in the following canonical form y0 = F (t; y) = (t; y)y ; y(0) 2 M; (1) for some function : R M ! g (see [28, 23, 5]). If (t; y) = (t), the equation is of Lie type or linear type. Otherwise it is of general type. Equations of Lie type can be handled more eciently than general type equations. If the domain M and the algebra g are provided in DiMan, or if they can be constructed using the functors in Section 5.1, the user only needs to supply the function . Otherwise DiMan can be extended by adding new domains. A large class of numerical algorithms are based on the assumption that the maps ((); m) can be computed eciently. The RKMK methods in [23] are based on () = exp(). More general 's are discussed in [5]. The following commutative diagram, which is thoroughly discussed in [5], illustrates the relations used as a framework these algorithms: 0 T Tg ,T ,,,,, ! TM du ,1 y0 x ? ? g y x? ?F (y)= y ,,,,! M y0 Here y0 2 M is the initial point, u 2 g , y0 (u) = (u; y0) and du,1 : g ! g is the right trivialized tangent of . If = exp, then d,u 1 = d expu ,1 can be expressed in terms of Lie brackets. For more general one needs an ecient algorithm for computing du,1 , see [5, 29] for dierent choices. This yields the following algorithm: Algorithm 2.1 Given a dierential equation in the form (1), this algorithm produces a q'th order approximation to the solution, using step size h: Find an approximation u1 u(t0 + h) by integrating the following dierential equation on g ,, u0 = d,u 1 t0 + t; ((u); y0) ; u(t0) = 0; from t0 to t0 + h using one step with a qth order Runge-Kutta method. Advance the solution on M to y1 = ((u1); y0) y(t0 + h). Repeat with new initial values y0 := y1 , t0 := t0 + h. 2.3 Examples We will here just provide the most basic examples of dierential equations written in the form (1). First, if M = Rn and (g; m) = g + m, then m = and (1) acquires the well known form y0 (t) = (t; y); 2 Rn: 5 Another example that is frequently used to present Lie group methods is the matrix case, G GL(n), a matrix group, M = G and (g; m) = gm then m = m (matrix products), hence (1) becomes y0 (t) = (t; m)y(t): In Appendix A.3 we give an example where G = SO(n), the set of orthogonal matrices, M = S n , the n-sphere (represented as n vectors with norm 1), and (g; m) = gm (matrix{vector product). This yields the form y0 (t) = (t; m)y(t); where is a skew{symmetric matrix. In [23] it is shown how isospectral ows, exponential integrators for sti systems, Riccati equations and the setting of rigid frames can be phrased in this form. Isospectral problems is discussed in detail in [35]. Lie{Poisson equations are discussed in [6] and the use of Toeplitz actions on heat equations in [26]. In forthcoming papers, we will see how many other systems can be viewed in this context. 3 Object orientation As of today there is really no tradition in the numerical analysis community for using object oriented languages in the development of numerical software. While object oriented languages have become ubiquitous in the computer science community, the classical computational mathematicians have been reluctant in adopting modern computer languages. We do not want to speculate about the reason for this, but merely observe that most industry based numerical software is coded in FORTRAN. The terms 'class' and 'object' are very likely to be among the rst words that you encounter in reading a book about object orientation. For a computer scientist these terms are as natural as the Butcher tableau is for a numerical analysist. In the next section we will give you a very rudimentary explanation of these terms along with ideas describing why modern computer languages are so well suited in capturing abstractions so omnipresent in pure mathematics. A somewhat random reference on object orientation found in the authors' library is [16]. 3.1 Object orientation and mathematics A class is simply a collection of 'elements' with equal properties. In mathematical terms one would say that a class is a set. The 'elements' of a class are usually called objects, and the common properties of the objects specify the class. Mathematically, the properties of a class can be stated as relations that the objects must satisfy in order to be a member of the class. A very important and interesting aspect of object orientation is that it allows for information hiding. An object typically consists of a public and a private part. The public part can be accessed from the outside of the class, whereas the private part cannot. This enables us to hide implementation specic issues for the particular class in the private part, and can easily make changes to it, without altering the public interface of the class. This then, naturally brings up the issue of specifying a class. A class specication can be divided up into a 'what' part and a 'how' part. 'what' describes the interaction of the class with the surroundings; what is the public interface of the class, what is the class supposed to do? 'How' the class is implemented is an issue related to the private section of the class. The surroundings do not need to know about implementational issues as long as the interaction of the class is as specied by the public interface. 