On the Iwasawa decomposition of a symplectic matrix

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Applied Mathematics Letters 20 (2007) 260 265 www.elsevier.com/locate/aml On the Iwasawa decomposition of a symplectic matrix Michele Benzi, Nader Razouk Department of Mathematics and Computer Science, Emory University, Atlanta, GA 30322, USA Received 28 February 2006; received in revised form 11 April 2006; accepted 20 April 2006 Abstract We consider the computation of the Iwasawa decomposition of a symplectic matrix via the QR factorization. The algorithms presented improve on the method recently described by T.-Y. Tam in [Computing Iwasawa decomposition of a symplectic matrix by Cholesky factorization, Appl. Math. Lett. (in press) doi:10.1016/j.aml.2006.03.001]. c 2006 Elsevier Ltd. All rights reserved. Keywords: Iwasawa decomposition; Symplectic group; QR factorization; Numerical stability; Cholesky factorization 1. Introduction AmatrixS R 2n 2n is called symplectic if it satisfies S t JS = J,where 0 In J =. I n 0 In this note we are concerned primarily with the real case; the complex case can be treated along similar lines. Under matrix multiplication, the symplectic matrices form a (non-compact) Lie group denoted by S = Sp(n, R) ={S SL 2n (R) : S t JS = J},whereSL 2n (R) denotes the group of 2n 2n matrices with unit determinant. The symplectic group is closed under transposition. Consider the following subgroups of S: { } K11 K K = K = : K K K 11 + ik U(n) = O(2n) Sp(n, R), 11 { } A11 0 A = 0 A 1 : A 11 positive diagonal, 11 { } N11 N N = 0 N11 t : N 11 unit upper triangular, N 11 N t = N N11 t. The first of these three subgroups is compact, the second is abelian, and the third is nilpotent. The decomposition S = KAN is called the Iwasawa decomposition of S. AnyS Sp(n, R) can be written as S = KAN,where Corresponding author. E-mail addresses: benzi@mathcs.emory.edu (M. Benzi), nrazouk@emory.edu (N. Razouk). 0893-9659/$ - see front matter c 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.aml.2006.04.004

M. Benzi, N. Razouk / Applied Mathematics Letters 20 (2007) 260 265 261 K K, A A, andn N ; moreover, this decomposition is unique. It is a special case of the general Iwasawa decomposition of a connected semisimple Lie group first given in [7]. For a more detailed discussion of the Iwasawa decomposition, see [9] or[5]. The importance of this decomposition is both theoretical and practical, in particular in the area of dynamical systems. Note that the factorization S = KAN (more precisely, the factorization S = KM with M = AN) differs from the QR factorization [10], since N is not upper triangular. (Note that the factors in the usual QR factorization of S are not symplectic, in general.) It is also not to be confused with the SR factorization (see, e.g., [3, p. 20]). Thedecomposition S = KM with M = AN is called aunitary SR decomposition in [2, pp. 68 69]. In the recent note [8], Tam presents a method for explicitly computing the Iwasawa decomposition S = K AN of a symplectic matrix using the Cholesky factorization of S t S. The approach in [8], however, does not take numerical stability considerations into account and may lead to inaccurate results in finite precision computations. It is also not very efficient in terms of operation count. Here we give a more accurate and efficient algorithm for computing the Iwasawa decomposition of a symplectic matrix. 2. Computing the Iwasawa decomposition The approach in [8] is based on the following result. ( S11 S A1 B 1 B1 t D 1 Theorem 2.1. Let S = S 21 ) S 22 S and S t S = (also in S). Let A 1 = U t HU be the root-free Cholesky factorization of the symmetric positive definite matrix A 1, where U is unit upper triangular and H is positive diagonal. Then S = KAN,where H 1 ( 2 0 U H A = 0 H 2 1, N = U t ) B 1 0 U t, and K = S(AN) 1, is the Iwasawa decomposition of S. On the basis of Theorem 2.1, the author of [8] proposes a Cholesky-based approach for explicitly determining the Iwasawa factors K, A, N of a given symplectic matrix. This approach has certain drawbacks. The main problem is that the approach in [8] may lead to inaccurate results in finite precision arithmetic. It is well known that forming the product S t S explicitly may lead to significant loss of information in finite precision computations; see [6, p. 386]. If S is ill-conditioned (which can happen, since the group S is not compact), forming S t S mayevenresultinlossof positive definiteness, with the consequent breakdown of the Cholesky factorization. It is also important to avoid the use of explicit matrix inverses when computing the factorization. Although efficiency is not the primary concern of this note, it is also worth noting that the approach in [8] has a rather high computational complexity. It is possible to extract from [2, pp. 64 69] a method for computing( the Iwasawa decomposition of a symplectic matrix, which proceeds as follows. Given a real symplectic matrix S = S11 S S 21 S 22 ), the following algorithm computes the factors K, A, N of the Iwasawa decomposition of S. Algorithm 2.2. 1. Compute the QR factorization of S 11 + is ; denote by U the unitary factor of S 11 + is. 2. Compute the Iwasawa factors K, A and N of S as follows: K 11 = 1 (U + Ū), 2 K = i (Ū U), 2 K11 K K =, K K 11 ˆN = K t S A = diag(ˆn 11,...,ˆn 2n,2n ), where ˆn ii are the diagonal entries of ˆN, N = A 1 ˆN. Note that Algorithm 2.2 necessitates complex arithmetic even if the symplectic matrix S and its factors are real, an undesirable feature. Motivated by this, we examine here another algorithm for computing the Iwasawa decomposition of a symplectic matrix. This approach is based on the thin QR factorization [4, p. 230] and does not require complex

