Numerische Mathematik

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1 Numer. Math. DOI /s Numerische Mathematik and numerical integrators Stein Krogstad Hans Z. Munthe-Kaas Antonella Zanna Received: 18 August 2008 / Revised: 12 May 2009 Springer-Verlag 2009 Abstract Motivated by developments in numerical Lie group integrators, we introduce a family of local coordinates on Lie groups denoted generalized polar coordinates. Fast algorithms are derived for the computation of the coordinate maps, their tangent maps and the inverse tangent maps. In particular we discuss algorithms for all the classical matrix Lie groups and optimal complexity integrators for n-spheres. Mathematics Subject Classification (2000) 22E15 General properties and structure of real Lie groups 17B40 Automorphisms, derivations, other operators 57S25 Groups acting on specific manifolds 1 Introduction Lie group methods for integration of differential equations have been an active area of research over more than a decade [3,10,21,26,28,30,32,33]. Such methods are constructed by composing Lie group actions on a manifold. Due to their accuracy and structure preserving properties, these methods have found applications in many areas of computational science, such as inverse eigenvalue problems [9], image processing [7], NMR spectroscopy [18], computational mechanics [24], neural learning [4], independent component analysis [35], micromagnetics [25], Sturm Liouville problems [16], Evans functions [27], non-linear dynamical systems [38] and image registration [23]. S. Krogstad SINTEF ICT, Forskningsveien 1, Oslo, Norway Stein.Krogstad@sintef.no H. Z. Munthe-Kaas A. Zanna (B) Matematisk Institutt, University of Bergen, Johannes Brunsgt 12, 5008 Bergen, Norway Antonella.Zanna@math.uib.no H. Z. Munthe-Kaas Hans.Munthe-Kaas@math.uib.no

2 S. Krogstad et al. In many applications the efficient computation of coordinate maps on Lie groups is an important issue. A family of numerical integrators for Lie groups, based on local coordinates, was presented in [14] (see Algorithm 1). These methods are built on two fundamental components: (i) a local coordinate map from a Lie algebra to a Lie group; and (ii) the inverse tangent map of. These two maps will be the focus of the present paper. Analytic coordinate maps include the exponential map [30], the Cayley map, and more generally diagonal Padé approximants to the exponential. It is well known that for certain groups (e.g. SL(n)), the only analytic map from the algebra to the group is the exponential mapping [15]. Matrix splitting techniques yield non-analytic coordinate maps. Among these, the coordinates of the second kind, studied by Owren and Marthinsen [34], show excellent computational cost for certain groups with semisimple Lie algebras over C. Their theory is based on the concept of admissible ordered bases (AOB), and, according to the authors, should be also able to handle particular real Lie algebras over R. However, the theory does not apply for instance to the case of real skew-symmetric matrices (corresponding to the real orthogonal group, one of the most recurrent in applications), in the sense that, the natural basis of skew-symmetric matrices, E ij = e i e T j e j ei T, does not give rise to an AOB. In this paper an other family of coordinates based on matrix splittings is studied. By recursively applying generalized polar decompositions of the Lie algebra [31,37], we obtain coordinates on all the classical matrix groups, where both the coordinate maps and the (forward and inverse) tangent maps can be computed efficiently. Let us briefly review the generalized polar coordinates as defined in Sect. 3. Consider a nested sequence of Lie algebras g = g 0 g 1 g k derived from a sequence of linear maps σ i : g i g i, i = 0,...,k 1, such that the following relations hold σ 2 i = I σ i ([X, Y ]) =[σ i (X), σ i (Y )] ( ) 1 Range 2 (I + σ i) = g i+1. An arbitrary element Z g is decomposed as Z = P 0 + P P k 1 + K k, (1) where the components are computed as follows. Let K 0 = Z and for i =0,...,k 1let K i+1 = 1 2 (I + σ i)k i g i+1 P i = K i K i+1 = 1 2 (I σ i)k i.

3 The coordinate maps to be studied in this paper are of the form (Z) = exp(p 0 ) exp(p 1 ) exp(p k 1 ) exp(k k ). (2) It is assumed that the computations in the last Lie algebra g k can be done fast, either because of low dimensionality or because it has a special structure (diagonal or block diagonal matrices). The computations involving the matrices P i and the tangent maps are relying on the structure of 2-cyclic matrices to be discussed in the sequel. The paper is structured as follows. In Sect. 2 we review basic results of linear algebra, matrix Lie theory and numerical Lie group integrators. Section 3 introduces the generalized polar coordinates, and develops the theory of their tangent maps, while in Sect. 4 the classical matrix groups are discussed in detail. These two sections are the core of the paper, in which the theory for the novel coordinates, for their tangent and inverse tangent maps, is studied in details and useful computational formulas are derived. In Sect. 5 we consider numerical examples, and finally, in Sect. 6 we present some short concluding remarks. 2 Preliminaries 2.1 Coordinates in Lie group integrators In order to motivate the theory in the sequel, we will briefly review a class of numerical integrators introduced in [29,30]. We will present the theory in the concrete context of matrix Lie groups. The generalization to general Lie groups is discussed in [20]. Let g denote a matrix Lie algebra, i.e. a family of square matrices closed under linear combinations and matrix commutators, [A, B] =AB BA. Let G denote the Lie group of g, defined as the set of matrices obtained by taking matrix exponentials of g and products of these exponentials. The Lie group is closed under matrix products and matrix inversions. Let M denote a linear space and : G M M an action of G on M, defined as a map satisfying g (h y) = (gh) y, for all g, h G, y M. The action induces a product :g M T M, where T M denotes the tangents of M, via V y = d dt exp(tv) y. t=0 We consider differential equations evolving on M, written in the form [30] where y(t) M and f : M g. y (t) = f (y(t)) y, (3)

