Scalar Levin-Type Sequence Transformations

Sequence transformations are important tools for the convergence acceleration of slowly convergent scalar sequences or series and for the summation of divergent series. Transformations that depend not only on the sequence elements or partial sums $s_n$ but also on an auxiliary sequence of so-called remainder estimates $\omega_n$ are of Levin-type if they are linear in the $s_n$, and nonlinear in the $\omega_n$. Known Levin-type sequence transformations are reviewed and put into a common theoretical framework. It is discussed how such transformations may be constructed by either a model sequence approach or by iteration of simple transformations. As illustration, two new sequence transformations are derived. Common properties and results on convergence acceleration and stability are given. For important special cases, extensions of the general results are presented. Also, guidelines for the application of Levin-type sequence transformations are discussed, and a few numerical examples are given.Comment: 59 pages, LaTeX, invited review for J. Comput. Applied Math., abstract shortene

Two-dimensional quantum black holes: Numerical methods

We present details of a new numerical code designed to study the formation and evaporation of 2-dimensional black holes within the CGHS model. We explain several elements of the scheme that are crucial to resolve the late-time behavior of the spacetime, including regularization of the field variables, compactification of the coordinates, the algebraic form of the discretized equations of motion, and the use of a modified Richardson extrapolation scheme to achieve high-order convergence. Physical interpretation of our results will be discussed in detail elsewhere