Numerical Solution of Isospectral

In this paper we are concerned with the problem of solving numerically isospectral ows. These ows are characterized by the diierential equation L 0 = B(L);L]; L(0) = L 0 ; where L 0 is a d d symmetric matrix, B(L) is a skew-symmetric matrix function of L and B;L] is the Lie bracket operator. We show that standard Runge{Kutta schemes fail in recovering the main qualitative feature of these ows, that is isospectrality, since they cannot recover arbitrary cubic conservation laws. This failure motivates us to introduce an alternative approach and establish a framework for generation of isospectral methods of arbitrarily high order. 1. Background and notation 1.1. Introduction. The interest in solving isospectral ows is motivated by their relevance in a wide range of applications, from molecular dynamics to micromag-netics to linear algebra. The general form of an isospectral ow is the diierential equation L 0 = B(L); L]; L(0) = L 0 ; (1) where L 0 is a given d d symmetric matrix, B(L) is a skew-symmetric matrix function of L and B(L); L] = B(L)L ? LB(L) is the commutator of B(L) and L. The choice of the matrix function B(L) characterizes the dynamics of the underlying ow L(t). Important special cases are the Toda lattice equations, double-bracket ows and KvM ows. Toda lattice equations in the Lax formulation (1) were considered by Toda T], Flaschka F] and Moser Mo] and their relation with the QR algorithm for nding eigenvalues by Symes Sy] and then extensively by Deift, Nanda, Tomei et al., Lagarias, in Na1], Na2] DNT], L], DRTW]. It has been nally generalized to the nonsymmetric case by Chu, Watkins and Elsner in Ch], W], WE]. The double bracket ow was introduced by Brockett in B1] and then investigated by Brockett et al.in BBR]. Its relation with the singular value decomposition (SVD) Driessel and Chu in ChD2] have also investigated another isospectral ow of the form (1) in relation with the inverse eigenvalue problem for Toeplitz symmetric matrices. Finally we mention the KvM