G-continued fractions and convergence acceleration in the solution of third-order linear recurrence relations of Poincaré-type (original) (raw)

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

Numerische Mathematik, 1989

In 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.

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

Journal of Computational and Applied Mathematics, 1990

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 general&d continued fractions and to accelerate the convergence of generalised continued fractions associated with a class of linear recurrence relations of Poincare-type.

Convergence Acceleration for Generalized Continued Fractions

Transactions of the American Mathematical Society, 1988

The main result in this paper is the proof of convergence acceleration for a suitable modification (as defined by de Bruin and Jacobsen) in the case of an n-fraction for which the underlying recurrence relation is of Perron-Kreuser type. It is assumed that the characteristic equations for this recurrence relation have only simple roots with differing absolute values.

Matrix continued fractions related to first-order linear recurrence systems

We introduce a matrix continued fraction associated with the first-order linear recurrence system Y k = θ k Y k−1 . A Pincherle type convergence theorem is proved. We show that the n-th order linear recurrence relation and previous generalizations of ordinary continued fractions form a special case. We give an application for the numerical computation of a non-dominant solution and discuss special cases where θ k is constant for all k and the limiting case where lim k→+∞ θ k is constant. Finally the notion of adjoint fraction is introduced which generalizes the notion of the adjoint of a recurrence relation of order n.

First-Order Linear Recurrence Systems and General N-Fractions

Nonlinear Numerical Methods and Rational Approximation II, 1994

In this paper we define a natural generalization of ordinary continued fractions and nfractions (de Bruin [2]). This so-called general n-fraction is associated with a first-order recurrence system, and we look at some of its applications: computation of eigenvectors, stable computation of non-dominant solutions of recurrence systems, and calculation of vector rational interpolants.

Some Properties of a Third Order Partial Recurrence Relation

Gazi University Journal of Science, 2021

This paper explores a connection between third order recursive sequences and generalized continued fractions by analogy with second order recursive sequences and ordinary two-dimensional continued fractions. It does this with a partial recurrence relation which is related to the original third order recurrence relation, and raises a related conjecture.