6 This distinction between 'what's and 'how's of objects (elements) is ubiquitous in pure mathematics. This is also the reason why abstract mathematical concepts are so well suited for implementation in object oriented programming languages, see [12, 25]. Coordinate free constructions in mathematics, e.g. tensors, try to capture what the operation of an object is regardless of the coordinate system. The tensor class is then specied by properties independent of the coordinate systems, and the dierent choices of coordinates used in an actual implementation on a computer is deferred to the private part of the class. Hence, a specication of a class emphasizes and extracts the important features of a class, and this conforms very well with algebraic techniques so rampant in pure mathematics. Thinking in these terms give rise to the rather contradictory term 'coordinate free numerics' [24, 25]. What is a coordinate free algorithm? The whole idea is to device algorithms specied by algebraic operations not depending on the particular representation of the object. All the methods in DiMan are dened on groups and they are all very good examples of coordinate free algorithms. The group elements can have very dierent representations, but the algorithms are all expressed through algebraically dened operations such as group multiplication and Lie-group actions. 3.2 Object orientation in MATLAB and DiMan The intention of this section is to give you an overview of the workings of the object oriented environment in MATLAB and how this is used in DiMan. This account is by no means complete, and we refer the interested reader to the MATLAB manual [20] and the DiMan manual [7] for more information. In MATLAB a class is dened by creating a directory @myclass, where myclass is the name of the class. The prex @ is needed in order to tell MATLAB that this is a class directory. All the public class functions are put in this directory, while the private class functions are put in a directory @myclass/private. Where the le is located within the class directory tree distinguishes the m-le from being a public or private function. Every class must have its own unique constructor. The constructor is implemented in an m-le called myclass.m, the same name as the class itself. In MATLAB a class object is represented as a MATLAB struct, where a struct is the same as a struct in C or a record in PASCAL. This struct can have an arbitrary number of elds. To turn a MATLAB struct into an object of @myclass the function class must be called within the constructor m-le: obj.field1 = n1; obj.field2 = n2; obj = class(obj,'myclass'); The user can not access the structure elds of the object directly in MATLAB. An attempt doing this will result in an error. Hence, the elds of the object struct can be viewed as part of the data representation of the object, and is private to the class. For the user to interact with the information contained in the elds of the object struct, the class must have implemented public m-les particularly doing this. Public functions making up the interface of a class are naturally divided up into three categories: constructors, observers, and generators. In MATLAB there is only one constructor, but in other object oriented programming languages, like C++, it is possible to have more than one constructor. The observers of a class are the public functions that extract information from the class objects without altering the object itself. The generators of a class are those public functions that change properties of the class objects, or create new objects of the same or other classes. In DiMan you will typically nd this partition of the public functions when reading a class specication. 7 In DiMan the object orientation is applied in several dierent ways. The domain points (elements of a manifold) are treated as members of a class. Depending on the specic properties of the domain, there are several types of domains implemented in DiMan. Each of the dierent types of domains are collections of algebraically similar domain classes. The integration methods used to solve the ODEs are called time steppers, and the dierent time stepper methods are treated as dierent classes. Flows and vector elds are also implemented as two classes. In DiMan 1.5 there are three categories of domains implemented: Homogeneous spaces, Lie algebras, and Lie groups. Each domain category is further divided up into domain classes of that particular type. Hence, each of the classes within a particular domain type have similar characteristics, but there are dierences that partition them into individual classes. These similar characteristics of the classes of a specic domain type are what denes the domain category. Trying to dene and specify the domain category is done through the introduction of a virtual superclass in each domain category. The virtual superclass denes and takes care of operations that are common to all the classes in the domain category. This is obtained through the concept of inheritance in object orientation. The virtual superclass is the parent class for all the other classes in the domain category, and all the child classes inherit the parents' functions. This means that one can apply the public functions of the virtual superclass to an object of a child class. If the child class needs specically implemented versions of any of the public functions of the parent class, this is achieved through overloading. Supply a public function to the child class with a matching name, and MATLAB will use this version of the public function instead of the one supplied by the parent class. 4 Objects in DiMan In DiMan you will encounter 4 dierent types of objects: domain, eld, time stepper, and ow objects. Each object is dened in such a way as to capture the mathematical essentials of a manifold, the eld dening the dierential equation, the numerical time stepper algorithm, and the ow operator, respectively. We will discuss each one of these objects in the following. 