4 S. Krogstad et al. Let : g G denote a smooth mapping from a Lie algebra into a Lie group such that (0) = e, where e is the identity in G.Letd Z denote the right trivialized tangent of at a point Z g, i.e. s (Z + sδz) = d Z (δz) (Z). s=0 It follows easily that d Z is a linear map from g to itself. We assume that d 0 = I, thus d Z is invertible for Z sufficiently close to 0. The map defines a diffeomorphism between a neighbourhood of 0 g and a neighbourhood of e G. Via the translations on G, we may extend to an atlas of coordinate charts covering the whole of G. Using the action of G on M, we may also via obtain a coordinate atlas on M. For numerical algorithms based on such coordinates, it is often essential that we can compute both and the inverse tangent map d 1 Z efficiently. As an example, consider the class of numerical Lie group integrators for (3), introduced and developed in [14,30]. Given y n M, a timestep h R, and the coefficients of an s-stage Runge Kutta method, { } s a i, j i, j=1 and { } s b j j=1, we step from y n y(t n ) to y n+1 y(t n + h) in the following manner: Algorithm 1 for i = 1, s, U i = s j=1 a i, j K j K i = h f ( (U i ) y n ) K i = d 1 U i (K i ) end y n+1 = ( s j=1 b j K j ) y n Various coordinate maps have been proposed and studied in the literature. The exponential mapping is used in [30] and the Cayley map in [26]. Owren and Marthinsen [34] develop the theory of coordinates of the second kind. Given a basis {V i } for a d-dimensional Lie algebra g, an element Z = d i=1 z i V i maps to (Z) = exp(z 1 V 1 ) exp(z 2 V 2 ) exp(z d V d ). Owren and Marthinsen introduce special classes of nice bases, so-called AOBs, and show that for such bases the map (Z) and d Z 1 can be computed in O(d 3/2 ) operations, d = dim g. Using the representation theory of semisimple Lie algebras, they show that for certain semisimple Lie algebras, AOBs can be obtained from Chevalley bases. In the cases where AOBs are found, they report favourable numerical experiments indicating that the resulting numerical algorithms are between two and six times faster than corresponding algorithms based on using (Z) = exp(z), see also [5,6]. Unfortunately, it is not known if AOBs can be found for all classical matrix Lie groups. In particular there are still open problems with several of the real matrix groups, such as the real orthogonal group SO(n, R) with algebra so(n, R), consisting of real skew symmetric matrices.

5 In this paper we will introduce coordinates based on generalized polar decompositions of g, and we will show that this framework leads to fast computable coordinates for many Lie algebras, among these all the classical matrix Lie algebras. The basis for our approach is some results from the theory of symmetric spaces, as presented in [31]. We will now review some linear algebra and basic elements of the theory of symmetric spaces needed for the present purpose. 2.2 Projections, involutions and 2-cyclic matrices By a projection matrix on a vector space V we mean a linear map : V V such that 2 =. Unless explicitly stated we will not require projections to be orthogonal (i.e. T = ). By an involutive matrix we mean a linear map S : V V such that S 2 = I. These two concepts are naturally linked by the following lemma, whose proof is trivial. Lemma 1 To any projection matrix there corresponds an involutive matrix S = I 2. Conversely, to any involutive S there corresponds two projection matrices S and + S defined by S = 1 (I S) 2 (4) + S = 1 (I + S). 2 (5) These satisfy the following relations S + + S = I (6) S + S = + S S = 0 (7) S S = S (8) S + S = + S. (9) Thus V splits in the direct sum of two subspaces, the ±1 eigenspaces of S, where ± S are projections onto these. Note that if S is involutive then also S is involutive, the latter corresponding to the opposite identification of the +1 and 1 eigenspaces. Thus, at the moment there seems to be no fundamental difference between these two subspaces. We will, however, later return to involutive automorphisms on Lie algebras where the +1 and 1 eigenspaces play fundamentally different roles. Definition 1 A matrix K : V V is block diagonal with respect to an involution S on V if SKS = K, (10)

6 S. Krogstad et al. and a matrix P : V V is 2-cyclic with respect to S if SPS = P. (11) Any matrix M can be split in a sum of a 2-cyclic P and a block diagonal K, M = P + K, (12) where P = 1 (M SMS) (13) 2 K = 1 (M + SMS). (14) 2 To understand the structure of these matrices, it is convenient to represent linear operators M : V V in 2 2 block partitioned form in the following manner. The matrix M splits naturally in four parts: M = ( S + ) ( + S M S + + ) S = S M S + S M + S + + S M S + + S M + S. (15) The partitioning of M with respect to S is defined by dividing V in an upper block corresponding to the range of S and a lower block corresponding to the range of + S. Thus ( ) M M + M = M + M ++, (16) where M ij = i S M j S restricted to the appropriate subspaces. In partitioned form S becomes ( I ) 0 S = 0 I. Thus, K consists of the diagonal blocks of M, while P is the off diagonal blocks. Efficient computation of analytic functions of 2-cyclic matrices will be crucial to our algorithms. Theorem 1 Let S P S = P, where S 2 = I. Let = P 2 S. For an analytic function ψ(x) we have ψ(p) = ψ(0)i + ψ 1 ( )P + Pψ 1 ( ) + Pψ 2 ( )P + ψ 2 ( ), (17)

