Technical Report CS0874

TR#:CS0874
Class:CS
Title: A COMPLETE CONVERGENCE AND STABILITY THEORY FOR A GENERALIZED RICHARDSON EXTRAPOLATION PROCESS.
Authors: A. Sidi
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Abstract: Let $A(y) \sim A+\mu^{\infty}_{k=1} Q_k (\log y) y^{\sigma_k}$ as $y \rightarrow )+$, where $y$ is a discrete or continuous variable and $Q_k(\xi)$ are polynomials in $\xi$. It is assumed that $\sigma_k$ and the degree of $Q_k(\xi)$ or an upper bound for it are known for each $k$, and that $A(y)$ is known for all possible $y \in (0,b)]$. The aim is to find $A$, whether it is the limit or antilimit of $A(y)$ for $y \rightarrow 0+$. A very effective way of doing this is by the generalized Richardson extrapolation. In this paper this procedure is described and a very efficient recursive algorithm for its implementation is given when the set of extrapolation points is $\{y_l = y_0 \omega^l, l = 0,1,...,\omega \in (0,1)\}$. In addition, a complete theory of convergence and stability for the columns and the diagonals of the corresponding extrapolation table is provided. Finally, two applications are considered in detail, one being to generalized Romberg integration of functions with algebraic and logarithmic end point singularities.
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