On the complexity of skew arithmetic

13 pagesIn this paper, we study the complexity of several basic operations on linear differential operators with polynomial coefficients. As in the case of ordinary polynomials, we show that these complexities can be expressed in terms of the cost of multiplication

Ball arithmetic

33 pagesThe Mathemagix project aims at the development of a ''computer analysis'' system, in which numerical computations can be done in a mathematically sound manner. A major challenge for such systems is to conceive algorithms which are both efficient, reliable and available at any working precision. In this paper, we survey several older and newer such algorithms. We mainly concentrate on the automatic and efficient computation of high quality error bounds, based on a variant of interval arithmetic which we like to call ''ball arithmetic''

Certifying trajectories of dynamical systems

This paper concerns the reliable integration of dynamical systems with a focus on the computation of one specific trajectory for a given initial condition at high precision. We describe several algorithmic tricks which allow for faster parallel computations and better error estimates. We also introduce so called "Lagrange models". These serve a similar purpose as the more classical Taylor models, but we will show that they allow for larger step sizes, especially when the truncation orders get large

Uniformization of multivariate power series

In this paper, we describe an algorithm for the ``uniformization'' of a multivariate power series. Let K[[T]] be the field of ``grid-based power series'' over a sufficiently large non archimedean ``monomial group'' (or value group) T, such as T= {t1^i1 ... tn^in : i1, ..., in in R} with the lexicographical ordering on i1, ..., in. We interpret power series f in K[[x1, .., xn]] as functions K[[T< K[[T

Efficient root counting for analytic functions on a disk

7 pagesIn this note, we present a variant of an algorithm by Schönhage for counting the number of zeros of a complex polynomial in a disk. Our algorithm implements a few optimizations and also applies to more general analytic functions

GNU TeXmacs

GNU TeXmacs is a free scientific editing platform with special features for mathematicians. The editor can be used to produce documents with a professional typesetting quality (better than TeX/LaTeX) via a user-friendly front-end. The editor can be used as a front-end to several computer algebra systems and includes a lot of additional facilities, like a presentation mode, a technical picture editor, a typed linking tool, etc. The editor can be extended by users in several ways: using style files, plug-ins or via the Scheme extension language. Converters exist for LaTeX, Xhtml and MathML

Faster relaxed multiplication

In previous work, we have introduced several fast algorithms for relaxed power series multiplication (also known under the name on-line multiplication) up till a given order n. The fastest currently known algorithm works over an effective base field K with sufficiently many 2^p-th roots of unity and has algebraic time complexity O(n log n exp (2 sqrt (log 2 log log n))). In this note, we will generalize this algorithm to the cases when K is replaced by an effective ring of positive characteristic or by an effective ring of characteristic zero, which is also torsion-free as a Z-module and comes with an additional algorithm for partial division by integers. We will also present an asymptotically faster algorithm for relaxed multiplication of p-adic numbers

Overview of the Mathemagix type system

The goal of the Mathemagix project is to develop a new and free software for computer algebra and computer analysis, based on a strongly typed and compiled language. In this paper, we focus on the underlying type system of this language, which allows for heavy overloading, including parameterized overloading with parameters in so called "categories". The exposition is informal and aims at giving the reader an overview of the main concepts, ideas and differences with existing languages. In a forthcoming paper, we intend to describe the formal semantics of the type system in more details