The discrete Moser Veselov algorithm for the free rigid body

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1 The discrete Moser Veselov algorithm for the free rigid body 1/100 Antonella Zanna and Robert McLachlan Report in Informatics nr. 255 (2003). To appear in Found. Comp. Math. Antonella Zanna University of Bergen, Norway anto

2 Report Documentation Page Form Approved OMB No Public reporting burden for the collection of information is estimated to average 1 hour per response, including the time for reviewing instructions, searching existing data sources, gathering and maintaining the data needed, and completing and reviewing the collection of information. Send comments regarding this burden estimate or any other aspect of this collection of information, including suggestions for reducing this burden, to Washington Headquarters Services, Directorate for Information Operations and Reports, 1215 Jefferson Davis Highway, Suite 1204, Arlington VA Respondents should be aware that notwithstanding any other provision of law, no person shall be subject to a penalty for failing to comply with a collection of information if it does not display a currently valid OMB control number. 1. REPORT DATE 03 JAN REPORT TYPE N/A 3. DATES COVERED - 4. TITLE AND SUBTITLE The discrete MoserVeselov algorithm for the free rigid body 5a. CONTRACT NUMBER 5b. GRANT NUMBER 5c. PROGRAM ELEMENT NUMBER 6. AUTHOR(S) 5d. PROJECT NUMBER 5e. TASK NUMBER 5f. WORK UNIT NUMBER 7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES) University of Bergen, Norway 8. PERFORMING ORGANIZATION REPORT NUMBER 9. SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES) 10. SPONSOR/MONITOR S ACRONYM(S) 12. DISTRIBUTION/AVAILABILITY STATEMENT Approved for public release, distribution unlimited 11. SPONSOR/MONITOR S REPORT NUMBER(S) 13. SUPPLEMENTARY NOTES See also ADM001749, Lie Group Methods And Control Theory Workshop Held on 28 June July 2004., The original document contains color images. 14. ABSTRACT 15. SUBJECT TERMS 16. SECURITY CLASSIFICATION OF: 17. LIMITATION OF ABSTRACT UU a. REPORT unclassified b. ABSTRACT unclassified c. THIS PAGE unclassified 18. NUMBER OF PAGES 31 19a. NAME OF RESPONSIBLE PERSON Standard Form 298 (Rev. 8-98) Prescribed by ANSI Std Z39-18

3 Overview The subject of this talk is the numerical solution of the free RB equations, M = [M, Ω], M = ΩJ + JΩ, 2/100 where M, Ω are skew-symmetric matrices and J is a diagonal matrix with positive entries. M is the matrix of body momenta Ω is the matrix of body angular velocity Often the above equations are associated with the equations that give the configuration of the body in the fixed frame, Q = QΩ, Q SO(N). The Discrete Moser Veselov description of the rigid body Integrability of the discrete algorithm ward error analysis of the the DMV algorithm Higher order integrable approximations Numerical experiments and comparisons with other methods Explicit methods for the 3 3 case

4 The Moser Veselov discrete version of the dynamics of a Rigid Body The Lagrangian of the continuous RB equations, is the kinetic energy, L = 1 2 tr(ω M) = 1 2 tr( Ω2 J ΩJΩ) = tr(ω JΩ), (1) where we take into account that Ω = Ω and that the trace is invariant under cyclic permutations. Following (Marsden, Pekarsky & Shkoller 1999), discretise Ω = X 1 Ẋ, where X SO(N) is the configuration of the body, using a finite difference approximation of the derivative, 3/100 Ω = X 1 Ẋ 1 h X k+1(x k+1 X k ), X k, X k+1 SO(N), which gives L 1 h 2 tr(j X k X k+1 J JX k+1x k X k X k+1 JX k+1x k ). Due to the orthogonality of the X k s and the cyclicity of the trace, the first and the last term cancel, and moreover, we can write L 2 h 2 tr(x kjx k+1). Up a scaling factor, this is precisely the discrete Lagrangian of Moser and Veselov.