262 M. Benzi, N. Razouk / Applied Mathematics Letters 20 (2007) 260 265 arithmetic. Let S be partitioned into four blocks as in Theorem 2.1. The following algorithm computes the factors K, A, N in the Iwasawa decomposition of S. ( Algorithm 2.3. 1. Let S 1 = S11 S 21 ). 2. Compute the thin QR factorization of S 1,whereQ = Q11 Q 21 and R = R 11. 3. Factor the upper triangular matrix R 11 as R 11 = HU with H diagonal and U unit upper triangular. Then R11 t R 11 = U t DU, whered = H 2. 4. Compute the Iwasawa factors A, K and N of S as follows: D 1 2 0 A = 0 D 1 A, 2 K 11 = Q 11 HD 1 2, K = Q 21 HD 2 1, K11 K K = K, K K 11 U N N N = N, where = A 0 N 22 N 1 K t S. 22 S 22 To see that Algorithm 2.3 computes the Iwasawa factors, first note that A A. Furthermore, we need to show that K 11 + ik is unitary. To this end, it suffices to verify the following two equalities: K t 11 K = K t K 11, K t 11 K 11 + K t K = I n. For the first equality it is enough to observe that Q t 11 Q 21 = Q t 21 Q 11.SinceS Sp(n, R) and hence S t 11 S 21 = S t 21 S 11, the first equation follows. The second equality can be rewritten as D 1 2 U t (S t 11 S 11 + S t 21 S 21)U 1 D 1 2 = In. Since S11 t S 11 + S21 t S 21 = U t DU we conclude that K 11 + ik U(n) and therefore K K. It can be easily seen using the symplecticity of S that N 22 = U t. Using the fact that A 1 1 B 1 = B1 t A t 1 where A 1 and B 1 are as in Theorem 2.1 one can easily show that UN t = N U t, and hence N N. Finally note that KAN = S. A few remarks are in order. Since D = H 2, the matrix HD 2 1 appearing in step 4 is just a signature matrix, i.e., a diagonal matrix with entries equal to ±1. The above algorithm requires no explicit matrix inverses except for that of a diagonal matrix. The cost of the algorithm is dominated by the computation of the QR factorization of S 1 and by the matrix products in the computation of N and N 22. We point out that the overall cost of Algorithm 2.3 is 40 3 n 3 + O(n 2 ) floating point operations. This is somewhat less than the cost of the method proposed in [8], which can be shown to be 48 3 n3 + O(n 2 ) operations. We also note that the use of complex arithmetic in Algorithm 2.2 makes this approach significantly more expensive than Algorithm 2.3 in the real case. It might be possible that even more efficient algorithms could be developed, for instance making use of the symplectic QR decomposition described in [1]. Here we restrict ourselves to algorithms that can be easily implemented in MATLAB using only built-in functions. 3. Numerical experiments We constructed a number of symplectic matrices of different dimensions by first constructing the symplectic Iwasawa factors K, A and N and then forming the product S = KAN. Specifically, we constructed the blocks for the factors as follows. First we generated a random positive diagonal matrix A 11 to form A A. ForN, we constructed a random n n upper triangular matrix N 11 with unit diagonal and set N = N 11. Finally, to form K we generated two random n n matrices X and Y and let C = X + iy. We then computed the QR factorization of C and let K 11 be the real part of Q and K be the imaginary part of Q. WetestedAlgorithms 2.2 and 2.3 on a large set of these matrices and observed a noticeable difference in the accuracy of the computed factors compared to the approach suggested in [8]. Algorithms 2.2 and 2.3 are more accurate, especially for matrices with relatively high condition numbers. In particular, the factor K may be far from being orthogonal when the method suggested in [8] is used. Also, with that method the computed N may not satisfy the symplecticity conditions to high relative accuracy.