7 where ψ 1 (x) = 1 2 ( ) ψ( x) ψ( x) x (18) ψ 2 (x) = 1 ( ) ψ( x) + ψ( x) 2ψ(0). 2x (19) Proof Involutivity of S implies SP = PS hence ± S P = P S. By induction it is now easy to verify that for any k 0wehave P 2k+1 = k P + P k, P 2k+2 = k+1 + P k P. Letting ψ(x) = i=0 α i x i, we find from these formulae that ψ(p)=α 0 I + α 2k+1 k P + P α 2k+1 k + α 2k+2 k+1 + P α 2k+2 k P. k=0 k=0 We define ψ 1 (x) = k=0 α 2k+1 x k and ψ 2 (x) = k=0 α 2k+2 x k and derive (18) and (19) by straightforward manipulation of the series for ψ(x). Note that if P is partitioned with respect to S as P = k=0 k=0 ( ) 0 B, (20) A 0 then Thus ( ) = P 2 BA 0 S =. (21) 0 0 ψ i ( ) = ( ) ψi (BA) 0. 0 ψ i (0)I Since BAis a p p matrix, p = Rank( S ), we are mainly interested in cases where p is small, and the case p = 1 is particularly important. Corollary 1 If p = 1 then = θ S, for a scalar θ, and where ψ 1 and ψ 2 are given in (18) (19). ψ(p) = ψ(0)i + ψ 1 (θ)p + ψ 2 (θ)p 2, (22)

8 S. Krogstad et al. Proof If = θ S then ψ 1 ( )P + Pψ 1 ( ) = ψ 1 (θ) ( S P + ) P S = ψ1 (θ)p. Similarly ( Pψ 2 ( )P + ψ 2 ( ) = ψ 2 (θ) P S P + S P2) = ψ 2 (θ)p Involutive automorphisms on Lie algebras An involutive automorphism on a Lie algebra g is an involutive map σ : g g such that σ ([U, V ]) = [σ(u), σ (V )]. (23) Corresponding to the automorphism σ on g there is an automorphism on the Lie group G, which plays an important role in the theory of symmetric spaces. In this paper we will, however, not need this automorphism on G and we omit this from the discussion. Note that if σ is an automorphism, then σ is not an automorphism. Thus in this case we can clearly distinguish between the +1 and 1 eigenspaces of σ,theyplay a fundamentally different role in the theory. Let ± σ be projections on ± eigenspaces of σ,givenin(4) (5). Denote these spaces by ( 1 p = Range (I σ) 2 ( 1 k = Range (I + σ) 2 ) = { P g σ(p) = P } ) = { K g σ(k ) = K }. The subspace k is a Lie subalgebra of g, while p forms a Lie triple system, meaning that it is closed under double brackets [P 1, [P 2, P 3 ]] p for all P 1, P 2, P 3 p. More generally, the spaces p and k satisfy the following odd even parity rules under brackets (compare to multiplication table of -1 and 1): [ ] k, k k (24) [ ] p, k p (25) [ ] p, p k. (26) The decomposition we have introduced is thus closely related to the Cartan decomposition, see[17]. However, the Cartan decomposition requires σ to be a Cartan involution, whose definition involves non-degeneracy of a bilinear form derived from the

9 Cartan-Killing form. For the applications in this paper the Cartan property of σ is not needed, thus we have considerable freedom in choosing a suitable σ. Corresponding to the additive splitting g = p k there exists a multiplicative splitting of G. Any element g G sufficiently close to I can be written as a product g = exp(p) exp(k ), where P p and K k. The element exp(p) G belongs to a symmetric space, while exp(k ) belongs to a Lie subgroup of G. For example, if G = GL(n) and σ(z) = Z T, then the splitting g = p + k corresponds to writing a general matrix as the sum of a symmetric and a skew matrix. The corresponding product splitting of G is the polar decomposition, where a matrix is written as a product of a symmetric positive definite matrix and an orthogonal matrix. In general, if σ is any involutive automorphism, we will refer to the decomposition as a generalized polar decomposition, see[31] for more details. 3 Generalized polar coordinates (GPC) The coordinates we are studying in this paper are recursively defined as follows. Definition 2 GPC denotes any coordinate map from a Lie algebra to a Lie group obtained by the following recursion: The exponential map exp : g G is GPC on G. Let σ be an involutive automorphism on g and let ± σ be defined by (4) (5). Let k = Range( + σ ) g and Gσ G be the corresponding sub-algebra and subgroup. If : k G σ is GPC on G σ then a map : g G defined as (Z) = exp ( σ Z) ( + σ Z), (27) is GPC on G. The coordinates are completely determined by a sequence of involutive automorphisms {σ i } i=0 k 1, giving rise to a sequence of subalgebras g = g 0 g 1 g k, where g i+1 = Range( + σ i ) and σ i : g i g i. By unfolding the recursion, we obtain the equivalent form of the coordinate map given in (1) (2). Note that we let the 1 eigenspace appear on the left and the +1 on the right in (27). This is important if we want to compute right trivialized tangents. If we instead want to work with left trivializations, we must reverse the definition and let the +1 eigenspace appear on the left. As a trivial example consider G = C (the multiplicative group of nonzero complex numbers), let σ(z) = z (complex conjugation) and let (K ) = exp(k ). If Z = X + iy then σ Z = X, + σ Z = iy and (Z) = exp(x) exp(iy) = r exp(iθ), where r = exp(x) and θ = Y. This yields (classical) polar coordinates on C.

10 S. Krogstad et al. 3.1 The coordinate map To obtain efficient algorithms for computing the coordinate map, we assume that σ is a low rank inner automorphism, defined as follows. Definition 3 An automorphism σ : g g of the form σ(z) = SZS, where S 2 = I (28) is called an inner automorphism. 1 By the rank of an inner automorphism we mean p = Rank( S ). If σ is given by (28) then P = σ (Z) = 2 1 (Z SZS) is 2-cyclic with respect to S, i.e. SPS = P. Theorem 1 yields the following formula for computing exp(p). Theorem 2 Let S P S = P, where S 2 = I. Let = P 2 S, then where exp(p) = I + ψ 1 ( )P + Pψ 1 ( ) + Pψ 2 ( )P + ψ 2 ( ), (29) { sinh( x) x = sin( x) x for x = 0 ψ 1 (x) = 1 for x = 0 { ψ 2 (x) = 2 sinh2 ( x/2) x = 2 sin2 ( x/2) x for x = for x = 0 (30) (31) Proof From (18) we find ψ 1 (x) = (exp( x) exp( x))/(2 x) = sinh( x)/ x. Similarly from (19) ψ2 (x) = (cosh( x) 1)/x, the equivalent form (31) being numerically better for small x. See notes after Theorem 1 for a discussion of the structure of. To compute ψ i ( ), practical algorithms can be devised using the theory of minimal polynomials of the underlying linear operator; these will be discussed in the sequel. For the rank 1 case we establish explicit formulae. Corollary 2 Let S P S = P, where p = Rank( S ) = 1. Let the scalar θ be given as S P2 = θ S. Then I + P P2 if θ = 0 exp(p) = I + sin( θ) θ P + 2 sin2 ( θ/2) θ P 2 if R θ<0 I + sinh( θ) P + 2 sinh2 ( θ/2) θ θ P 2 if R θ>0or if θ is complex. (32) 1 If G is a matrix group, then σ induces the group automorphism G g SgS G. Although it is not necessary that S G, it must be an element of some larger group containing G as a subgroup, and on this larger group the automorphism is properly of inner type. We stick to the name inner also when S G.

