The goal of this session is to increase the interaction between those who apply computer algebra to science and technology and specialists in the systems. We hope that developers of computer algebra systems will learn about new applications of their products and we hope that the users will be able to discuss their computational problems with experts. Talks by those who use computer algebra while studying other problems in science, technology or industry are welcome.
Abstract:
Automatic code generation facilities within a computer algebra system allow users to work within a symbolic environment while producing code that can be incorporated into numeric systems for scientific applications.
We will show how Maple's code generation tools fit into the process of modelling scientific problems and prototyping algorithms. Furthermore, we will show how the tools may be customized for specific target languages and environments. Finally, we will discuss issues and challenges related to source-to-source translation in this context.
Abstract:
Software tools for design, modeling, analysis, and simulation can greatly improve the performance and efficiency of mechanic al systems. Symofros, an environment for modeling and simulation, is used for all robots and experimental systems at the Canad ian Space Agency R&D Robotics Laboratory. In particular, Symofros is the core of the SPDM (Special Purpose Dextrous Manipulato r) Tasks Verification Facilities[1] where a ground robot is used to emulate the contact task of a space manipulator (here the S pace Station manipulator SPDM). In SMP[2], a generic simulator for astronauts training has been developed based on Symofros and other tools. SMP is used to develop situation awareness, provides kinematics and dynamics understandings of robotic systems an d increases the dexterity of the operator. SMP is now on board of the International Space Station.
Symofros is composed of several modules for mechanical system description, modeling and simulation. The modeling module of Symofros, the Symbolic Model Generator, is built within Maple. This module is composed of several Maple modules: topology, pre processing, kinematics, non-linear dynamics, linear dynamics and dynamics identification. It automatically derives a set of ove r 90 functions, called basic functions, which represent the kinematics and dynamics. These functions are symbolic expressions written in terms of variables (time dependent) and parameters (constant with time). Variables comprise positions, velocities, a ccelerations and system inputs while parameters can be the mass and the length of the links. The functions are translated int o C for simulation purposes.
Abstract:
Recently, element preconditioning has been proposed as an improvement for the algebraic multigrid method (AGM) for solving elliptic boundary value problems.
The sub-problem of finding the closest M-matrix to a given symmetric and positive matrix has to solved for many instances (namely, for each element). The numerical solution of all these optimization problems can be very expensive. We achieve a speed-up by solving the optimization problem once and for all symbolically, and then instantiating the solution by the local data.
Theoretically, it is possible to give a closed form solution in terms of field operations and square roots, and -- in one particular case -- roots of higher degree polynomials. Such a closed form would be too large to be useful, so we prefer to give a ``formula'' consisting of a program with arithmetic or square root (and in one case higher order root) assignments and {\bf if then else} branches, but no loops. Using these formula, we can compute the optimal preconditioners faster and more accurately than by standard numerical optimization techniques.
The optimization problem can be reduced to a quantifier elimination problem over real closed fields. However, the problem is too complex for a solution by a general method like Gröbner bases, resultants, or cylindrical algebraic decomposition. It is necessary to exploit the specific structure of the problem. We use some techniques from linear algebra and geometry.