Paul Levrie

A note on Thiele n-fractions

In this paper we construct ann-fraction which is a generalization of a Thiele continued fraction. We prove that, under certain conditions, themth approximant of thisn-fraction solves the vector case of the rational interpolation problem.

Convergence Acceleration for Miller’s Algorithm

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.

Pringsheim's theory for generalized continued fractions

Abstract In this paper we present a generalization to generalized continued fractions of Pringsheims theorem on the convergence of ordinary continued fractions.

Convergence acceleration for the numerical solution of second-order linear recurrence relations

The concept of modification used for accelerating the convergence of ordinary continued fractions is adapted to the case of the Gautschi–Aggarwal–Burgmeier algorithm for the computation of nondominant solutions of nonhomogeneous second-order linear recurrence relations.

Improving a method for computing non-dominant solutions of certain second-order recurrence relations of Poincaré-type

SummaryIn this paper we present a method of convergence acceleration for the calculation of non-dominant solutions of second-order linear recurrence relations for which the coefficients satisfy certain asymptotic conditions. It represents an improvement of the method recently proposed by Jacobsen and Waadeland [3, 4] for limit periodic continued fractions. For continued fractions the method corresponds to a repeated application of the Bauer-Muir transformation. Some examples and a generalization to non-homogeneous recurrence relations are given.

Pringsheim's theorem revisited

Abstract In this paper we prove a generalization to higher-order linear recurrence relations of Pringsheims theorem on the convergence of ordinary continued fractions.

A note on two convergence acceleration methods for ordinary continued fractions

Abstract In this note we compare two recent methods of convergence acceleration for ordinary continued fractions, the first one introduced by Thron and Waadeland [13], and further developed by Brezinski [1], the second one by Jacobsen and Waadeland [4,5].

Some identities for G -continued fractions and generalized continued fractions

Abstract In this paper we use the invariance property of generalized linear fractional transformations to derive some useful identities for generalized continued fractions and G -continued fractions, two types of generalizations of ordinary continued fractions associated with higher-order recurrence relations. Furthermore, the relationship between these two kinds of continued fractions is discussed.

G -continued fractions and convergence acceleration in the solution of third-order linear recurrence relations of Poincare´-type

Abstract In this paper we present a method of convergence acceleration for the calculation of nondominant solutions of third-order homogeneous recurrence relations of Poincare-type for which the coefficients satisfy certain asymptotic conditions. In this method the value of the G -continued fraction associated with such a recurrence relation is calculated using a generalization of the convergence acceleration method for ordinary continued fractions discussed in [7].

On the relationship between generalised continued fractions and G -continued fractions

In this paper the connection between generalised continued fractions (de Bruin 1974)) and G-continued fractions (Levrie (1988)) is studied. This connection is used to prove a convergence theorem for generalised continued fractions and to accelerate the convergence of generalised continued fractions associated with a class of linear recurrence relations of Poincare-type.