11 It should be noted that this is exactly the same formula as the Rodrigues formula for skew 3 3 matrices. Indeed, let 0 x 3 x 2 P = x = x 3 0 x 1 x 2 x 1 0 and let S = I 2zz T /z T z, where z is a vector such that z T x = 0. Then SPS = P. One finds that θ = x T x, from which Rodrigues formula follows. 3.2 The tangent map In this section we will develop the formulae for computing the tangent map of and its inverse. Whereas the theory of the previous section regarded elements of g as matrices (linear operators in R n n acting on R n ), we are now concerned with d Z and d 1 Z which belong to the space of all linear operators from g to itself, denoted End(g). Ifg is represented as n n matrices, then End(g) could be represented as n 2 n 2 matrices. Inversion of such linear operators by Gaussian elimination would cost O(n 6 ) operations, but we are seeking algorithms of complexity at most O(n 3 ). To achieve this, we can not rely on a matrix representation of End(g), but rather work directly with operators (projections, involutions) as outlined in Sect The theory of this section relies on the odd even splitting of the subspaces g = p k in (24) (26). We will develop several formulae for the tangent map of and its inverse. In the following, let σ be an involutive automorphism on g, and let P p g, i.e. σ(p) = P. Define the linear operator ad P : g g as ad P (Z) =[P, Z]. (33) Theorem 3 Let Z = P + K = σ Z + + σ Z. The right trivialized tangent of and its inverse are given as where u = ad P. d Z = d exp p σ + Ad exp(p)d K + σ ( ) exp(u) 1 ( ) = σ u + exp(u) + σ σ + d K + σ ) d 1 Z = ( σ + d 1 K + σ (34) ( ) 1 + (u 1) cosh(u) σ sinh(u) + (1 u) + σ, (35)

12 S. Krogstad et al. Proof The first form of (34) follows from d Z (δz) (Z) = s exp(p + sδ P) (K + sδk ) s=0 = d exp P (δ P) exp(p) (K ) + exp(p)d K (δk ) (K ) ( ) = d exp P (δ P) + exp(p)d K (δk ) exp( P) (Z). Let u = ad P.Usingd exp P = (exp(u) 1)/u,Ad exp(p) = exp(ad P ), 2 σ = σ and + σ σ = 0, we obtain the second form of (34). To verify (35), we observe from (24)to (26) that if ψ(x) is an analytic function with odd and even parts ψ(x) = ψ o (x)+ψ e (x), ψ o (x) = 1 (ψ(x) ψ( x)), 2 ψe (x) = 1 (ψ(x) + ψ( x)), 2 then σ ψ(u) σ = ψe (u) σ, + σ ψ(u) σ = ψo (u) σ σ ψ(u) + σ = ψo (u) + σ, + σ ψ(u) + σ = ψe (u) + σ. Thus σ + σ exp(u) 1 σ u = sinh(u) σ u exp(u) 1 σ u = cosh(u) 1 σ u σ exp(u) + σ = sinh(u) + σ + σ exp(u) + σ = cosh(u) + σ. Using these formulae, we find that ( )( ) 1+(u 1) cosh(u) exp(u) 1 σ sinh u +(1 u) + σ σ u +exp(u) + σ = σ + + σ = I. Since is acting only on the subalgebra k we have σ d k + σ = 0 and + σ d k + σ = d K + σ from which we get )( ) ( σ + d 1 K + σ σ + d K + σ = σ + + σ = I, thus (35) is verified. Using the same odd even parity argument as in the proof of Theorem 3 we obtain the following result:

13 Corollary 3 The partitioning of d Z and d 1 Z with respect to σ is ( sinh(u) )( ) u sinh(u) I 0 d Z = (36) cosh(u) 1 u cosh(u) 0 d K ( )( I 0 u cosh(u) ) d 1 sinh(u) u Z = 0 d 1, (37) K 1 cosh(u) sinh(u) 1 where u = ad P and P = σ Z restricted to the appropriate subspace. For efficient computation of the tangent map, it is essential to develop fast algorithms for computing analytic functions of ad P. First we use the theory of 2-cyclic matrices to simplify (35). Lemma 2 If σ(p) = P, then ad P is 2-cyclic with respect to σ, i.e. Proof Let Z g be arbitrary. We have σ ad P σ(z) = σ ([P,σ(Z)]) = σ ad P σ = ad P. (38) [ ] σ(p), σ 2 (Z) = [ P, Z] = ad P (Z). Theorem 4 Let σ be an arbitrary involutive automorphism. Let Z = P + K where P = σ Z, let u = ad p and = ad 2 P σ. Then where d Z 1 = ( ) σ + d 1 K + (I ( σ + u ψ1 ( ) σ ) + σ + ψ2 ( ) ) σ, (39) ψ 1 (x) = ψ 2 (x) = { tanh( x/2) x = tan( x/2) x for x = for x = 0 { x tanh( x) 1 = x tan( 1 for x = 0 x) 0 for x = 0. (40) (41) Proof Starting from (35), we use Lemma 2 with Theorem 1. Since u σ = + σ u, and + σ σ = 0, all terms of the form ψ i ( ) u σ vanish, and the result follows by a straightforward symbolic computation. To compute terms of the type ψ i ( )W, good complexity algorithms can be obtained using the theory of minimal polynomials and the knowledge of the eigenspace structure of. We commence by mentioning the following result, that is an immediate consequence of Theorem 5 below.