4.1 The domain object Every domain object (e.g. objects of the type Lie algebra, Lie group, or homogeneous space) is built up as a MATLAB struct with two elds: shape and data. Generically, every domain object is represented as: domainobject = shape: data: A domain object species a specic point in a specic manifold. It is often useful to create a single class for representing a family of manifolds, e.g. all Lie algebras gl(n) are represented by the same class @lagl. The shape species the particular manifold in the family (in this case n), while the data part represents a particular point in this manifold (in this case n by n matrices. The shape is in computer science called a dynamic subtyping of the class. If an object has an empty data eld, it is taken to just represent the space (the subtype). A second example is the dynamic subtyping of the homogeneous space @hmlie. This is the homogeneous space obtained by any Lie group acting on itself by left multiplication. Considering all the dierent Lie groups and Lie algebras, the shape is chosen to be an object of the particular group or algebra. Since all Lie groups and Lie algebras themselves are dynamically subtyped, the 8 shape of @hmlie must be a Lie group or Lie algebra object with a preset shape. This is because we need to know a 'size' measure on the domain objects that are acting on themselves. The user cannot directly access the contents of the shape and data elds of a domain object, since the elds belong to the private part of the domain class. In order to do this the user must use the public functions getshape and getdata to return the contents of the elds, and setshape and setdata to update the values of the private elds. 4.2 The eld object A eld is dened over a manifold. Some examples of elds are vector elds, tensor elds, and divergence free vector elds. A vector eld is a mathematical construction that to every element of the manifold assigns a vector. Likewise; for a tensor eld a tensor is assigned to each element of the manifold. From this it is natural to conclude, since the output from dierent elds is not similar, that a generic eld object only contains information about the manifold over which the eld is dened: fieldobject = domain: To dene an ordinary dierential equation we only need the notion of a vector eld. Tensor elds are mainly used in partial dierential equations. DiMan 1.5 is only devoted to the solution of ODEs evolving on manifolds. Hence, the only eld class implemented is @vectorfield. The generic representation of a vector eld object is: vectorfieldobject = domain: eqntype: fm2g: Compared to the above eld object two more struct elds have been added in the vector eld object. In DiMan 1.5 every vector eld over a manifold is represented by a function : R M ! g as in (1). This function is called fm2g, (function from M to g). If the function fm2g is only depending on time, the ODE is said to be linear or of Lie type, and general if the function fm2g depends on both time and conguration. The eqntype provides this information. 4.3 The time stepper object This is where all the numerics is hidden. A time stepper is the numerical algorithm used in advancing the solution of the dierential equation one step along the integral curve of the ow. There are dierent approaches in constructing these time steppers and a list of the available time steppers in DiMan 1.5 is found in Appendix C. 4.4 The ow object Mathematically, the ow is an operator dened by the vector eld. Given the ordinary dierential equation y0 = F (y); y(0) = p 2 M; 9 the ow operator of this dierential equation is the operator F;t : M ! M satisfying d dt F;t (p) = F(F;t(p)): The classical solution of a dierential equation is an integral curve of the vector eld generating the ow operator. This integral curve is found by evaluating the ow operator in the initial point on the manifold. The generic representation of a ow object is: flowobject = vectorfield: timestepper: defaults: The ow object must of course know the vector eld dening it. Next, it needs to know a time stepper object. The choice of time stepper species the numerical algorithm to be used in the solution of the dierential equation. This is really all the information that the ow object needs to know. However, for convenience, constants used in variable time stepping and nonlinear equation solving is collected in the ow object in the third eld defaults. Unless the user changes any of these constants with the setdefaults function, the default values set by the ow constructor will be used. 5 The structure of DiMan The philosophy used in designing DiMan is to respect the underlying continuous mathematics to an as large extent as possible. Finding the continuous ow of a eld is equivalent of nding the numerical solution of a dierential equation. The continuous ingredients of a dierential equation are: domain, eld, and ow. The relationships among them are that the eld is dened over the domain, and the ow is dened by the eld. This general structure in the continuous case is reected in DiMan in the way the directories are organized. In the DiMan root directory you will nd three directories each corresponding to domain, eld, and ow, see Figure 1. There is also a fourth directory; auxiliary, which collects things related to the non-mathematical workings of DiMan. The structure is very modular and therefore it is very easy to add new classes. The domain directory contains the domain categories, very important building blocks of DiMan. Creating a domain category is done by creating a subdirectory in the domain directory. In DiMan 1.5 there are 3 domain categories implemented: homogeneous spaces, Lie algebras, and Lie groups. The classes of a domain category are put in this subdirectory along with a virtual superclass specifying the type of domain. The field directory contains eld classes dened over domain classes. Think of this directory as the eld category. In DiMan 1.5 there is only one eld class implemented: @vectorfield. This is the only eld class needed in the solution of ordinary dierential equations. In order to solve some partial dierential equations it is interesting to be able to dene tensor elds over manifolds. When implemented the tensor eld class will be placed in the field directory in DiMan. The flow directory collects classes pertinent to the continuous ow. The numerical methods are treated as time steppers, and they are all placed in the subdirectory timesteppers found in the flow directory. The ow class is a virtual superclass with no subclasses since all the numerics is placed within the time steppers. The reason for this distinction between ow and time stepper is an attempt to isolate features common to all the numerical methods (the time steppers), e.g. variable time stepping, and place these features in the ow class. 