Partial Cylindrical Algebraic Decomposition Research Articles

Dynamics of a Cournot game with bounded rational firms and various scale effects.

In this paper, we focus on a dynamic Cournot game in the market with a nonlinear (isoelastic) demand function. In our model, there are N competing firms featured by nonlinear cost functions, which enhances our model's resemblance to real-world scenarios. Firstly, we focus on the homogeneous case where firms' marginal costs change at equal rates. We establish analytical expressions of the market supply at equilibrium and perform comparative static analysis. In addition, we investigate the local stability under different economies of scale and show that there could be multiple stable equilibria if firms face economies of scale. The heterogeneous case where firms' marginal costs change at distinct rates is much more complex, thus we investigate the duopoly game with only two firms involved. Methods of symbolic computations such as triangular decomposition and partial cylindrical algebraic decomposition are employed in the analytical investigations of the equilibrium, which is nearly impossible by using the pencil-and-paper approach since the closed-form equilibrium is quite complicated. According to the computational results, we derive that two stable positive equilibria may coexist if both firms face economies of scale. Additionally, we conduct preliminary numerical simulations and find two different types of complex dynamics of the model considered in this paper: complex trajectories such as periodic and chaotic orbits may appear; the topological structure of the basins of attraction may be complex.

Real zeros of the zero-dimensional parametric piecewise algebraic variety

The piecewise algebraic variety is the set of all common zeros of multivariate splines. We show that solving a parametric piecewise algebraic variety amounts to solve a finite number of parametric polynomial systems containing strict inequalities. With the regular decomposition of semi-algebraic systems and the partial cylindrical algebraic decomposition method, we give a method to compute the supremum of the number of torsion-free real zeros of a given zero-dimensional parametric piecewise algebraic variety, and to get distributions of the number of real zeros in every n-dimensional cell when the number reaches the supremum. This method also produces corresponding necessary and sufficient conditions for reaching the supremum and its distributions. We also present an algorithm to produce a necessary and sufficient condition for a given zero-dimensional parametric piecewise algebraic variety to have a given number of distinct torsion-free real zeros in every n-cell in the n-complex.

QEPCAD B

This report introduces Q EPCAD , B, a program for computing with real algebraic sets using cylindrical algebraic decomposition (CAD). QEPCAD B both extends and improves upon the Q EPCAD system for quantifier elimination by partial cylindrical algebraic decomposition written by Hoon Hong in the early 1990s. This paper briefly discusses some of the improvements in the implementation of CAD and quantifier elimination via CAD, and provides somewhat more detail on extensions to the system that go beyond quantifier elimination. The author is responsible for most of the extended features of Q EPCAD B, but improvements to the basic CAD implementation and to the SACLIB library on which Q EPCAD is based are the results of many people's work, including: George E. Collins, Mark J. Encarnación, Hoon Hong, Jeremy Johnson, Werner Krandick, Richard Liska, Scott McCallum, Nicolas Robidoux, and Stanly Steinberg. Source code, documentation and installation instructions for QEPCAD B are all available at www.cs.usna.edu/~qepcad.

A complete algorithm for automated discovering of a class of inequality-type theorems

Making use of the discriminant sequence for polynomials, WR algorithm, Wu’s elimination and a partial cylindrical algebraic decomposition, we present here a practical algorithm for automated inequality discovering which can discover new inequalities automatically without requiring to put forward any conjectures beforehand. That is complete for an extensive class of inequality-type theorems. Also this algorithm is applied to the classification of the real physical solutions of geometric constraint problems. Many inequalities with various backgrounds have been discovered or rediscovered by our program, DISCOVERER, which implements the algorithm in Maple.

Dead beat controllability of polynomial systems: symbolic computation approaches

State and output dead beat controllability tests for a very large class of polynomial systems with rational coefficients may be based on the quantifier elimination by partial cylindrical algebraic decomposition (QEPCAD) symbolic computation program. The method is unified for a very large class of systems and can handle one- or two-sided control constraints. Families of minimum time state/output dead beat controllers are obtained. The computational complexity of the test is prohibitive for general polynomial systems, but by constraining the structure of the system we may beat the curse of complexity. A computationally less expensive algebraic test for output dead beat controllability for a class of odd polynomial systems is presented. Necessary and sufficient conditions are given. They are still very difficult to check. Therefore, a number of easier-to-check sufficient conditions are also provided. The latter are based on the Grobner basis method and QEPCAD. It is shown on a subclass of odd polynomial systems how it is possible to further reduce the computational complexity by exploiting the structure of the system.

Applying Quantifier Elimination to Stability Analysis of Difference Schemes

Stability analysis is an important tool for constructing time-stepping finite difference schemes for partial differential equations. This paper describes how von Neumann stability analysis can be reduced to a quantifier elimination problem over the reals. We report our experience in analysing some difference schemes by using a quantifier package based on the partial cylindrical algebraic decomposition algorithm.

Partial Cylindrical Algebraic Decomposition for quantifier elimination

The Cylindrical Algebraic Decomposition method (CAD) decomposes Rr into regions over which given polynomials have constant signs. An important application of CAD is quantifier elimination in elementary algebra and geometry. In this paper we present a method which intermingles CAD construction with truth evaluation so that parts of the CAD are constructed only as needed to further truth evaluation and aborts CAD construction as soon as no more truth evaluation is needed. The truth evaluation utilizes in an essential way any quantifiers which are present and additionally takes account of atomic formulas from which some variables are absent. Preliminary observations show that the new method is always more efficient than the original, and often significantly more efficient.