14 S. Krogstad et al. Corollary 4 Let σ(p) = SPS = P, where Rank( S ) = p. Let = P2 S be nondefect of rank p, with d p distinct non-zero eigenvalues {µ i } d i=1. Then = ad 2 P σ is non-defect with the 1 + d + d2 distinct eigenvalues 0, {µ i } i, { µi µ j }i> j, { µi + } µ j i j. (42) Thus, the minimal polynomial of is the monic (1 + d + d 2 )-degree polynomial with zeros in the points (42). In the rank-1 case, we can obtain explicit formulae. If the scalar θ given by P 2 S = θ S is nonzero, then the minimal polynomial for = ad2 p σ becomes From (39) we obtain: q(x) = x(x θ)(x 4θ). Corollary 5 Let σ be a rank-1 involutive automorphism. Let Z = P + K where P = σ Z, let u = ad p and let θ be the scalar P 2 S = θ S, as in Corollary 2. If θ = 0 then )( ( ) d 1 Z = ( σ + d 1 K + σ I + u (ψ 1 (θ) + ψ 2 (θ)u 2 ) σ + σ ( ψ 3 (θ)u 2 + ψ 4 (θ)u 4) ) σ (43) where ψ 1 (θ) = 8 tanh( θ/2) + tanh( θ) 6 θ ψ 2 (θ) = 2 tanh( θ/2) tanh( θ) 6θ 3 2 ψ 3 (θ) = 15 + θ(tanh( θ) 15 coth( θ)) 12θ ψ 4 (θ) = 3 + θ( tanh( θ)+ 3 coth( θ)) 12θ 2. It should be noted that u 2 and u 4 act on p = Range ( ) σ, yielding also a result in p. Ifg R n n then the cost of computing the terms u 2 and u 4 is O(n). Themain work is the computation of the single u acting on p k, with a cost of O(n 2 ). Now we return to the general rank p case and the computation of ψ( ) via eigenspace projections. Let P = σ Z and = ad2 P S. For our present purpose it is not important whether or not µ i are distinct, so we just assume that is non-defect with p eigenvalues µ i for i = 1,...,p. Let the corresponding left and right eigenvectors

15 be denoted y T i and x i, yi T = µ i yi T x i = µ i x i normalized such that yi T x j = δ i, j. The following statement is verified by straightforward computation. Lemma 3 P has 2p nonzero left and right eigenvectors given as v ±i = 1 2 ( xi ± P x i / µ i ) w T ±i = 1 2 ( y T i ± y T i P/ µ i ) for i = 1,...,p (44) for i = 1,...,p, (45) satisfying Pv ±i =± µ i v ±i w T ±i P =± µ i w T ±i w T j v k = δ j,k for j, k {±1,...,±p}. Lemma 4 If = P 2 S non-defect and of rank p, then P is non-defect. Proof In Lemma 3, we found 2p independent right eigenvectors v ±i. Referring to the partitioned form of P in (20), we have that B is a p (n p) matrix with linearly independent rows. Thus there are n 2p zero right eigenvectors of P of the form ( 0 x T ) T, where x Ker(B). We have found n independent right eigenvectors of P. For i = 1,...,d let the eigenspace projections of P be given as ±i = v ±i w T ±i (46) 0 = I d ( i + i ). (47) i=1 Now we are ready to formulate the main theorem describing the eigenspace structure of. Note that k ker( ) and (p) p, thus we can restrict the discussion to the action of on p. Theorem 5 Let σ be a rank p inner automorphism, and let σ(p) = P. If = P 2 S is non-defect and of rank p, then = ad2 P σ is non-defect with a complete

16 S. Krogstad et al. list of eigenspace projections i, j : p p expressed in terms of an arbitrary W p as 0,0 (W ) = 0 W 0 (48) i,0 (W ) = ( i + i ) W W ( i + i ) for i = 1,...,p (49) i, j (W ) = i W j + i W j for i, j = 1,...,p and i > j (50) i, j (W ) = i W j + i W j for i, j = 1,...,p and i j. (51) The corresponding eigenvalues are given via Proof For i, j we compute 0,0 (W ) = 0 (52) i,0 (W ) = µ i i,0 (W ) (53) i, j (W ) = ( µ i µ j ) 2 i, j (W ) (54) i, j (W ) = ( µ i + µ j ) 2 i, j (W ). (55) ad P ( i W j ) = P i W j i W j P = ( µ i µ j ) i W j. Hence under the action of ad 2 P the two projections W i W j and W i W j correspond to the same eigenvalue ( µi ) 2. µ j These can be combined into the single projection i, j (W ) = i W j + i W j. Now, since P ±i =± µ i ±i and σ(p) = P, we find that σ( ±i ) = i.if W p we have σ(w ) = W and hence σ( i, j (W )) = σ( i W j + i W j ) = i ( W ) j + i ( W ) j = i, j (W ). Thus W p i, j (W ) p. We conclude that σ i, j(w ) = i, j (W ), which yields i, j (W ) = ( µ i µ j ) 2 i, j (W ). The others follow by a similar computation. To check that we have a complete list of eigenspace projections, we use p j= p j = I to decompose W as W = p j j= p W p k k= p each term j W k is appearing exactly once in (48) (51).,