M. Benzi, N. Razouk / Applied Mathematics Letters 20 (2007) 260 265 263 Table 1 Results for the three approaches Tam s algorithm Algorithm 2.2 Algorithm 2.3 10 10 matrix with condition number 3 10 1 K t K I 2 1 10 15 6 10 16 7 10 16 K 11 K 22 2 9 10 16 0 0 K + K 21 2 8 10 15 0 0 K K 2 9 10 16 4 10 16 4 10 16 UN t N U t 2 3 10 15 1 10 15 2 10 15 UN 22 t I 2 U 2 0 2 10 16 5 10 16 N N 2 N 2 8 10 16 5 10 16 1 10 15 Ā A 2 A 3 10 16 2 10 16 2 10 16 S K Ā N 2 S 2 3 10 16 4 10 16 5 10 16 100 100 matrix with condition number 7 10 4 K t K I 2 1 10 10 2 10 15 8 10 14 K 11 K 22 2 6 10 11 0 0 K + K 21 2 6 10 11 0 0 K K 2 7 10 11 7 10 14 8 10 14 UN t N U t 2 3 10 09 2 10 2 10 11 UN 22 t I 2 U 2 2 10 15 7 10 15 3 10 14 N N 2 N 2 7 10 11 6 10 3 10 Ā A 2 A 2 10 11 2 10 15 5 10 15 S K Ā N 2 S 2 6 10 15 1 10 15 7 10 14 4 4matrixS in (3.1) with t = 8; condition number 10 7 K t K I 2 3 10 03 2 10 16 2 10 16 K 11 K 22 2 3 10 03 0 0 K + K 21 2 1 10 03 0 0 UN t N U t 2 6 10 08 2 10 10 5 10 10 UN 22 t I 2 U 2 0 4 10 10 1 10 10 S K Ā N 2 S 2 2 10 10 3 10 16 3 10 16 Table 1 shows some sample computational results comparing the three algorithms. The first two examples use random matrices constructed as described above. For those instances we compare the computed factors K, Ā and N to the factors K, A and N used to construct the symplectic matrix S. In addition, we compute errors to measure the departure of the computed factors from satisfying the simplecticity conditions. For the last example we use the following symplectic matrix: cosht sinh t 0 sinht S = sinh t cosht sinh t 0 0 0 cosht sinht, t R. (3.1) 0 0 sinht cosh t For the first example, which is well conditioned, all three approaches yield good approximations to the Iwasawa factors (by any measure). When the symplectic matrix S to be factored is larger and/or has a higher condition number, as in the second example, we begin to notice some loss of (forward) accuracy in some of the factors computed using Tam s method. As may be expected, the effect is also present with the other two methods, but is less pronounced.