5 Consider the fuctional S(X) determined by S = k tr(x k JX k+1) where X = {X k } with X k O(N) and J is a diagonal matrix. To obtain the stationary points of S, we consider tr(x k JXk+1) 1 tr(λ k (X k Xk I)), 2 (where Λ k = Λ k k is a Lagrange multiplier), and δs = 0 becomes k 4/100 X k+1 J + X k 1 J = Λ k X k, from which, multiplying by Xk on the right and taking into consideration the symmetry of Λ k, X k+1 JXk + X k 1 JXk = Λ k = Λ k = X k JXk+1 + X k JXk 1, (2) hence, the discrete analogue of the angular momentum in space, is conserved. m k = X k JX k 1 X k 1 JX k,

6 In the body variables, setting ω k = X k X k 1 O(N) and M k = X 1 k 1 m kx k 1 = ω k J Jω k so(n) (angular momentum w.r.t. the body), (2) becomes M k+1 = ω k M k ωk M k = ωk J Jω k. the discrete Euler Arnold equation. In the continuous limit: when t k = t 0 + kε, k = 0, 1, 2,..., X k = X(t k ) ω k = Xk X k 1 I εω(t k ), M k ε(jω + ΩJ) = εm(t k ), (3) 5/100 letting ε 0, one obtains the familiar Euler Arnold equations for the motions of the N-dimensional rigid body, M = [M, Ω] M = JΩ + ΩJ, Ω so(n).

7 To solve the discrete Euler Arnold equations (3): For k = 0, 1, 2,..., find ω k SO(N) such that M k = ω k J Jω k. Update M k+1 = ω k M k ω k. By construction, this algorithm preserves exactly momentum and energy (integrable map) is a second order approximation to the continuous rigid body preserves the standard Poisson structure of T so(n), 6/100 {f, g} = tr(m[f M, g M ]), f, g C (so(n)), where f M = ( f/ M i,j ). Note that Also the IMR is second order, preserves all the integrals of the continuous rigid body. Another much used method is a Lie Poisson integrator of McLachlan and Reich (LP2). For the 3 3 RB, it consists in splitting the Hamiltonian H = H 1 + H 2 + H 3 = m2 1 J 2 + J 3 + m2 2 J 1 + J 3 + m2 3 J 1 + J 2 and integrating explicitly (a la Strang) the vector fields of each split Hamiltonian. The method is second order, explicit, preserves the Poisson structure but does not preserve H.

8 Hamiltonian error Hamiltonian error 4 x IMR time 2 x McL splitting time 3 x MVlu Hamiltonian error time MVlu McL splitting IMR MVqr /100 Error in the Hamiltonian function H in the interval [0, 100] and for h = 1 2. The components of the vector m k for h = m = m 1 m 2 m 3 ˆm = M = 0 m 3 m 2 m 3 0 m 1 m 2 m 1 0

9 Integrability of the Moser Veselov equation Recall the Moser Veselov equation, 8/100 M k = ωk J Jω k, Mk = M k, ωk ω k = I. (4) in tandem with the update M k+1 = ω k M k ωk. The Moser Veselov equation (4) has not a unique solution; Lemma 1 (Moser and Veselov) Equation (4) is equivalent to the factorization (I λm k λ 2 J 2 ) = (ωk + λj)(ω k λj) Rewrite the above factorization as (I λm k λ 2 J 2 ) = A k (λ)a k (λ) A k (λ) = (ω k λj) and define the mapping with image point M k+1 such that By construction, (I λm k+1 λ 2 J 2 ) = A k (λ)a k (λ). (I λm k+1 λ 2 J 2 ) = A k (λ)(i λm k λ 2 J 2 )A 1 k (λ), hence the map is an integrable as it is an isospectral flow.

10 BEA for DMV Recall the DMV equations and the continuous RB equations 9/100 M k+1 = ω k M k ω k, M = [M, Ω] M k = ω k J Jω k, M = ΩJ + JΩ, where ω k I hω(t k ). We wish to write M k+1 = Φ h (M k ) = M k + h[m k, Ω k ] + h 2 d 2 + h 3 d 3 + h 4 d 4 +, and find the modified vector field M = [ M, Ω] + hf 2 ( M, Ω) + h 2 f 3 ( M, Ω) + h 3 f 4 ( M, Ω) + (5) such that Φ h (M k ) equals the solution M(t k+1 ) at time t k+1 = t 0 + (k + 1)h of the modified vector field (5). To find Φ k (h), we write ω k = exp( hω 0 h 2 Ω 1 h 3 Ω 2 h 4 Ω 3 h 5 Ω 4 + ), (6) where Ω 0, Ω 1, Ω 2,..., are skew-symmetric matrices computed so that ω k J Jω k = h(ω(t k )J + JΩ(t k )). (7)