10 The auxiliary directory includes 4 subdirectories that contain the DiMan documentation, command line examples, demos, and utility functions. As the name of this directory reects, the content is not vital to the workings of DiMan. DiffMan flow @flow timestepper @timestepper @tscg . . . @tsrkmk field @vectorfield domain hmanifold @hmanifold @hmisospec . . . @hmtop liealgebra @liealgebra @ladirprod . . . @lasp liegroup @liegroup @lgdirprod . . . @lgsp auxiliary demos documentation examples utilities Figure 1: Schematic picture of the structure in DiMan. 5.1 Functor classes In DiMan the dierent domain types are viewed as categories. A function on a category is called a functor. Examples of internal functors are the direct product, semi-direct product, and tangent map. The direct product functor will take n domain classes as input, and create a new domain object; the direct product of the domains. The semi-direct functor works in an analogous way. The tangent functor takes a domain manifold and turns it into the tangent bundle of that manifold. The tangent bundle is also a manifold; hence, it is a domain. In DiMan we call classes that automatically generate new domains from other ones functor 11 classes. The choice of name should be clear from the above discussion. In DiMan 1.5 you will nd one of the above functorial constructors implemented; the direct product of domains. The functor classes in question are @ladirprod and @lgdirprod. A future release of DiMan will include the semi-direct product functor and the tangent functor. 5.2 Free Lie algebras Of all the Lie algebra classes in DiMan, the class @lafree (free Lie algebra) plays a special role. It can be viewed as a `symbolic computation engine', capable of simplifying expressions involving Lie brackets, by systematic use of skew-symmetry and the Jacobi identity. When a formal expression is simplied, it may subsequently be evaluated in a concrete Lie algebra. Thus, this class is very useful for developing and simplifying algorithms in Lie algebras. Furthermore, in Appendix B we show that @lafree is very useful for another type of optimization, related to the fact that calls to overloaded operators are handled very ineciently in MATLAB 5. More details on numerical computations in free Lie algebras, complexity results, and applications to Lie group integrators are found in [27]. Given an arbitrary index set I, either nite or countably innite. The following denition is equivalent to the one in [32]. Denition 5.1 A Lie algebra g is free over the set I if: i) For every i 2 I there corresponds an element Xi 2 g. ii) For any Lie algebra h and any function i 7! Yi 2 h, there exists a unique Lie algebra homomorphism : g ! h satisfying (Xi ) = Yi for all i 2 I . In category theory, g is said to be a universal object, i.e. it contains a structure that is common to all Lie algebras, but nothing more. Furthermore, computations in g can be applied in any concrete Lie algebra h via the homomorphism . Computationally it is useful to represent a free Lie algebra in terms of a basis. The most used bases are the (classical) Hall basis and the Lyndon basis. They can both be regarded as generalized Hall type bases [31]. The DiMan @lafree class is implemented using a Hall basis. Example 5.2 If I = f1; 2; 3g, the (classical) Hall basis consisting of elements with length 4 is given as: X1 X2 X3 [X1 ; X2 ] [X1 ; X3 ] [X2 ; X3 ] [X1 ; [X1 ; X2 ]] [X1; [X1 ; X3 ]] [X2 ; [X1; X2 ]] [X2 ; [X1 ; X3 ]] [X2 ; [X2 ; X3 ]] [X3; [X1 ; X2 ]] [X3 ; [X1; X3 ]] [X3 ; [X2 ; X3 ]] [X1 ; [X1; [X1 ; X2 ]]] [X2 ; [X1; [X1 ; X3 ]]] [X2 ; [X2; [X2 ; X3 ]]] [X3 ; [X2; [X1 ; X2 ]]] [X3 ; [X3; [X1 ; X2 ]]] [[X1; X2 ]; [X1 ; X3 ]] [X1; [X1 ; [X1 ; X3 ]]] [X2; [X2 ; [X1 ; X2 ]]] [X3; [X1 ; [X1 ; X2 ]]] [X3; [X2 ; [X1 ; X3 ]]] [X3; [X3 ; [X1 ; X3 ]]] [[X1; X2 ]; [X2; X3 ]] 12 [X2 ; [X1 ; [X1 ; X2 ]]] [X2 ; [X2 ; [X1 ; X3 ]]] [X3 ; [X1 ; [X1 ; X3 ]]] [X3 ; [X2 ; [X2 ; X3 ]]] [X3 ; [X3 ; [X2 ; X3 ]]] [[X1 ; X3 ]; [X2; X3 ]] Free Lie algebras are innite dimensional algebras, which for many numerical applications need to be truncated according to some measure of the size of the terms. If is some small parameter and each generator Xi has an order Xi = O(w ), then [Xi ; Xj ] = O(w +w ). Now we might want to neglect all terms of order q or higher. Intuitively we form all brackets in a Hall basis and remove all brackets of suciently high order. Mathematically this is a quotient construction on a graded free Lie algebra. Let X1 ; : : : ; Xs be generators for a free Lie algebra g and let w1; : : : ; ws be positive integers. We dene a grading of the Hall basis of g by grade(Xi ) = wi grade([hi; hj ]) = grade(hi ) + grade(hj ) for all hi ; hj 2 H, where H denotes the Hall basis. Now let gq = spanf h 2 H j grade(h) q g. Clearly gq is an ideal of g, and g=gq is the nite dimensional Lie algebra obtained by dropping all high order terms. This construction can be dened independently on a particular choice of basis for g. In DiMan we construct the object g=gq by issuing the command objname = lafree(f[s; q];[w1; : : : ; ws]g): In [27] we establish Witt-type formulas for computing the dimension of g=gq . A tutorial example on using @lafree is given in Appendix B, where we apply this class in the construction of RKGL type Lie group integrators for solving equations of Lie type. i i j A How to solve ODEs in DiMan This appendix will teach you the basics of solving dierential equations in