Computer Algebra in Scientific Computing

We are proud to present the invited speakers of CASC conference:

• Marc Moreno Maza, Professor in the Computer Science Department and the Applied Mathematics Department at the University of Western Ontario, Canada
Title of the talk: Implementation Techniques for Power, Laurent, and Puiseux Series in Several Variables
Abstract: While computer algebra systems can perform highly sophisticated algebraic tasks, they are much less equipped for solving problems from mathematical analysis in a symbolic manner. Elementary problems in analysis, such as the manipulation of Taylor series and the calculation of limits of univariate functions, are supported, with some limitations, in general-purpose computer algebra systems such as Maple and Mathematica. However, limits of multivariate functions and more advanced notions of limits, like topological closures, are almost absent from such systems. For instance, Maple is not always capable of computing finite limits of a multivariate rational function at a zero of the denominator that is not an isolated pole.
Many fundamental concepts in mathematics are defined in terms of limits and it is highly desirable for computer algebra systems to implement these concepts. However, limits are, by their essence, hard to compute, in the sense of performing finitely many rational operations on polynomials or matrices. A first helper tool is the famous Weierstrass preparation theorem (and its extensions) which essentially reduces the local study of analytic functions (and more general functions) to the local study of polynomials via the manipulation of power series. A second helper tool is the famous Newton-Puiseux algorithm (and its extensions) which essentially allows for the local study of curves (separating their branches about a point) via the manipulation of Laurent and and Puiseux series.
The Newton-Puiseux algorithm, and more generally the factorization of univariate polynomials over power, Laurent and Puiseux series, has been a very active research area in the computer algebra community in the past 50 years. This research effort, however, has been mainly focusing on the development of algorithms and the analysis of their algebraic complexity. Relatively little has been done in terms of implementation, except for the case of univariate series.
In this talk, we will discuss recent findings on (1) the implementation of power, Laurent and Puiseux series in several variables and, (2) its application to the factorization of univariate polynomials over such series. We will start with the implementation of arithmetic operations which can be sometimes easier than one may think (for instance, the substitution of unit formal power series into formal power series) and sometimes harder (for instance, the inversion of multivariate Laurent series). We will also discuss the impact of the implementation environment on the implementation techniques, considering both interpreted and compiled code. We will illustrate our points with Maple's MultivariatePowerSeries package and the Basic Polynomial Algebra Subroutines. In that latter environment, we will show how different parallel programming patterns can be used to obtain efficient multithreaded implementation of arithmetic operations on power series and factorization of univariate polynomials over such series.
• Michael Nikitas Vrahatis, Director of the Institute of Artificial Intelligence, University Research Center of Patras, Greece
  Title of the talk: Generalizations of the Intermediate Value Theorem and Applications
  Abstract: Generalizations of the intermediate value theorem in several variables are presented. These theorems are very useful in various approaches including, among others, the existence of solutions of systems of nonlinear algebraic and/or transcendental equations, the existence of fixed points of continuous functions, the localization of extrema of objective functions, as well as the localization of periodic orbits of nonlinear mappings and periodic orbits (fixed points) on Poincare's surface of section (Poincare map). These theorems are of major importance for tackling problems with imprecise (not exactly known) information.
  Based on the corresponding existence criteria emanated by the above theorems, methods, named “generalized bisection methods'', are given. The only computable information required by the generalized bisection methods is the algebraic sign of the function value which is the minimum possible information (one bit of information) necessary for the purpose needed, and not any additional information. Thus, these methods are of major importance for studying and tackling problems with imprecise (not exactly known) information. This kind of problems occurs in various scientific fields. This is so, because, in a large variety of applications, precise function values are either impossible or time consuming and computationally expensive to obtain. Furthermore, these methods are particularly useful for studying and tackling various problems where the corresponding functions obtain very large and/or very small values.
  It is well known that algebraic equations are very important in studying and solving problems in a variety of scientific fields. Regarding the algebraic signs of algebraic expressions there are various efficient approaches in obtaining this information.
  Applications related to systems of nonlinear algebraic and/or transcendental equations, as well as fixed points of continuous functions are presented. Furthermore, an application is presented which concerns the computation of all the periodic orbits (stable and unstable) of any period and accuracy which occur, among others, in the study of beam dynamics in circular particle accelerators like the Large Hadron Collider (LHC) machine at the European Organization for Nuclear Research (CERN).