Reports in Informatics Recurrence Relation for the Factors in the Polar Decomposition on Lie Groups Recurrence Relation for the Factors in the Polar Decomposition on Lie Groups

In this paper we consider the polar decomposition on a Lie group G endowed with an involutive automorphism , as considered in (Munthe-Kaas, Quispel & Zanna 2000). In that paper, among other things, it was shown that z = exp(tZ) 2 G, Z 2 g, can be decomposed as z = xy, where x = exp(X(t)) and y = exp(Y (t)), where X(t) 2 p and Y (t) 2 k, g = p k being a Cartan decomposition of the algebra g induced by. The authors also derived a recurrence relation for the series expansion of X(t). This paper is devoted to the derivation of the analogous formula for the function Y (t). We obtain an explicit (but rather complicated) recurrence relation for the coeecients of the series expansion. We also point to some Matlab software for the computation of the series expansion of X(t) and Y (t).