DiMan. The rst section shows you how to initialize DiMan and get the toolbox up and running. The next section describes a 5-step procedure to be followed when solving dierential equation in DiMan. Finally, the last section takes you through a very detailed example showing you the 5-step procedure in practice. A.1 How to get started The very rst thing to do is to initialize the DiMan toolbox. Make sure that you are located in the DiMan directory, or that this directory is included in the MATLAB path. You can easily include the following command in your startup.m le, or issue it at the MATLAB prompt: >> addpath('/local/path/on/your/machine/DiffMan'); Initializing DiMan is done simply by typing the command: >> dminit The result of this command is that all necessary paths are set and DiMan is ready for use. The DiMan facility dmtutorial will launch a window where you can choose to run dierent kinds of tutorials. One of these tutorials will guide you through 'How to solve ODEs in DiMan '. This is the 5-step procedure presented in the next section. The other tutorials will guide you through other important aspects of the DiMan toolbox essential to the user. dmhelp is a substantially improved version of the helpwin facility in MATLAB. dmhelp will launch a DiMan help window where you can get help on every function and class in DiMan, and also every other function in MATLAB. Hence, when working with DiMan you are urged to use dmhelp instead of the MATLAB functions helpwin and help. 13 The MATLAB demo utility will include DiMan among its toolboxes. Running the DiMan demos is another convenient way of launching the DiMan tutorials, and running all the DiMan command line examples. A.2 How to solve dierential equations in DiMan { A 5-step procedure Once DiMan is initialized you can start solving dierential equations. In DiMan we are exclusively working with objects, and these objects are members of dierent classes. To create an object of a particular class, invoke the constructor of that class. The constructor always has the same name as the class. The 5-step procedure for solving dierential equations in DiMan is the following: 1) Construct an initial domain object y in a homogeneous space. In order to solve an initial value problem, DiMan needs to know an initial condition. The initial domain object serves this purpose. 2) Construct a vector eld object vf over the domain object y. DiMan nds numerically the integral curve of this vector eld through the initial domain object. A vector eld object consists of three parts: domain, eqntype, and fm2g. Set these properties of the vector eld object by the functions setdomain, seteqntype, and setfm2g. You retrieve them with the corresponding get functions. 3) Construct a time stepper object ts. The time stepper class determines the numerical method used to advance the numerical solution along the integral line. A time stepper object consists of two parts: coordinate and method. Set these properties of the time stepper object by the functions setcoordinate and setmethod. You retrieve them with the corresponding get functions. 4) Construct a ow object f. The ow object is dened by the vector eld object. Since we are doing numerical computations the ow object also needs to know how to step forward, hence the ow object f also needs to know the time stepper object. To set the two properties of the ow object use the functions setvectorfield and settimestepper. You retrieve them with the corresponding get functions. 5) Solve the ODE. Solving the ODE dened by the ow object in DiMan is simply done by evaluating the ow object at the initial domain object, start time, end time, and step size: >> output = (y,tstart,tfinal,h); Variable step size is indicated by using negative values for h. The initial step then be of length jhj. The struct output consists of three elds: output.y is a vector of domain objects, output.t is a vector of time points, and output.rej is a vector indicating rejection of a time step in variable time stepping. Detailed mathematical information and denitions of ows and vector elds are found in Section 2. The 5-step procedure will be detailed out on an example in the next section. 14 A.3 A detailed example Consider solving the following dierential equation on the sphere S 2 : 2 3 203 t ,0:4 cos(t) dy = 4 ,0t 0 0:1t 5 y(t); y(0) = 405 ; dt 0:4 cos(t) ,0:1t 0 1 (2) where y 2 R3 is a vector of unit length, and the matrix on the right hand side is a map from R into so(3). The homogeneous manifold in question is @hmnsphere which consists of the sphere manifold S 2 , the Lie algebra of O(3) which is so(3), and the action : (v; m) ! exp(v) m of so(3) on S 2 . The elements of the manifold S 2 are vectors of unit length. Step #1: Construct an initial domain object y in @hmnsphere The initial domain object is created by calling the constructor of @hmnsphere. This constructor can take an integer or an laso object as an argument and thereby specifying the shape of the manifold object. >> y = hmnsphere(3) y = Class: hmnsphere Shape-object information: Class: laso Shape: 3 The shape of an object in @hmnsphere consists of an object in the Lie algebra laso. The integer supplied to the constructor sets the shape of this Lie algebra object that comprises the shape of the @hmnsphere object. If an argument to the constructor is not supplied, the shape can be set later by use of the setshape function. As mentioned in the beginning of this Section, the data representation of an object in @hmnsphere is a vector of unit length. If the initial condition for the ODE on the sphere is the North pole, the data of the initial object must be set equal to the North pole vector. >> setdata(y,[0 0 1]'); >> y y = Class: hmnsphere Shape-object information: Class: laso Shape: 3 Data: 0 0 1 The rst step is now completed. Step #2: Construct a vector eld object vf over the domain object y