264 M. Benzi, N. Razouk / Applied Mathematics Letters 20 (2007) 260 265 Finally, the third example shows that accuracy can be seriously compromised when Tam s method is used. Similar trends were noticed in all our numerical experiments. 4. Implementation For completeness, we include the MATLAB code we used to test Algorithm 2.3. function [K,A,N] = ifactor(s); This function computes the Iwasawa decomposition of a real symplectic matrix of order 2n. Input: a real symplectic matrix [S_11 S_; S_21 S_22] Output: K = 2n-by-2n orthogonal symplectic matrix A = 2n-by-2n positive diagonal symplectic matrix N = 2n-by-2n "triangular" symplectic matrix s.t. S = K*A*N n_2 n = size(s); = n_2/2; Compute thin QR factorization of S1 = [S_11; S_21]. S_11 S_21 = S(1:n,1:n); = S(n+1:n_2,1:n); S1 = [S_11; S_21]; [Q,R] = qr(s1,0); Q_11 Q_21 = Q(1:n,1:n); = Q(n+1:n_2,1:n); Compute U and D from given R where U is unit upper triangular and H is a diagonal matrix such that R = H*U and R *R = U *H^2*U. H U = diag(diag(r)); = H\R; Compute blocks for the factors K, A, N. h SQRT_D SQRT_D_inv h h = sign(diag(h)); = diag(h.*diag(h)); = diag(1./diag(sqrt_d)); = h ; = h(ones(1,n),:);

M. Benzi, N. Razouk / Applied Mathematics Letters 20 (2007) 260 265 265 K_11 K_ = Q_11.*h; = -Q_21.*h; Form the Iwasawa factors K, A, N. A K S1 N1 N = [SQRT_D zeros(n) ; zeros(n) SQRT_D_inv]; = [K_11 K_; -K_ K_11]; = S(1:n_2,n+1:n_2); = A\(K *S1); = [U N1(1:n,1:n); zeros(n) N1(n+1:n_2,1:n)]; Acknowledgements Thanks to Daniel Kressner and Valeria Simoncini for their comments on an earlier version of this note. The work of Michele Benzi was supported in part by the National Science Foundation grant DMS-0511336. References [1] P. Benner, D. Kressner, Algorithm 854: Fortran 77 subroutines for computing the eigenvalues of Hamiltonian matrices II, ACM Trans. Math. Software 32 (2006) 1 22. [2] A. Bunse-Gerstner, Matrix factorizations for symplectic QR-like methods, Linear Algebra Appl. 83 (1986) 49 77. [3] H. Fassbender, Symplectic Methods for the Symplectic Eigenproblem, Kluwer Academic, Plenum Publishers, New York, 2000. [4] G.H. Golub, C. van Loan, Matrix Computations, third ed., John Hopkins University Press, Baltimore, London, 1996. [5] S. Helgason, Differential Geometry, Lie Groups, and Symmetric Spaces, Academic Press, New York, 1978. [6] N.J. Higham, Accuracy and Stability of Numerical Algorithms, second ed., Society for Industrial and Applied Mathematics, Philadelphia, PA, 2002. [7] K. Iwasawa, On some types of topological groups, Ann. of Math. 50 (1949) 507 558. [8] T.-Y. Tam, Computing Iwasawa decomposition of a symplectic matrix by Cholesky factorization, Appl. Math. Lett. (in press). doi:10.1016/j.aml.2006.03.001. [9] A. Terras, Harmonic Analysis on Symmetric Spaces and Applications II, Springer-Verlag, Berlin, 1988. [10] L.N. Trefethen, D. Bau, III, Numerical Linear Algebra, Society for Industrial and Applied Mathematics, Philadelphia, PA, 1997.

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