17 The computation of these projections should be done with care in order to keep a favourable complexity. It should be noted that for W p then i, j (W ) = i W j + i W j = 2 σ ( i W j ). If g consists of n n matrices, then when i, j = 0 the computation of i, j (W ) is O(np). The 0-projections 0 are typically high rank matrices, and their computation must be done indirectly. After having computed i, j (W ) and i, j (W ) for non-zero i, j, we find the remaining projections as W = W i> j i, j (W ) i j i, j (W ) k,0 (W ) = ( k + k ) W 0,k (W ) = W ( k + k ) p ( 0,0 (W ) = W k,0 (W ) + 0,k (W ) ). k=1 The total complexity of computing ψ i ( )W becomes O(np 3 ), which is a minor contribution to the total cost as long as p < n. If the matrix P is close to singular or defect, a Schur decomposition can be used instead. 4 GPCs for matrix groups and symmetric spaces For the classical matrix groups we can obtain GPCs with the computational complexity O(n 3 ) by recursively applying the splitting (27). For symmetric spaces our techniques may yield algorithms with optimal complexity, e.g. O(n) for problems on n-spheres. We illustrate this by some examples. In Definition 2 and the following discussion, we assumed the existence of a nested family of algebras g g 1 g k and automorphisms σ i : g i g i. For matrix algebras g gl(n) it may be convenient to define σ i on g rather than on g i, in which case we must make sure that {σ i } define a nested sequence of Lie algebras. Lemma 5 Let g gl(n) be a Lie algebra of n n matrices. Let S i gl(n), i = 1,...,k be involutive matrices such that σ i (Z) = S i ZS i are involutive automorphisms on g and {S i } i=1 k commute S i S j = S j S i for i, j {1,...,k}. (56) Then the projections + σ i = 1 2 (I + σ i) define a nested sequence of Lie algebras g g 1 g k where g i = + σ i 1 + σ i 2 + σ 1 (g) such that σ i g i are automorphisms on g i.

18 S. Krogstad et al. Proof If {S i } i commute, then also { σi }i commute. Hence σ i σ j σi = σ j σi, and we conclude that g i g i 1. The rest of the lemma is obvious. Important examples of such nested sequences are S i = I 2e i e T i, which defines automorphisms for the Lie algebras gl(n) and so(n), and S i = I 2(e i e T i + e i+m e T i+m ), which yields automorphisms on the symplectic algebra sp(2m). 4.1 Low rank GPC for matrix Lie groups In this section we give explicit formulae for the tangent maps of some low rank generalized polar coordinate maps, and comment on their arithmetical complexity. For details on the implementation of the coordinate map we refer to [37]. The simplest approach is to use rank-1 splittings, in particular the one resulting from the inner automorphisms σ i (M) = S i MS i with S i = I 2e i ei T. We thus obtain a splitting of Z g: Z = P 1 + P P n 1 + K n 1, where for K 0 := Z and for i = 1,...,n 1wehave P i = σ i (K i 1 ) and K i = + σ i (K i 1 ). We thus use the map (Z) = exp(p 1 ) exp(p 2 ) exp(p n 1 ) exp(k n 1 ). (57) Note, however, that this approach does not work for every Lie group, i.e. one needs to make sure that the summands P i and K i both lie in the Lie algebra, such that the coordinate map (Z) resides in the Lie group. As we shall see in the sequel, the symplectic group requires a rank two splitting in order for the coordinate map to reside in the Lie group. For simplicity we consider the first step in the recursive splitting above. Let Z, Ẑ g, and denote their splittings Z = P + K and Ẑ = P + K, and further ( ) 0 b T P = a 0 ( ) 0 d T P =. (58) c 0

19 Then by Theorem 3, for the coordinate map (Z) = exp(p) (K ), its inverse tangent applied to Ẑ, splits as follows: p part : (I + ψ 2 (ad 2 P )) P ad P K (59) k part : d 1 K ( K + ad P (ψ 1 (ad 2 P ) P)), (60) where ψ 1 and ψ 2 are given in (40) (41). Using the eigenspace projections of Theorem 5 for = ad 2 P σ, we obtain the following expression for ψ i(ad 2 P ) P: where ( ) ψ i (ad 2 0 y P ) T P = x 0 (61) x = ψ i (θ)c + ( 2φ i (4θ)(b T c d T a) φ i (θ)b T c)a (62) y = ψ i (θ)d + (2φ i (4θ)(b T c d T a) φ i (θ)d T a)b. (63) Here θ = b T a and φ i, i = 1, 2 are the functions φ i (x) = (ψ i (x) ψ i (0))/x. (φ i (0) = ψ i (0) is defined since ψ i is analytic.) Letting [x, y] =fad2(ψ, a, b, c, d) denote the function giving the vectors x and y above, we get the following algorithm for the tangent map (in pseudo-code). Algorithm 2 % In: n n matrices Z and Ẑ % Out: Ẑ overwritten as d 1 Z Ẑ for k = 1 : n 1 a := Z(k + 1 : n, k); b := Z(k, k + 1 : n) T ; c := Ẑ(k + 1 : n, k); d := Ẑ(k, k + 1 : n) T ; [x 1, y 1 ]:=fad2(ψ 1, a, b, c, d); [x 2, y 2 ]:=fad2(ψ 2, a, b, c, d); Ẑ(k + 1 : n, k) := c + x 2 aẑ(k, k) + Ẑ(k + 1 : n, k + 1 : n)a; Ẑ(k, k + 1 : n) := d T + y2 T bt Ẑ(k + 1 : n, k + 1 : n) + Ẑ(k, k)b T ; Ẑ(k, k) := Ẑ(k, k) + b T x 1 y1 T a; Ẑ(k + 1 : n, k + 1 : n) := Ẑ(k + 1 : n, k + 1 : n) + ay1 T x 1b T ; end The general linear group and the special linear group Recall that the general linear group GL(n) consists of the n n matrices with nonzero determinant, and that the special linear group SL(n) consists of the matrices with determinant equal to one. Their corresponding Lie algebras gl(n) and sl(n) consist of the n n matrices and the n n matrices with zero trace respectively. For both Lie algebras it is easy to see that for a rank one splitting Z = P + K with S as above, both P and K lie in the Lie algebra, and thus also the coordinate map resides in the Lie group. By studying Algorithm 2 one sees that at stage k, we perform two matrix vector products, two outer products of vectors and two matrix additions which amounts to