A vector eld is dened over a domain. The constructor of @vectorfield is called with the domain object as input: >> vf = vectorfield(y) 15 vf = Class: vectorfield Domain: hmnsphere Shape-object information: Class: laso Shape: 3 Eqn type: General Already, vf contains a lot of information. Since the domain object was supplied as an argument for the vector eld constructor, the domain information is already set. The shape of the @hmnsphere object is an laso object, and the information about this Lie algebra object is displayed as Shape-object information. Further, the equation type of the generator map for the vector eld is set to be 'General'. This is the default value. However, equation (2) is of Lie type, so the type should be changed to 'Linear' in order to speed up the calculations. >> seteqntype(vf,'Linear'); The generator map of equation (2) is the matrix on the right hand side of the equation. The m-le vfex5.m contains the necessary MATLAB code to implement the generator map. >> setfm2g(vf,'vfex5'); How does this m-le vfex5.m look like? To view the le you can type type vfex5.m at the MATLAB prompt, or use dmhelp and push the button View src. Any way the output is: function [la] = vfex5(t,y) % VFEX5 - Generator map from RxM to liealgebra. Linear type. la = liealgebra(y); dat = [0 t -0.4*cos(t); -t 0 .1*t; .4*cos(t) -.1*t 0]; setdata(la,dat); return; All the generator map function les that you write on your own must have this generic structure: The le must support two arguments; the rst is a scalar { time, and the second is a domain object from the homogeneous space. Output must be a Lie algebra object. To nd the correct Lie algebra of the domain object call liealgebra(y), which will return an object in the correct Lie algebra with preset shape information. Edit the data dat, and call setdata(la,dat) in order to set the data representation of the Lie algebra object. Now the vector eld object vf displays as: >> vf vf = Class: vectorfield Domain: hmnsphere Shape-object information: Class: laso Shape: 3 Map fm2g: vfex5 Eqn type: Linear 16 Step #3: Construct a time stepper object ts The time stepper class decides which numerical method to use for advancing along the integral curve of the vector eld. Calling any of the time stepper constructors will return a time stepper object with default coordinate and method. If the user prefers other coordinates or another method, these can be changed through the functions setcoordinate and setmethod. To get an overview of the dierent time steppers type dmhelp timestepper in MATLAB. In our example we want to use an RKMK method: >> ts = ts = Class: Coord.: Method: tsrkmk tsrkmk exp RK4 In case of @tsrkmk the default coordinate is exp and the default method is RK4. For a discussion of the possible choices of coordinates, see [7]. For each time stepper class there is a whole lot of methods to choose from. None of these methods can be used for all the dierent time stepper classes and DiMan will issue an error message if a wrong selection is made. In our example we are not satised with only the standard 4th-order RK4 method, we want the more accurate answer supplied by the 6th-order Butcher method: >> setmethod(ts,'butcher6') >> ts ts = Class: tsrkmk Coord.: exp Method: butcher6 To get information about the dierent methods while running DiMan, type dmhelp setmethod. Step #4: Construct a ow object f The ow object is constructed from the vector eld object vf and the time stepper object ts already created. Just calling the @flow constructor will create an object with a default time stepper. The default time stepper preset in the ow object f is only a matter of convenience, and must not be confused with the time stepper object created in Step #3. >> f = flow f = Class: flow Timestepper class: tsrkmk Coordinates: exp Method: RK4 In our example we have created another time stepper object ts that we want to use instead of the default time stepper object supplied by the @flow constructor. To change the time stepper of the ow object f to ts, call the function settimestepper: >> settimestepper(f,ts) A ow is dened as the ow of some vector eld. Hence, our ow object f must have information about this vector eld. 17 >> setvectorfield(f,vf) >> f f = Class: flow Vector field information: Domain: hmnsphere Equation type: Linear Map defining DE: vfex5 Timestepper class: tsrkmk Coordinates: exp Method: butcher6 Now the ow object has the necessary information and we can go on to the next, and nal, step in the 5-step solution procedure. Step #5: Solve the ODE Solving equation (2) with the RKMK method is done by evaluating the ow object with four arguments: initial domain object, start time, end time, and step size. >> curve curve = y: t: rej: = f(y,0,5,0.05) [1x61 hmnsphere] [1x61 double ] [1x61 double ] The output curve is a MATLAB struct with the three elds: y, t, and rej. curve.y is a vector of objects from the homogeneous space upon which the problem is modeled. curve.t is a vector of scalars, the time points. curve.rej is a vector of integers indicating if a step was rejected or not. In our example curve.rej is the zero vector, since we did not use variable time stepping. Calling getdata(curve.y) will access the actual data representations of all the @hmnsphere objects. In this case this output will be a vector three times the length of the scalar vector curve.t. To get the 3-vectors corresponding to each time point, the output from getdata(curve.y) must be reshaped into a 3length(t) matrix where each column corresponds to a time point. To plot the data we can do the following: >> >> >> >> t = curve.t; a = getdata(curve.y); a = reshape(a,3,length(t)); comet3(a(1,:),a(2,:),a(3,:)); In Figure 2 the solution of the problem is plotted. This detailed example is found as Example 5 in the DiMan toolbox and runs by typing: >> dmex5 What's next?: Solve the same problem with a