Changbo Chen

Truth Table Invariant Cylindrical Algebraic Decomposition by Regular Chains

A new algorithm to compute cylindrical algebraic decompositions (CADs) is presented, building on two recent advances. Firstly, the output is truth table invariant (a TTICAD) meaning given formulae have constant truth value on each cell of the decomposition. Secondly, the computation uses regular chains theory to first build a cylindrical decomposition of complex space (CCD) incrementally by polynomial. Significant modification of the regular chains technology was used to achieve the more sophisticated invariance criteria. Experimental results on an implementation in the RegularChains Library for Maple verify that combining these advances gives an algorithm superior to its individual components and competitive with the state of the art.

Cylindrical Algebraic Decomposition in the RegularChains Library

Cylindrical algebraic decomposition (CAD) is a fundamental tool in computational real algebraic geometry and has been implemented in several software. While existing implementations are all based on Collins’ projection-lifting scheme and its subsequent ameliorations, the implementation of CAD in the RegularChains library is based on triangular decomposition of polynomial systems and real root isolation of regular chains. The function in the RegularChains library for computing CAD is called CylindricalAlgebraicDecompose. In this paper, we illustrate by examples the functionality, the underlying theory and algorithm, as well the implementation techniques of CylindricalAlgebraicDecompose. An application of it is also provided.

A Numerical Method for Computing Border Curves of Bi-parametric Real Polynomial Systems and Applications

For a bi-parametric real polynomial system with parameter values restricted to a finite rectangular region, under certain assumptions, we introduce the notion of border curve. We propose a numerical method to compute the border curve, and provide a numerical error estimation.

Quantifier elimination by cylindrical algebraic decomposition based on regular chains

A quantifier elimination algorithm by cylindrical algebraic decomposition based on regular chains is presented. The main idea is to refine a complex cylindrical tree until the signs of polynomials appearing in the tree are sufficient to distinguish the true and false cells. We report an implementation of our algorithm in the RegularChains library in Maple and illustrate its effectiveness by examples.

Regular Chains under Linear Changes of Coordinates and Applications

Given a regular chain, we are interested in questions like computing the limit points of its quasi-component, or equivalently, computing the variety of its saturated ideal. We propose techniques relying on linear changes of coordinates and we consider strategies where these changes can be either generic or guided by the input.

The basic polynomial algebra subprograms

The Basic Polynomial Algebra Subprograms (BPAS) provides arithmetic operations (multiplication, division, root isolation, etc.) for univariate and multivariate polynomials over prime fields or with integer coefficients. The code is mainly written in CilkPlus [11] targeting multicore processors. The current distribution focuses on dense polynomials and the sparse case is work in progress. A strong emphasis is to put on adaptive algorithms as the library aims at supporting a wide variety of situations in terms of problem sizes and available computing resources. One of the purposes of the BPAS project is to take advantage of hardware accelerators in the development of polynomial systems solvers. The BPAS library is publicly available in source at www.bpaslib.org.

Real Quantifier Elimination in the RegularChains Library

Quantifier elimination (QE) over real closed fields has found numerous applications. Cylindrical algebraic decomposition (CAD) is one of the main tools for handling quantifier elimination of nonlinear input formulas. Despite of its worst case doubly exponential complexity, CAD-based quantifier elimination remains interesting for handling general quantified formulas and producing simple quantifier-free formulas.

The Basic Polynomial Algebra Subprograms.

The Basic Polynomial Algebra Subprograms (BPAS) provides arithmetic operations (multiplication, division, root isolation, etc.) for univariate and multivariate polynomials over prime fields or with integer coefficients. The code is mainly written in CilkPlus [10] targeting multicore processors. The current distribution focuses on dense polynomials and the sparse case is work in progress. A strong emphasis is put on adaptive algorithms as the library aims at supporting a wide variety of situations in terms of problem sizes and available computing resources. One of the purposes of the BPAS project is to take advantage of hardware accelerators in the development of polynomial systems solvers. The BPAS library is publicly available in source at www.bpaslib.org .

Penalty Function Based Critical Point Approach to Compute Real Witness Solution Points of Polynomial Systems

We present a critical point method based on a penalty function for finding certain solution (witness) points on real solutions components of general real polynomial systems. Unlike other existing numerical methods, the new method does not require the input polynomial system to have pure dimension or satisfy certain regularity conditions.

Full Rank Representation of Real Algebraic Sets and Applications

We introduce the notion of the full rank representation of a real algebraic set, which represents it as the projection of a union of real algebraic manifolds (V_{mathbb {R}}(F_i)) of (mathbb {R}^m), (mge n), such that the rank of the Jacobian matrix of each (F_i) at any point of (V_{mathbb {R}}(F_i)) is the same as the number of polynomials in (F_i).