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Bican Xia

Publications and source records attributed to Bican Xia.

44 records · Page 3Linked to original sources
arXiv2012

Stability of Triangular Decomposition and Comprehensive Triangular Decomposition

A new concept, decomposition-unstable (DU) variety of a parametric polynomial system, is introduced in this paper and the stabilities of several triangular decomposition methods, such as characteristic set decomposition, relatively simplicial decomposition and regular chain decomposition, for parametric polynomial systems are discussed in detail. The concept leads to a definition of weakly comprehensive triangular decomposition (WCTD) and a new algorithm for computing comprehensive triangular decomposition (CTD) which was first introduced in [4] for computing an analogue of comprehensive Groebner systems for parametric polynomial systems. Our algorithm takes advantage of a hierarchical solving strategy and a self-adaptive order of parameters. The algorithm has been implemented with Maple 15 and experimented with a number of benchmarks from the literature. Comparison with the Maple package RegularChains, which contains an implementation of the algorithm in [4], is provided and the results illustrate that the time costs by our program for computing CTDs of most examples are no more than those by RegularChains.

cs.SC↗
arXiv2012

Non-Termination Sets of Simple Linear Loops

A simple linear loop is a simple while loop with linear assignments and linear loop guards. If a simple linear loop has only two program variables, we give a complete algorithm for computing the set of all the inputs on which the loop does not terminate. For the case of more program variables, we show that the non-termination set cannot be described by Tarski formulae in general

cs.LO↗
arXiv2010

Triangular Decomposition of Semi-algebraic Systems

Regular chains and triangular decompositions are fundamental and well-developed tools for describing the complex solutions of polynomial systems. This paper proposes adaptations of these tools focusing on solutions of the real analogue: semi-algebraic systems. We show that any such system can be decomposed into finitely many {\em regular semi-algebraic systems}. We propose two specifications of such a decomposition and present corresponding algorithms. Under some assumptions, one type of decomposition can be computed in singly exponential time w.r.t.\ the number of variables. We implement our algorithms and the experimental results illustrate their effectiveness.

cs.SC↗
arXiv2009

Real Solution Isolation with Multiplicity of Zero-Dimensional Triangular Systems

Existing algorithms for isolating real solutions of zero-dimensional polynomial systems do not compute the multiplicities of the solutions. In this paper, we define in a natural way the multiplicity of solutions of zero-dimensional triangular polynomial systems and prove that our definition is equivalent to the classical definition of local (intersection) multiplicity. Then we present an effective and complete algorithm for isolating real solutions with multiplicities of zero-dimensional triangular polynomial systems using our definition. The algorithm is based on interval arithmetic and square-free factorization of polynomials with real algebraic coefficients. The computational results on some examples from the literature are presented.

cs.SC↗
arXiv2009

Termination of Linear Programs with Nonlinear Constraints

Tiwari proved that termination of linear programs (loops with linear loop conditions and updates) over the reals is decidable through Jordan forms and eigenvectors computation. Braverman proved that it is also decidable over the integers. In this paper, we consider the termination of loops with polynomial loop conditions and linear updates over the reals and integers. First, we prove that the termination of such loops over the integers is undecidable. Second, with an assumption, we provide an complete algorithm to decide the termination of a class of such programs over the reals. Our method is similar to that of Tiwari in spirit but uses different techniques. Finally, we conjecture that the termination of linear programs with polynomial loop conditions over the reals is undecidable in general by %constructing a loop and reducing the problem to another decision problem related to number theory and ergodic theory, which we guess undecidable.

cs.LO↗
arXiv2009

Computing Cylindrical Algebraic Decomposition via Triangular Decomposition

Cylindrical algebraic decomposition is one of the most important tools for computing with semi-algebraic sets, while triangular decomposition is among the most important approaches for manipulating constructible sets. In this paper, for an arbitrary finite set $F \subset {\R}[y_1, ..., y_n]$ we apply comprehensive triangular decomposition in order to obtain an $F$-invariant cylindrical decomposition of the $n$-dimensional complex space, from which we extract an $F$-invariant cylindrical algebraic decomposition of the $n$-dimensional real space. We report on an implementation of this new approach for constructing cylindrical algebraic decompositions.

cs.SC↗
arXiv2006

Solution to the Generalized Champagne Problem on Simultaneous Stabilization of Linear Systems

The well-known "Generalized Champagne Problem" on simultaneous stabilization of linear systems is solved by using complex analysis \cite{A73,C78,G69,N52,R87} and Blondel's technique \cite{B94,BG93,BGMR94}. We give a complete answer to the open problem proposed by Patel et al. \cite{P99,PDV02}, which automatically includes the solution to the original "Champagne Problem" \cite{BG93,BGMR94,BSVW99,LKZ99,P99,PDV02}. Based on the recent development in automated inequality-type theorem proving \cite{Y98,Y99,YHX01,YX05,YX03}, a new stabilizing controller design method is established. Our numerical examples significantly improve the relevant results in the literature \cite{LKZ99,PDV02}.

math.DS↗