Appendix: the theory of space with introduction, comments, and addenda

The epoch-making work of János Bolyai is presented here, together with a supplement outlining Hungarian political and science history to help the reader to get acquainted with the miserable fate of János Bolyai and with his intellectual world. A facsimile of a copy of Bolyai''s original 1831 Scientia Spatii (also known as the Appendix) is included, together with a translation. Comments and notes, and a survey of the effects of his work, complete the volume

The genesis and early developments of Aitken\u2019s process, Shanks\u2019 transformation, the \u3b5\u2013algorithm, and related fixed point methods

In this paper, we trace back the genesis of Aitken\u2019s \u3942 process and Shanks\u2019 sequence transformation. These methods, which are extrapolation methods, are used for accelerating the convergence of sequences of scalars, vectors, matrices, and tensors. They had, and still have, many important applications in numerical analysis and in applied mathematics. They are related to continued fractions and Pad\ue9 approximants. We go back to the roots of these methods and analyze the original contributions. New and detailed explanations on the building and properties of Shanks\u2019 transformation and its kernel are provided. We then review their historical algebraic and algorithmic developments. We also analyze how they were involved in the solution of systems of linear and nonlinear equations, in particular in the methods of Steffensen, Pulay, and Anderson. Testimonies by various actors of the domain are given. The paper can also serve as an introduction to this domain of numerical analysis