dierent time stepper To use another time stepper to solve equation (2) we must repeat steps #3 through #5. We must create a new time stepper object, put this into the existing ow object, and evaluate the ow again. To use the Crouch-Grossman method we do the following: >> ts2 = tscg 18 Figure 2: Plot of the solution of (2). ts2 = Class: tscg Coord.: exp Method: CG3a >> settimestepper(f,ts2) >> f f = Class: flow Vector field information: Domain: hmnsphere Equation type: Linear Map defining DE: vfex5 Timestepper class: tscg Coordinates: exp Method: CG3a >> curve = f(y,0,5,0.05); To plot the solution we just repeat the above plotting commands. The solution is the same, except from the fact that we have used a third-order method. This solution is not as accurate as the solution obtained by the 6th-order Butcher method used in the RKMK time stepper. B An example using the free Lie algebra The commands given below reect the basic operations of Denition 5.1. : : : ,wp]}); Generate a free Lie algebra from p symbols with grades w1,w2, : : : ,wp. All terms of total grade greater than q are set to 0. >> fla = lafree({[p,q],[w1,w2, >> Xi = basis(fla,i); i Return the 'th Hall basis element in fla. If 1 i p, return the i'th generator Xi . >> X+Y; r*X; [X,Y]; Basic computations in the free Lie algebra. 19 : : : ,Yp); If is an element of a free Lie algebra, and Y1,Y2, : : : ,Yp are the elements of any DiMan Lie algebra, this will evaluate the expression E, using the data set Y1,Y2, : : : ,Yp in place of the generating set. This corresponds to the homomorphism : g ! h in Denition 5.1. One may also write Z = eval(E,Y); where Y(i) = Yi;. >> Z = eval(E,Y1,Y2, E As an example of the use of this class we will consider the construction of Runge{Kutta{Gauss{ Legendre (RKGL) type integrators for equations of Lie type (or linear type). To make things simple, consider a matrix equation y0 (t) = f(t)y(t); y(t0 ) = y0 ; (3) where y(t) is a curve on a matrix Lie group and f(t) is a curve in the Lie algebra. It is well known [23] that the solution for suciently small t can be written as y(t) = exp((t))y0 ; where (t) satisfy 0(t) = dexp,(1t)(f(t)); (t0 ) = 0: (4) RKGL methods are s-stage implicit methods of order 2s. For equations of Lie type, where f = f(t), the problem of solving implicit equations can be solved once and for all by a computation in a free Lie algebra. We seek an expression for (t0 + h) in terms of the s quantities ki = hf(t0 + ci ). To develop a general integration scheme, we may regard ki as being generators of a free Lie algebra, and nd an expression for (t0 + h) by solving the implicit RK equations in the free Lie algebra. If this is done in a straightforward manner, the number of commutators involved in the expression for (t0 + h) turns out to be very large. In [27], we show that this can be overcome by changing the basis for the free Lie algebra. We introduce a new set of generators X1 ; : : : ; Xs as X ki = Vi;j Xj ; j where Vi;j = (ci , 1=2)j ,1 is a Vandermonde matrix. After this change of variables, it can be shown that Xi = O(hi ), thus X1 ; : : : ; Xs generate a graded free Lie algebra with the grading wi = i. In Figure 3 we show the computation of . A 6'th order method is obtained by [sig; c; W; s] = rkgllmethod(6); which yields (numerically) the result: 1 X3 , 1 [X1 ; X2] + 1 [X2 ; X3] + 1 [X1; [X1; X3 ]] , : : : sig = X1 + 12 12 240 360 1 [X2; [X1; X2 ]] + 1 [X1; [X1; [X1; X2 ]]] , 240 5,p15 1 5+p15720T c= 0 100 21 100 1 B p5 0 p155 CA W = @ , 15 10 3 , 203 10 3 =3 Figure 4 shows how these data are used in a simple timestepper. A major part of the computation is the evaluation of the expression sig on the data X, s = eval(sig; X);. Once the expression for sig is known, this line could have been written out explicitly, e.g. as s = X(1) + X(3) (1=12) , [X(1); X(2)] (1=12) + : s 20 % RKGLLMETHOD - Construct q'th order RKGL-Lie integrator function [sig,c,W,s] = rkgllmethod(q) % coeffs. for classical s-stage q'th order RKGL A,b,c,s = rkglcoeff(q); % change basis with Vandermonde matrix V = vander(c-1/2); V = V(:,s:-1:1); W = inv(V); laf = lafree( [s,q],[1:s] ); for i = 1:s, k(i) = zero(laf); for j = 1:s, k(i) = k(i)+V(i,j)*basis(laf,j); end; end; % solve implicit RK step in laf by fixpoint iteration u = zeros(laf,s); for ord = 1:q, for i = 1:s, ktild(i) = dexpinv(u(i),k(i),q); end; for i = 1:s, u(i) = A(i,:)*ktild; end; end; sig = b*ktild; return; [ ] f g Figure 3: Construction of q'th order RKGL-Lie integrator. However, the eval function allows optimizations that are lost in an explicit expansion. When eval is called, the expression is analyzed and the computation is optimized. So, if a given commutator 1 [X2 ; [X1; X2]], the eval appears several places in the same expression, like 121 [X1; X2 ] + , 240 function will only compute [X1 ; X2] once. An even more signicant optimization is related to the overhead of calling overloaded operators. If a bracket [X1 ; X2] is called in MATLAB, then the system rst has to gure out in which Lie algebra Xi belongs, and then call the corresponding commutator. In the current version of MATLAB, there is a large overhead associated with this handling of overloaded operators. The eval function, on the other hand, is implemented such that the job of guring out which algebra is involved is only done once. We have seen that for small matrices this optimization trick yields codes running up to ten times faster! C The numerical time steppers in DiMan The development of DiMan is an ongoing project, and the numerical time steppers implemented are subject to change in the future. Version 1.5 of DiMan includes the following time steppers: tscg: the Crouch-Grossman methods [4, 30]. tsfer: the Fer expansion methods [10, 13, 34]. tsmagnus: the Magnus series methods [17, 14, 34]. tsqq: quadrature methods for quadratic Lie groups [19]. tsrk: classical Runge-Kutta methods. These methods are well-known to the ordinary dierential equation community and the classical references include [3, 11]. 