A hyperbolic tangent identity and the geometry of Padé sign function iterations

The rational iterations obtained from certain Padé approximations associated with computing the matrix sign function are shown to be equivalent to iterations involving the hyperbolic tangent and its inverse. Using this equivalent formulation many results about these Padé iterations, such as global convergence, the semi-group property under composition, and explicit partial fraction decompositions can be obtained easily. In the second part of the paper it is shown that the behavior of points under the Padé iterations can be expressed, via the Cayley transform, as the combined result of a completely regular iteration and a chaotic iteration. These two iterations are decoupled, with the chaotic iteration taking the form of a multiplicative linear congruential random number generator where the multiplier is equal to the order of the Padé approximation.This research was supported in part by the National Science Foundation under Grant No. ECS-9120643, the Air Force Office of Scientific Research under Grant no. F49620-94-1-0104DEF, and the Office of Naval Research under Grant No. N00014-92-J-1706.