11 we obtain h(ω(t k )J + JΩ(t k )) = h(ω 0 J + JΩ 0 ) + h 2 (Ω 1 J + JΩ (Ω2 0J JΩ 2 0)) + h 3 (Ω 2 J + JΩ [(Ω 0Ω 1 + Ω 1 Ω 0 ), J] (Ω3 0J + JΩ 3 0)) +. Comparing left and right-hand-sides, it is trivially observed that the order-h term disappears if Ω 0 = Ω (to simplify notation, we omit the dependence of Ω on t k ). In order to annihilate the h 2 -term, we require that Ω 1 J + JΩ (Ω2 0J JΩ 2 0) = 0. 10/100 Recall that M = ΩJ + JΩ and hence M = Ω J + JΩ. On the other hand, M = [M, Ω] = (Ω 2 J JΩ 2 ). Hence we can write O = Ω 1 J + JΩ M = Ω 1 J + JΩ (Ω J + JΩ ) and the identity is satisfied by if and only if Ω 1 = 1 2 Ω. (8) In general, the algorithm to derive Ω i, for i = 1, 2,..., is 1. Find the coefficient of h i+1 in (7) and set it equal to zero. This will give an equation of the type Ω i J +JΩ i = C i J +JC i +[D i, J]. Note that the terms C i J +JC i have an odd occurrence of the Ω j s, while the terms of the type [D i, J] have an even occurrence of the Ω j s. 2. Use the derivatives of M and Ω to express the term [D i, J] as C i J + J C i.

12 3. Deduce Ω i = C i + C i. Ω 0 = Ω Ω 1 = 1 2 Ω Ω 2 = 1 4 Ω 1 6 Ω3 Ω 3 = 1 8 Ω 1 24 (5Ω2 Ω + 2ΩΩ Ω + 5Ω Ω 2 ) 11/100 The functions Ω i Once the Ω i s are known, substituting back in M k+1 = ω k M kω k and using the well known identity exp(x)y exp( X) = exp adx Y = k=0 1 k! adk X(Y ), where ad X (Y ) = [X, Y ] and, recursively, ad k X(Y ) = [X, ad k 1 X (Y )], we find the expressions for the functions d i in terms of the Ω i 1, Ω i 2,..., Ω 0, d i = i ( 1) j j! j=1 k 1 +k 2 + +k j =i j ad Ωk1 ad Ωk2 ad Ωkj M, k 1,... k j {0, 1,..., i 1}. (9) d 2 = 1([M, 2 Ω ] + [[M, Ω], Ω]), d 3 = 1[M, 4 Ω ] + 1[[M, 4 Ω ], Ω] + 1[[M, Ω], 4 Ω ] + 1[[[M, Ω], Ω], Ω] 1[M, 6 6 Ω3 ], (10) d 4 =...,

13 Taylor expansion of the solution of the modified equation Consider d dtỹ = f(ỹ) + hf 2(ỹ) + h 2 f 3 (ỹ) +, where f(m) = [M, Ω] = [M, J 1 M] is the original vector field of the RB equations, where J is a linear operator, defined such that J Ω = ΩJ + JΩ = M. Putting ỹ(t) = M(t), we expand the solution of the above equation in a Taylor series and collect corresponding powers of h, ỹ(t + h) = M(t) + hf(m) + h (f 2 2 (M) + 1 ) 2! f f(m) + h (f 3 3 (M) + 1 2! (f f 2 (M) + f 2f(M)) + 1 ) 3! (f (f, f)(m) + f f f(m)) +, 12/100 where f is considered as a linear operator, f as a bilinear operator and so on and so forth. In our case, f (z)(m) = [z, J 1 M] + [M, J 1 z] = [z, Ω] + [M, J 1 z] f (z 1, z 2 )(M) = 2[z 1, J 1 z 2 ], and, since f is quadratic, f and all the other higher derivatives equal zero. At this point it is important to stress an important difference between the expressions for the modified vector field of (Hairer, Lubich & Wanner 2002) and ours. While the vector field discussed in (Hairer et al. 2002) is in R n, hence the f is a symmetric quadratic operator, this is not the case for our vector field which is on matrices, thus f (f f, f) f (f, f f).

