A SORT-JACOBI ALGORITHM ON SEMISIMPLE LIE ALGEBRAS∗ M. KLEINSTEUBER† Abstract. A structure preserving Sort-Jacobi algorithm for computing eigenvalues or singular values is presented. The proposed method applies to an arbitrary semisimple Lie algebra on its ( 1)-eigenspace of the Cartan involution. Local quadratic convergence for arbitrary cyclic schemes is− shown for the regular case. The proposed method is independent of the representation of the underlying Lie algebra and generalizes well-known normal form problems such as the symmetric, Hermitian, skew-symmetric, symmetric and skew-symmetric R-Hamiltonian eigenvalue problem and the singular value decomposition. Due to the well-known classification of semisimple Lie algebras, the approach yields structure-preserving Jacobi eigenvalue methods for so far unstudied problems as Takagi’s and a Takagi-like factorization, the Hermitian quaternion, the Hermitian C-Hamiltonian eigenvalue decomposition and a symplectic singular value decomposition. The algorithm is exempli- fied for an exceptional Lie algebra. Key words. Jacobi algorithm, Cartan decomposition, semisimple Lie algebra, restricted-root space, structured eigenvalue decomposition, structured singular value decomposition, Takagi’s fac- torization, local quadratic convergence AMS subject classifications. 16D70, 45C05, 65F15, 15A18, 17B20, 74P20, 41A25 1. Introduction. Structured eigenvalue and singular value problems frequently arise in areas such as control theory and signal processing, where e.g. optimal control and signal estimation tasks require the fast computation of eigenvalues for Hamil- tonian and symmetric matrices, respectively. We refer to [4] for an extensive list of structured eigenvalue problems and relevant literature. Such problems cannot be solved easily by using standard software packages, as these eigenvalue algorithms do not necessarily preserve the underlying matrix structures and therefore may suffer by accumulation of rounding errors or numerical instabilities. In contrast, structure preserving eigenvalue methods have the potential of combining memory savings with high accuracy requirements and therefore are the methods of choice. Most of the previous approaches to structured problems have been on a case-to-case basis, by try- ing to adapt either known solution methods or software packages to each individual problem. However, such an ad–hoc approach is bound to fail if the problem of interest cannot be easily reformulated as a standard problem from numerical linear algebra. Thus a more flexible kind of algorithm is needed, that enables one to tackle whole classes of interesting eigenvalue problems. Clearly, one cannot expect to find one method that works for all structured problems. Nevertheless, it seems reasonable to develop methods that allow a unified approach to sufficiently rich classes of interesting eigenvalue problems. The structure preserving Jacobi-type methods proposed here are of this kind. Their inherent parallelizability, cf. [2], and high computational accuracy, cf. [11, 39] make them also useful for large scale matrix computations. They have moreover the potential to compete in convergence speed with state-of-the-art eigenvalue tools. Re- cently, a modified version of the one-sided Jacobi method outperforms the established QR-algorithm for computing the singular value decomposition of a dense matrix, cf. [9, 10]. ∗This paper contains results from the author’s PhD thesis, [34]. †National ICT Australia, Canberra Research Laboratory, SEACS Program, Locked Bag 8001, Canberra ACT 2601, Australia and Department of Information Engineering, RSISE, The Australian National University, Canberra, ACT 0200, Australia ([email protected]). 1 Variants of the Jacobi algorithm have been applied to many structured eigen- value problems (EVP), including e.g. the real skew-symmetric eigenvalue decompo- sition (EVD), [20, 31, 40], computations of the singular value decomposition (SVD) [36], non-symmetric EVPs [6, 14, 43, 46], complex symmetric EVPs [15], and the computation of eigenvalues of normal matrices [18]. For extensions to different types of generalized EVPs, we refer to [7, 47]. For applications of Jacobi–type methods to problems in systems theory, see [25, 26, 27]. In contrast to earlier work, we extend the classical concept of a Jacobi-algorithm towards a new, unified Lie algebraic approach to structured EVDs, where structure of a matrix is defined by being an element of a Lie algebra (or of a suitably defined sub-structure). To the best of the author’s knowledge, Wildberger [48] has been the first who pro- posed a generalization of the non-cyclic classical Jacobi algorithm on arbitrary com- pact Lie algebras and succeeded in proving global convergence of the algorithm. The well–known classification of compact Lie algebras shows that this approach essentially includes structure preserving Jacobi-type methods for the real skew-symmetric, the complex skew-Hermitian, the real skew-symmetric Hamiltonian, the complex skew- Hermitian Hamiltonian eigenvalue problem, and some exceptional