21 Given: f, y0, t0, h, q. [sig,c,W,s] = rkgllmethod(q) t = t0; y = y0; for istp = 1:nstp, for i = 1:s, k(i) = h*f(t+c(i)*h); end; for i = 1:s, X(i) = W(i,:)*k; end; s = eval(sig,X); y = exp(s)*y; t = t+h; end; Figure 4: Simple RKGL-Lie timestepper. tsrkmk: Runge-Kutta methods of Munthe-Kaas type. A large number of papers discuss these methods, but the trilogy where the methods were derived is [21, 22, 23]. In future releases, general linear methods as well as linear multistep methods will be included in DiMan, both as classical methods [11] and in a generalized setting [9]. References [1] R. Abraham and J. E. Marsden. Foundations of Mechanics. Addison-Wesley, second edition, 1978. [2] R. Abraham, J. E. Marsden, and T. S. Ratiu. Manifolds, Tensor Analysis, and Applications. AMS 75. Springer-Verlag, second edition, 1988. [3] J. C. Butcher. The Numerical Analysis of Ordinary Dierential Equations. Wiley, 1987. [4] P. E. Crouch and R. Grossman. Numerical integration of ordinary dierential equations on manifolds. J. Nonlinear Sci., 3:1{33, 1993. [5] K. Eng. On the construction of geometric integrators in the RKMK class. Technical Report No. 158, Department of Informatics, University of Bergen, Norway, 1998. [6] K. Eng and S. Faltinsen. Integrating Lie{Poisson systems with the RKMK method. In preparation. 1999. [7] K. Eng, A. Marthinsen, and H. Munthe-Kaas. DiMan | an object oriented MATLAB toolbox for solving dierential equations on manifolds: User's Guide. Version 1.5. Available on WWW at http://www.math.ntnu.no/num/diffman/. [8] K. Eng, H. Z. Munthe-Kaas, and A. Zanna. On the time symmetry of Lie group integrators. In preparation. 1999. [9] S. Faltinsen, A. Marthinsen, and H. Munthe-Kaas. Multistep methods integrating ordinary dierential equations on manifolds. Manuscript, 1999. [10] F. Fer. Resolution del l`equation matricielle U_ = pU par produit inni d`exponentielles matricielles. Bull. Classe des Sci. Acad. Royal Belg., 44:818{829, 1958. 22 [11] E. Hairer, S. P. Nrsett, and G. Wanner. Solving Ordinary Dierential Equations I, Nonsti Problems. Springer-Verlag, second revised edition, 1993. [12] M. Haveraaen, V. Madsen, and H. Munthe-Kaas. Algebraic programming technology for partial dierential equations. In Proceedings of Norsk Informatikk Konferanse (NIK), Trondheim, Norway, 1992. Tapir. [13] A. Iserles. Solving linear ordinary dierential equations by exponentials of iterated commutators. Numer. Math., 45:183{199, 1984. [14] A. Iserles and S. P. Nrsett. On the solution of linear dierential equations in Lie groups. Technical Report 1997/NA3, Department of Applied Mathematics and Theoretical Physics, University of Cambridge, England, 1997. To appear in Philosophical Transactions of the Royal Society. [15] D. Lewis and J. C. Simo. Conserving algorithms for the dynamics of Hamiltonian systems of Lie groups. J. Nonlinear Sci., 4:253{299, 1994. [16] S. B. Lippman. C++ Primer. Addison Wesley, second edition, 1991. [17] W. Magnus. On the exponential solution of dierential equations for a linear operator. Comm. Pure and Appl. Math., VII:649{673, 1954. [18] J. E. Marsden and T. S. Ratiu. Introduction to Mechanics and Symmetry. Springer-Verlag, 1994. [19] A. Marthinsen and B. Owren. Quadrature methods based on the Cayley transform. Technical Report Numerics No. 1/1999, The Norwegian University of Science and Technology, Trondheim, Norway, 1999. [20] The MathWorks, Inc., 24 Prime Park Way, Natick, MA 01760-1500. Using MATLAB, 5.2 edition, April 1998. [21] H. Munthe-Kaas. Lie{Butcher theory for Runge{Kutta methods. BIT, 35(4):572{587, 1995. [22] H. Munthe-Kaas. Runge{Kutta methods on Lie groups. BIT, 38(1):92{111, 1998. [23] H. Munthe-Kaas. High order Runge{Kutta methods on manifolds. Applied Numerical Mathematics, 29:115{127, 1999. [24] H. Munthe-Kaas and M. Haveraaen. Coordinate free numerics; Part 1: How to avoid indexwrestling in tensor computations. Technical Report No. 101, Department of Informatics, University of Bergen, Norway, 1995. [25] H. Munthe-Kaas and M. Haveraaen. Coordinate free numerics | Closing the gap between 'Pure' and 'Applied' mathematics? In Proceedings of ICIAM-95, Zeitschrift fur Angewandte Mathematik und Mechanik (ZAMM), Berlin, 1996. Akademie Verlag. [26] H. Munthe-Kaas and E. Lodden. Explicit Lie group integrators for heat equations. In preparation. 1999. [27] H. Munthe-Kaas and B. Owren. Computations in a free Lie algebra. Technical Report No. 148, Department of Informatics, University of Bergen, Norway, 1998. To appear in Philosophical Transactions of the Royal Society. [28] H. Munthe-Kaas and A. Zanna. Numerical integration of dierential equations on homogeneous manifolds. In F. Cucker and M. Shub, editors, Foundations of Computational Mathematics, pages 305{315. Springer Verlag, 1997. 23 [29] B. Owren and A. Marthinsen. Lie group integrators based on canonical coordinates of the second kind. In preparation. 1999. [30] B. Owren and A. Marthinsen. Runge{Kutta methods adapted to manifolds and based on rigid frames. BIT, 39(1):116{142, 1999. [31] C. Reutenauer. Free Lie Algebras. Number 7 in London Mathematical Society Monographs, New Series. London Mathematical Society, 1993. [32] V. S. Varadarajan. Lie Groups, Lie Algebras, and Their Representations. GTM 102. SpringerVerlag, 1984. [33] F. W. Warner. Foundations of Dierentiable Manifolds and Lie Groups. GTM 94. SpringerVerlag, 1983. [34] A. Zanna. Collocation and relaxed collocation for the Fer and the Magnus expansions. Technical Report 1997/NA17, Department of Applied Mathematics and Theoretical Physics, University of Cambridge, England, 1997. [35] A. Zanna. On the Numerical Solution of Isospectral Flows. PhD thesis, Cambridge University, 1998. 24

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