14 This non-commutative case is discussed with more generality in (Munthe-Kaas & Krogstad 2002). However, we observe that all the terms containing combinations of f, f and f correspond simply to higher derivatives of f. The mixed terms are treated instead specifically. After some algebra, we have f 2 = d 2 1 2! f f(m) = O, 13/100 f 3 = d 3 1 3! (f (f, f)(m) + f f f(m)) = 1 12 [M, Ω [Ω, Ω ] 2Ω 3 ], f 4 = d 4 1 4! M (iv) 1 2! (f f 3 + f 3f) = O, (11) f 5 = d 5 1 M (v) 1 (f f 5! 2! 4 + f 4 f + 1 d (f 2! dt 3f + f f 3 )) = 1 [M, 80 Ω(iv) ] 1 [M, [Ω, 80 Ω ]] + 3 [M, 40 Ω5 Ω ΩΩ ] + 1 [M, 80 [Ω, Ω ]] 1 [M, 40 ΩΩ Ω] 1 [M, 20 Ω2 Ω + Ω Ω 2 ] + 1 [M, 20 [Ω3, Ω ]] 1 [M, 40 Ω 2 Ω + ΩΩ 2 + Ω[Ω, Ω ]Ω].

15 1 m / m m time The DMV solution of the RB equations (dotted line), the exact solution (solid line) and the trajectories corresponding to the modified vector fields f + h 2 f 3 (dashed line) and f + h 2 f 3 + h 4 f 5 (dash-dotted line) in the interval [0, 50] with h = 8 10.

16 Some important results about DMV Theorem 2 The DMV is time-reversible, hence f 2i = 0, i = 1, 2,.... Theorem 3 (Moser Veselov) In the 3 3 case, the DMV is a time-reparamtetrisation of the flow of the original vector field of the rigid body. Since the mapping preserves the underlying Poisson structure and all the integrals F i = c i of the system, it commutes with all commuting Hamiltonian flows generated by the F i s, M = {M, F i }. The nonsingular compact level sets T c = i (F i = c i ) consists of a finite union of 1-dimensional tori and on each torus the DMV mapping is a shift along the trajectory depending on the integral quantity H 2. Hence, the DMV solves the modified equation 15/100 M = (1 + h 2 τ 3 + h 4 τ h 2i τ 2i+1 + )[M, Ω], where h is the stepsize of integration and the τ 2i+1, for i = 1, 2,..., are constants.

17 Introduce the constants C J,i,j = J i 1J j 2 + J i 1J j 3 + J i 2J j 3, C J,i = C J,i,i C J = C J,1 = (J 1 + J 2 )(J 1 + J 3 )(J 2 + J 3 ) H 2 = (J 1 + J 2 )(J 1 + J 3 )(J 2 + J 3 )H C J m 2. 16/100 Theorem 4 One has and τ 5 = τ 3 = ((3 det(j)tr(j) + C J,2) m (3C J + tr(j 2 ))H 2 ), 1 ( (3tr(J 4 ) + 27C 40 4 J,2 + 15tr(J 2 )C J + 45 det(j)tr(j))h2 2 + (10C J, det(j)tr(j)c J + 10 det(j)tr(j)tr(j 2 ) + 2C J,2 tr(j 2 ) 28 det(j 2 )) m 2 2H 2 ) + (60 det(j 2 )C J + 3C J, det(j 2 )tr(j 2 ) + 15 det(j)(c J,2,3 + C J,3,2 )) m 4 2. Proof. By direct computation of f 3 and f 5.

18 Higher-order integrable methods For the original RB equations, scaling the initial condition is equivalent to scaling time. In our case, we know that DMV is a time-rescaling of the original RB equation. Therefore we wish to rescale the initial condition to obtain a better approximation of the unscaled original RB. 17/100 I.C. DMV h(ω(t k )J + JΩ(t k )) New I.C. DMV h(ω(t k )J+JΩ(t k )) 1+ τ 3 h 2 + τ 5 h 4 + We perform again the backward error analysis. We set now ω = exp( h Ω 0 h 2 Ω1 + ) and solve for the Ω i s as the skew-symmetric matrices that solve h(1 τ 3 h 2 + ( τ 2 3 τ 5 )h 4 + )(ΩJ + JΩ) = ω J J ω. (12) f 3 = d 3 1 3! M = τ 3 [M, Ω] + d 3 1 3! M = τ 3 [M, Ω] + f 3 = ( τ 3 + τ 3 )[M, Ω], hence, in order to have an order-four scheme, we must set f 3 = 0 which corresponds to the choice τ 3 = τ 3.