20 S. Krogstad et al. a complexity of 8(n k) 2. Moreover the operation count of order n k can with a small effort at least be reduced to 19. Summing from k = 1ton, the overall cost of computing the tangent map is 8 3 n n2 + O(n) The orthogonal group For the orthogonal group O(n), and its corresponding Lie algebra so(n), the formulas above simplifies considerably. Let P and P be the matrices (58), but with b and d replaced with a and c respectively. Then ψ i (ad 2 P ) P have the form: ( ) ψ i (ad 2 0 x P ) T P = x 0 (64) with x = ψ i (θ)c + φ i (θ)(a T c)a. (65) Also the overall algorithm simplifies. By exploiting the skew symmetry of the output matrix, one sees that at stage k one needs one matrix vector product, one vector outer product, and two half matrix additions, i.e. 4(n k) 2 operations. The (n k) coefficient can be reduced to 8, and summing up we achieve the computational cost 4 3 n 3 + 2n 2 + O(n) Upper triangular group We also include the upper triangular group in this discussion, although the formulas turn out rather trivial. By setting the vectors a and c in the matrices (58) equal to zero, it is clear that ad P P = 0, and thus ψ i (ad P ) P = P. Atstepk in the algorithm there is only one triangular matrix vector product which may be carried out in (n k) 2 operations. The rest of the computation requires about 3(n k) operations, and overall we end up with cost 1 3 n n2 + O(n) The symplectic group Recall that the symplectic group SP(2m) are the matrices X satisfying the property XJX T = J, J = ( ) 0 I. I 0 Its Lie algebra sp(2m) is characterized by the matrices Z which obeys ZJ+ JZ T = 0. Note that a rank-1 splitting of the matrix Z will not fulfill this property, i.e. for Z = P + K we will in general have PJ JP = 0. Instead consider the involutive S = I 2e 1 e1 T 2e m+1em+1 T. It is easily seen that the splitting Z = P + K according to S has the desired property, and that by applying suitable permutation matrices

21 (exchanging row/column 2 and m + 1), the matrix P has the form ( ) 0 B T P =, A 0 where A and B are (2m 2) 2 matrices. Note that, because of the symmetries in P, the matrix B is completely determined by A. Furthermore one sees that B T A is just a scalar times the 2 2 identity matrix, i.e. B T A = θ I, and thus ψ(ad P ) P is given where ( ) ψ(ad 2 0 Y P ) T P =, (66) X 0 X = ψ i (θ)c + A( 2φ i (4θ)(B T C D T A) φ i (θ)b T C) (67) Y = ψ i (θ)d + B(2φ i (4θ)(B T C D T A) φ i (θ)d T A). (68) Omitting the details we can also in this case exploit the symmetry of the matrices involved, and obtain the same computational cost for the tangent map as for the orthogonal group, i.e. 4 3 n3 + O(n 2 ) for n = 2m. Summarizing we obtain the following table for the leading term of the computational complexity for the maps and d Z 1 applied to the most common real matrix Lie algebras. sl(n), gl(n) o(n) tu(n) sp(n) Tangent map Coordinate map Vector Matrix 8 3 n3 3n 2 2n n 3 3n 2 2n n3 1 2 n3 4 3 n 3 2n 3 5 Numerical experiments 5.1 Finding a solution of the Schur Horn majorization problem As an application, we consider an inverse eigenvalue problem with prescribed entries along the main diagonal [8] which arises in conjunction with the Schur Horn theorem in linear algebra. Before presenting the theorem, we recall that a vector a R n is said to majorize λ R n if, assuming the ordering a j1 a j2 a jn, λ m1 λ m2 λ mn,

22 S. Krogstad et al. the following relationship holds: k λ mi i=1 and with equality for k = n [19]. k a ji, for k = 1, 2,...,n, (69) i=1 Theorem 6 (Schur Horn [19]) An Hermitian matrix H with eigenvalues λ and diagonal elements a exists if and only if a majorizes λ. Moreover, if a majorizes λ the matrix H can be chosen to be symmetric. Given λ and a majorizing λ, we wish to find such a symmetric matrix H. One possibility is to construct an isospectral flow (a matrix flow which leaves the eigenvalues of the initial matrix unchanged) which converges to a target symmetric matrix H, similar to diag(λ) and with a as diagonal elements. Setting = diag(λ), onthe isospectral manifold M λ ={X : X = U U T, UU T = I } of symmetric matrices with eigenvalues λ the problem can be reformulated as an optimization problem: find amatrixh that minimizes the quadratic function φ(x) = (x 1,1 a j1 ) 2 + +(x n,n a jn ) 2. (70) Following [36], we consider φ as a function of U and notice that φ(u) = tr[(x A)diag(X A)], where, for convenience, we have set A = diag(a). By direct computation, φ U = 2[X, diag(x A)]U, (here the square bracket denotes the usual matrix commutator, which leads to the double-bracket isospectral flow X = 2[[X, diag(x A)], X]. (71) It is not a good idea to solve (71) numerically directly, since it is well known that standard ODE methods cannot preserve isospectrality [2]. Instead, assuming that X n M λ is known, in each interval [t n, t n+1 ] one solves for the matrix U, obeying the Lie-group differential equation U = 2[X, diag(x A)]U, U(t n ) = I, (72) in tandem with the transformation X = U(t)X n U(t) T. As long as the solution of (72) is orthogonal (or skew-hermitian, in the complex setting), the numerical approximation X n has eigenvalues λ and it converges to a solution H minimizing (70). In the numerical experiments we chose λ and a from random symmetric matrices with distribution N (0, 1) obtained from the Matlab function randn. The

