An algorithm for primary decomposition in polynomial rings over the integers

We present an algorithm to compute a primary decomposition of an ideal in a polynomial ring over the integers. For this purpose we use algorithms for primary decomposition in polynomial rings over the rationals resp. over finite fields, and the idea of Shimoyama-Yokoyama resp. Eisenbud-Hunecke-Vasconcelos to extract primary ideals from pseudo-primary ideals. A parallelized version of the algorithm is implemented in SINGULAR. Examples and timings are given at the end of the article.Comment: 8 page

Numerical homotopies to compute generic points on positive dimensional algebraic sets

Many applications modeled by polynomial systems have positive dimensional solution components (e.g., the path synthesis problems for four-bar mechanisms) that are challenging to compute numerically by homotopy continuation methods. A procedure of A. Sommese and C. Wampler consists in slicing the components with linear subspaces in general position to obtain generic points of the components as the isolated solutions of an auxiliary system. Since this requires the solution of a number of larger overdetermined systems, the procedure is computationally expensive and also wasteful because many solution paths diverge. In this article an embedding of the original polynomial system is presented, which leads to a sequence of homotopies, with solution paths leading to generic points of all components as the isolated solutions of an auxiliary system. The new procedure significantly reduces the number of paths to solutions that need to be followed. This approach has been implemented and applied to various polynomial systems, such as the cyclic n-roots problem

Design and Optimization of Explicit Runge-Kutta Formulas

A model of the pretzel knot is described. Explicit Runge-Kutta methods have been studied for over a century and have applications in the sciences as well as mathematical software such as Matlab\u27s ode45 solver. We have taken a new look at fourth- and fifth-order Runge-Kutta methods by utilizing techniques based on Gröbner bases to design explicit fourth-order Runge-Kutta formulas with step doubling and a family of (4,5) formula pairs that minimize the higher-order truncation error. Gröbner bases, useful tools for eliminating variables, also helped to reveal patterns among the error terms. A Matlab program based on step doubling was then developed to compare the accuracy and efficiency of fourth-order Runge-Kutta formulas with that of ode45

Real-time Dynamic Simulation of Constrained Multibody Systems using Symbolic Computation

The main objective of this research is the development of a framework for the automatic generation of systems of kinematic and dynamic equations that are suitable for real-time applications. In particular, the efficient simulation of constrained multibody systems is addressed. When modelled with ideal joints, many mechanical systems of practical interest contain closed kinematic chains, or kinematic loops, and are most conveniently modelled using a set of generalized coordinates of cardinality exceeding the degrees-of-freedom of the system. Dependent generalized coordinates add nonlinear algebraic constraint equations to the ordinary differential equations of motion, thereby producing a set of differential-algebraic equations that may be difficult to solve in an efficient yet precise manner. Several methods have been proposed for simulating such systems in real time, including index reduction, model simplification, and constraint stabilization techniques. In this work, the equations of motion are formulated symbolically using linear graph theory. The embedding technique is applied to eliminate the Lagrange multipliers from the dynamic equations and obtain one ordinary differential equation for each independent acceleration. The theory of Gröbner bases is then used to triangularize the kinematic constraint equations, thereby producing recursively solvable systems for calculating the dependent generalized coordinates given values of the independent coordinates. For systems that can be fully triangularized, the kinematic constraints are always satisfied exactly and in a fixed amount of time. Where full triangularization is not possible, a block-triangular form can be obtained that still results in more efficient simulations than existing iterative and constraint stabilization techniques. The proposed approach is applied to the kinematic and dynamic simulation of several mechanical systems, including six-bar mechanisms, parallel robots, and two vehicle suspensions: a five-link and a double-wishbone. The efficient kinematic solution generated for the latter is used in the real-time simulation of a vehicle with double-wishbone suspensions on both axles, which is implemented in a hardware- and operator-in-the-loop driving simulator. The Gröbner basis approach is particularly suitable for situations requiring very efficient simulations of multibody systems whose parameters are constant, such as the plant models in model-predictive control strategies and the vehicle models in driving simulators

Noncommutative Involutive Bases

The theory of Groebner Bases originated in the work of Buchberger and is now considered to be one of the most important and useful areas of symbolic computation. A great deal of effort has been put into improving Buchberger's algorithm for computing a Groebner Basis, and indeed in finding alternative methods of computing Groebner Bases. Two of these methods include the Groebner Walk method and the computation of Involutive Bases. By the mid 1980's, Buchberger's work had been generalised for noncommutative polynomial rings by Bergman and Mora. This thesis provides the corresponding generalisation for Involutive Bases and (to a lesser extent) the Groebner Walk, with the main results being as follows. (1) Algorithms for several new noncommutative involutive divisions are given, including strong; weak; global and local divisions. (2) An algorithm for computing a noncommutative Involutive Basis is given. When used with one of the aforementioned involutive divisions, it is shown that this algorithm returns a noncommutative Groebner Basis on termination. (3) An algorithm for a noncommutative Groebner Walk is given, in the case of conversion between two harmonious monomial orderings. It is shown that this algorithm generalises to give an algorithm for performing a noncommutative Involutive Walk, again in the case of conversion between two harmonious monomial orderings. (4) Two new properties of commutative involutive divisions are introduced (stability and extendibility), respectively ensuring the termination of the Involutive Basis algorithm and the applicability (under certain conditions) of homogeneous methods of computing Involutive Bases.Comment: 378+x+I Pages; PhD Thesis (University of Wales, Bangor); Code available at http://www.dilan4.freeserve.co.uk/maths