19 After further computations, one has f 5 = 0 τ 5 = τ 5 2τ 2 3. This value of τ 5 gives indeed a method of order six. The new proposed algorithms of order four and six are described below. The DMV4 algorithm: 18/ Compute τ 3 and set M 0 := M(t 0 )h/(1 + h 2 τ 3 ). 2. For k = 0, 1,..., n 1, find the unique w k as above such that M k = ω k J Jω k set M k+1 = ω k M k ω k end 3. Reconstruct M n := M n (1 + h 2 τ 3 )/h M(t n ). The DMV6 algorithm: 1. Compute τ 3, τ 5 and set τ 5 = τ 5 2τ 2 3 and M 0 := M(t 0 )h/(1 + h 2 τ 3 + h 4 τ 5 ). 2. For k = 0, 1,..., n 1, find the unique w k as above such that M k = ω k J Jω k set M k+1 = ω k M k ω k end 3. Reconstruct M n := M n (1 + h 2 τ 3 + h 4 τ 5 )/h M(t n ).

20 Some numerical experiments We consider with initial condition and matrix J given as J = m 0 = and compare the DMV explicit scheme with the Hamiltonian-splitting method LP2 of (McLachlan 1993) and the Implicit Midpoint Rule (IMR), H = m2 1 J 2 + J 3 + m2 2 J 1 + J 3 + m2 3 J 1 + J 2 = H 1 + H 2 + H 3 19/100 m k+1 = m k + hf( m k + m k+1 ), 2 where f(m) = m ( J) 1 m, J = J 2 + J J 1 + J J 1 + J 2.

21 / Error at T= LP2 IMR DMV DMV4 DMV Step size h Error versus step size computed at T = 100 for the methods LP2, IMR, DMV, DMV4, DMV6.

22 LP2 IMR DMV DMV4 DMV /100 Error at T= Floating point operations Floating point operations versus accuracy (T = 100) for the methods LP2, IMR, DMV, DMV4, DMV6. The roots of P (λ) are recomputed at each step, use QR with pivoting, (DMV 22 LP2 per step).

23 10 2 LP2 IMR 10 0 DMV DMV4 DMV6 RATTLE / Floating point operations versus accuracy (T = 100) for the methods LP2, IMR, DMV, DMV4, DMV6 and RATTLE6. The roots of P (λ) are computed once, use LU instead of QR (DMV 19

24 LP2 per step). Method h = 1 16 h = 1 2 h = 1.2 h = 2.2 h = 2.5 h = 4 LP e e e e e e+00 IMR e e e e e e-01 DMV e e-01 NaN NaN NaN NaN DMV e e e e e e-01 DMV e e e-02 NaN e e-01 23/100 Error for the various methods and selected step sizes

25 Connections with matrix Ricatti equations Consider the matrix equation M = XJ JX. (13) Cardoso & Leite (2003) shown that every solution of (13) (not necessarily orthogonal) is of the form X = (M/2 + S)J 1, for some symmetric matrix S. Furthermore, X is a orthogonal solution of (13) if and only if S is a symmetric solution of the Riccati equation S 2 + S(M/2) + (M/2) S (M 2 /4 + J 2 ) = 0. (14) Riccati equations are associated to symplectic matrices. In our case, the symplectic matrix is [ ] M I H sympl = 2 M 2 + J 2 M. (15) 4 2 If M J 2 is positive definite, it has been shown in (Cardoso & Leite 2003) that (14) has a unique solution S which is symmetric, positive definite, and such that the eigenvalues of W = M/2 + S have positive real parts. This matrix W is precisely the same matrix in Moser & Veselov (1991), from which one obtains ω = W J 1. 24/100

26 Algorithm(Cardoso & Leite 2003): Compute X, the unique solution of (13) in the special orthogonal group SO(n). 1. Find a real Schur form of H sympl, Q H sympl Q = [ T11 T 12 O T 22 ], (16) 25/100 where T 11 and T 22 are block upper-triangular matrices such that the real parts of the spectrum of T 11 are positive and the real parts of the spectrum of T 22 are negative definite. 2. Partition Q accordingly, Then, compute 3. Compute [ Q11 Q Q = 12 Q 21 Q 22 X = S = Q 21 Q ]. ( ) M 2 + S J 1. Some computational details Compute real Schur forms by QR iterations for eigenvalues (Golub & van Loan 1989) Cost: O((2N) 3 ) operations (implicit methods for ODEs: O(N 3 )) N being the dimension of M.

27 The case N = 3 26/100 In this case, it is possible to find an explicit spectral decomposition of H sympl (without the QR eigenvalue method) construct the real Schur decomposition (16) and hence X from the eigenstructure of H sympl. This yields an explicit numerical method for the reduced RB equations.

