Convergence Acceleration for Miller’s Algorithm

Abstract

In this paper Miller’s recurrence algorithm for calculating a minimal solution of a p-th order linear homogeneous recurrence relation is modified with the intention of avoiding the occurrence of overflow and underflow. This algorithm is a generalisation of Gautschi’s continued fraction-algorithm for second-order recurrence relations. It uses a generalisation of a continued fraction which will be called a G-continued fraction. Convergence of this G-continued fraction is defined and some convergence results are given. The concept of modification of a G-continued fraction is introduced. The main result in this paper is the proof of convergence acceleration for a suitable modification in the case of a recurrence relation of Perron-Kreuser type. It is assumed that the characteristic equations for this recurrence relation have only simple roots with differing absolute values.