# Construction of New Generalizations of Wynn's Epsilon and Rho Algorithm by Solving Finite Difference Equations in the Transformation Order

## TL;DR

This paper introduces new sequence transformation algorithms that generalize Wynn's epsilon and rho methods, unifying existing algorithms and demonstrating improved convergence acceleration through numerical experiments.

## Contribution

The paper develops a new class of sequence transformations that encompass Wynn's epsilon, rho, and Osada's generalized rho algorithms, offering a unified framework.

## Key findings

- Algorithms effectively accelerate convergence of various sequences.
- Numerical results show improved performance over existing methods.
- The new transformations handle both convergent and divergent series.

## Abstract

We construct new sequence transformations based on Wynn's epsilon and rho algorithms. The recursions of the new algorithms include the recursions of Wynn's epsilon and rho algorithm and of Osada's generalized rho algorithm as special cases. We demonstrate the performance of our algorithms numerically by applying them to some linearly and logarithmically convergent sequences as well as some divergent series.