cases. Wildberger’s work has been subsequently extended by Kleinsteuber et al. [33], where local quadratic convergence for general cyclic Jacobi schemes is shown to hold for arbitrary regular elements in a compact Lie algebra. In this paper we propose and analyze the Sort-Jacobi algorithm for a large class of structured matrices that, besides the compact Lie algebra cases, essentially includes the normal form problems quoted in Table 5. These cases arise from the well-known classification of simple Lie algebras. Cyclic Jacobi-type methods for some of these cases have been discussed earlier, e.g. for the symmetric/Hermitian EVD see [17, 31], for the skew-symmetric EVD see [20, 31, 40], for the real and complex SVD see [31, 36], for the real symmetric and skew-symmetric EVD see [16], for the Hermitian R-Hamiltonian EVD see [6], and for the perplectic EVD see [38]. Thus we achieve a simultaneous generalization of all this prior work. Note that the methods proposed in this paper exclusively use one-parameter transformations and therefore slightly differ from the algorithms in [6, 16, 20, 31, 40], where block Jacobi methods are used, i.e. multiparameter transformations that annihilate more than one pair of off-diagonal elements at the same time. Explicitly, our setting is the following. Let G be a semisimple Lie group and g = k p be the Cartan decomposition of its Lie algebra. We propose a Jacobi–type method⊕ that ”diagonalizes” an element S p by conjugation with some k K, where K G is the Lie subgroup corresponding∈ to k. Of course, if A is a Hermitian∈ matrix, then⊂ iA is skew-Hermitian and therefore the Hermitian and the skew-Hermitian EVD are equivalent. A similar trick extends to arbitrary compact Lie algebras. In this sense it becomes clear that the present work immediately includes the corresponding results for compact Lie algebras in [33]. In fact, if k is a compact semisimple Lie algebra, then g0 := k ik is the Cartan decomposition of g0. Thus the analysis of Jacobi-type methods on⊕ compact Lie algebras appears as a special case of our result. A characteristic feature of all known Jacobi-type methods is that they act to minimize the distance to diagonality while preserving the eigenvalues. Thus different measures to diagonality can be used to design different types of Jacobi algorithms. Conventional Jacobi algorithms, like classical cyclic Jacobi algorithms for symmet- ric matrices, Kogbetliantz’s method for the SVD, cf. [36], methods for the skew- symmetric EVD, cf. [20], and for the Hamiltonian EVD, cf. [37], as well as recent 2 methods for the perplectic EVD, cf. [38], are all based on reducing the sum of squares of off-diagonal entries (the so-called off-norm). Although local quadratic convergence to the diagonalization has been shown in some of these cases at least for generic situ- ations, the analysis of such conventional Jacobi methods becomes considerably more complicate for clustered eigenvalues. This difficulty is unavoidable for conventional Jacobi methods and is due to the fact, that the off-norm function that is to be min- imized has a complicated critical point structure and several global minima. Thus this difficulty might be remedied by a better choice of cost function that measures the distance to diagonality. In fact, Brockett’s trace function turns out to be a more ap- propriate distance measure than the off-norm function. In [3], R.W. Brockett showed that the gradient flow of the trace function can be used to diagonalize a symmetric matrix and simultaneously sort the eigenvalues. This trace function has also been considered by e.g. M.T. Chu, [8] associated with gradient methods for matrix factor- izations, and subsequently by many others. For a systematic critical point analysis of the trace function in a Lie group setting, we refer to [13]. See also [44] for more recent results on this topic in the framework of reductive Lie groups. In his PhD thesis [31], K. H¨uper has been the first who realized, that Brockett’s trace function can be effectively used to design a new kind of Jacobi algorithm for symmetric matrix diagonalization, that automatically sorts the eigenvalues in any prescribed order. This Sort-Jacobi algorithm uses Givens rotations that do not only annihilate the off-diagonal element, but also sort the two corresponding elements on the diagonal. The idea of combining sorting with eigenvalue computations can be carried over to the SVD as well, cf. [31]. It is this sorting property that distinguishes the Sort-Jacobi algorithm from the known conventional schemes and leads both to improved convergence properties as well as to a simplified theory. Lie theory provides us both with a unified treatment and a coordinate free ap- proach of Jacobi-type methods. In particular, it allows a formulation of Jacobi meth- ods that is independent of the underlying matrix representation. Together with the above specific matrix cases, all isomorphic types can be simultaneously treated. Thus we can completely avoid the often tiring case-by-case analysis. We illustrate this process in Subsection 5.2.