28 The eigenvalues of the matrix H sympl, H sympl = [ M 2 I M J 2 M 2 are the solutions of the quadratic eigenvalue problem P (λ) = det(λ 2 I λm J 2 ) = 0. Without loss of generality, we assume that J is diagonal, with entries J 1, J 2, J 3. Then, P (λ) = λ 6 λ 4 (J J J 2 3 m 2 12 m 2 13 m 2 23) + λ 2 (J 2 1 J J 2 1 J J 2 2 J 2 3 m 2 12J 2 3 m 2 13J 2 2 m 2 23J 2 1 ) J 2 1 J 2 2 J 2 3 = λ 6 λ 4 (tr(j 2 ) m 2 ) + λ 2 (C J,2 H 2 ) det(j 2 ). ] (17) (18) 27/100 C J,i,j = J i 1J j 2 + J i 1J j 3 + J i 2J j 3, C J,i = C J,i,i C J = C J,1 H 2 = (J 1 + J 2 )(J 1 + J 3 )(J 2 + J 3 )H C J m 2. Reduce to a cubic equation (compute the roots explicitely)

29 Schematical procedure Compute eigenvalues/eigenvectors of H sympl : H sympl [ Y1 Y 2 ] = [ Y1 Y 2 ] [ Λ + Λ ], Re Λ + 0, (the eigenvectors need not be orthogonal and may be complex). Y 1, Y 2 R 6 3, Λ ± R /100 Orthogonalize the eigenvectors (by Grahm-Schmidt or QR), [Y 1, Y 2 ] = QR, so that H sympl Q = QRΛR 1 is the complex Schur form. Reduce to a real Schur form by considering real/imaginary part (complex Givens rotation). Compute S = Q 21 Q 1 11, X = (M/2 + S)J 1. We don t need all the eigenvectors, just Y 1. Don t need R. Avoid complex arithmetic alltogether.

30 The numerical DMV algorithm m 0 hm 0 = h ˆm 0. Compute the eigenvalues of H sympl = H(M k ) solving for P (λ) = 0 as in (18). For t k = t 0 + kh, k = 0, 1, /100 Compute the (real) eigenvectors corresponding to Λ + Compute 3 quadratic eigenvectors (3 matrix factorizations, LU/QR, with pivoting). No need to compute explicitely L or Q. Compute the dependent eigenvectors. Orthogonalize the eigenspace By (modified) Grahm Schmidt or QR. Only the Q factor is needed. Compute S = Q 21 Q 1 11, ω k = (M/2 + S)J 1 Update M k+1 = ω k M kω k Rescale m N M N /h. This algorithm produces an explicit method that is about times more expensive than LP2, the explicit method of McLachlan and Reich.

31 Concluding remarks Explicit algorithms to solve for the N = 3 free rigid body The methods are up to 6th order, completely integrable, possible to increase to arbitrary order 30/100 The cost of the method is about times more expensive than the explicit LP2. The cheaper versions seem to be less stable expecially for large step-size and long time computations Reconstruction equations: Find the configuration X k+1 = X k ω k. The complexive order is still 2 but the error is generally smaller. ward error analysis reveals that some error terms cancel, however the components are reparametrized with 3 different time scales. To obtain higher order reconstructions, one can use interpolation of the vector field XΩ and its higher derivatives using the computed values of Ω k. Optimal step-size?

32 References Cardoso, J. R. & Leite, F. S. (2003), The Moser Veselov equation, Lin. Alg. Applic. Golub, G. H. & van Loan, C. F. (1989), Matrix Computations, 2nd edn, John Hopkins, Baltimore. Hairer, E., Lubich, C. & Wanner, G. (2002), Geometric Numerical Integration, number 31 in Springer Series in Computational Mathematics, Springer, Berlin. Marsden, J. E., Pekarsky, S. & Shkoller, S. (1999), Discrete Euler Poincaré and Lie Poisson equations, Nonlinearity 12, McLachlan, R. I. (1993), Explicit Lie Poisson integration and the Euler equations, Physical Review Letters 71, Moser, J. & Veselov, A. P. (1991), Discrete Version of Some Classical Integrable Systems and Factorization of Matrix Polynomials, Commun. Math. Phys. 139, Munthe-Kaas, H. Z. & Krogstad, S. (2002), On enumeration problems in Lie Butcher theory, Report in informatics, Institutt for informatikk, University of Bergen, Norway. To appear in Future Generation Computer Systems, Special issue. 31/100