23 diag(x n ) a Frequencies Iterations Iterations Fig. 1 Left: Convergence of the isospectral matrix X n = U n X n 1 Un T to a prescribed diagonal matrix for five randomly chosen initial data λ, a R 25. The orthogonal matrix U n is computed solving the ODE (72) with the GPC method. Right: Histogram of the iterations for convergence for the same problem, for a sample of 1,000 initial data, (λ, a) i= initial value X 0,issettoX 0 = Q T Q where Q is a randomly chosen orthogonal matrix. Since we are interested in convergence to a fixed point of (72), the local error is not of concern, and thus a first order method works as well as a higher order method. We have used the Lie-Euler method with constant stepsize h = We use the GPC coordinate map (57) and also the matrix exponential for comparisons. Running the two methods on a set {λ, a} i, i = 1, 2,...,1,000, we found that the difference in rate of convergence for using the GPC map and the exponential map was negligible. In Fig. 1 on the right the histogram plot for the 1,000 trials are presented. Among the trials there were two cases outside the region of the plot, with a maximum number of 550 iterations until convergence. In this particular case with matrices the evaluation of the exponential map applied to a matrix requires more than six times the number of operations required for the GPC map, and thus considerable savings are obtained. 5.2 Computing all Lyapunov exponents for a ring of oscillators As a numerical example where the tangent map is needed, we consider computing all Lyapunov exponents of the following system ( ) ÿ = α y 2 1 ẏ ω 2 y ẍ 1 = d 1 ẋ 1 β [ V (x 1 x n ) V (x 2 x 1 ) ] + σ y ẍ i = d i ẋ i β [ V (x i x i 1 ) V (x i+1 x i ) ], i = 2,...,n. It describes a ring of n damped oscillators with amplitudes x i with periodic boundary conditions x n+1 = x 1. The ring is forced externally by y(t), the periodic space coordinate of the limit cycle of a van der Pol oscillator. The parameters α, β, ω, σ and d i are chosen as in [1] to obtain several positive exponents, i.e α = 1, β = 1, ω = 1.82 and σ = 4. The damping parameters are set to d i = for i odd, and d i = for i even. The potential function V is given V (x) = x 2 /2 + x 4 /4. The experiment is done with n = 5. The Lyapunov exponents give the rates of exponential divergence

24 S. Krogstad et al. or convergence of initial nearby orbits, and can be found by considering the system ẋ = f (x) linearized about a trajectory x(t): Ẏ = A(t)Y, Y (0) = Y 0, (73) where A(t) = df(x(t)). The Lyapunov exponents are now given as the logarithms of the eigenvalues of the Oseledec-matrix x = lim t (Y (t)t Y (t)) 1 2t. (74) A technique using continuous QR-factorization [11,12] attempts to calculate the orthonormal factor Q(t) in the QR-decomposition of Y (t). It can be shown that Q(t) obeys the (Lie group) differential equation Q = QH(t, Q), H(t, Q) = tril(q T AQ) tril(q T AQ) T. (75) Here tril(m) denotes the function setting the upper triangular part of M to zero. Since the columns of Q(t) are drawn towards the direction of the largest Lyapunov exponents it is crucial that the numerical solution stays orthonormal, and using standard methods the numerical solution will typically blow up after some time. Thus it is a good idea to use methods which conserves the orthogonality automatically. Given Q(t), it can be shown that the Lyapunov exponents can be obtained from the diagonal elements of the limit matrix 1 t lim Q(τ) T A(τ)Q(τ)dτ. (76) t t 0 In numerical computations it is necessary to truncate the above expression at some finite time T, to obtain approximations to the exponents. We have used the trapezoidal rule to approximate the integral (76). Since the tangent vectors in (75) are represented by an element in the Lie algebra multiplied from the left rather than the right, we use a left version of the coordinate map and the corresponding left trivialized tangent map. Letting be the map (57), we define for Z so(n) (Z) = ( Z) 1 = exp K n 1 exp(p n 1 ) exp(p 1 ). (77) The left trivialized tangent d Z of the map is then given by the relation (Z)d Z (δz) = s (Z + sδz) = s=0 s ( Z sδz) 1 (78) s=0 = ( Z) 1 d Z ( δz) ( Z) ( Z) 1 (79) = (Z)d Z (δz),

25 Time Fig. 2 Lyapunov exponents for a ring of five damped oscillators where in step (78) (79) wehaveused d dt (M 1 ) = M 1 ( d dt M)M 1. Thus the left trivialized tangent of the map is simply given as d Z = d Z. Also by noting that ad 2 P = ad2 P, we see that d Z 1 is given by the formula in Theorem 4 with just a single sign change in front of u. This is similar to the left trivialized tangent of the exponential map given as d exp Z. We use the classical fourth order Runge Kutta method both for the computation of the trajectory and as the underlying method for the Lie group integrator for solving (75) with step sizes h = and h = 0.01 respectively. A randomly chosen x 0 is used as initial value, and the system is integrated from t = 0tot = 4,000. For our choice of damping parameters d i, the sum of the exponents should add up to 2n k=1 λ k = div f (x) = n d j. j=1 In the numerical computations the error of the sum is of order 10 8.Moreoveritis shown in [13] that for constant d, the exponents are distributed symmetrically around d. In our case one can also show that the exponents come in pairs (λ i, λ i c i ) i = 1,...,n, where each c i satisfies d min c i d max. This property is clearly seen in Fig. 2. We also performed the experiment using the matrix exponential and its left trivialized tangent giving similar qualitative results. However the cost of the overall algorithm increased dramatically. Counting floating point operations, the overall cost when using the GPC-map was flops while for the exponential map flops. For comparison we also implemented the Cayley map (see [20]), and obtained flops.

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