Hoon Hong

Testing Stability by Quantifier Elimination

For initial and initial-boundary value problems described by differential equations, stability requires the solutions to behave well for large times. For linear constant-coefficient problems, Fourier and Laplace transforms are used to convert stability problems to questions about roots of polynomials. Many of these questions can be viewed, in a natural way, as quantifier-elimination problems. The Tarski?Seidenberg theorem shows that quantifier-elimination problems are solvable in a finite number of steps. However, the complexity of this algorithm makes it impractical for even the simplest problems. The newer Quantifier Elimination by Partial Algebraic Decomposition (QEPCAD) algorithm is far more practical, allowing the solution of some non-trivial problems. In this paper, we show how to write all common stability problems as quantifier-elimination problems, and develop a set of computer-algebra tools that allows us to find analytic solutions to simple stability problems in a few seconds, and to solve some interesting problems in from a few minutes to a few hours.

An efficient method for analyzing the topology of plane real algebraic curves

A practically efficient algorithm for analyzing the topology of plane real algebraic curves is given. Given a bivariate polynomial, the algorithm produces a planar graph which is topologically equivalent to the real variety of the polynomial on the Euclidean plane.

Simple solution formula construction in cylindrical algebraic decomposition based quantifier elimination

Since Tarski (1951) gave the first quantifier elimination algorithm for real closed fields, various improvements and new methods have been devised and analyzed (Arnon 1981, 1988b; Ben-Or et al. 1986; Boge 1980; Buchberger and Hong 1991; Canny 1988; Cohen 1969; Collins 1975; Collins and Hong 1991; Fitchas et al. 1990a; Grigor’ev 1988; Grigor’ev and Vorobjov 1988; Heintz et al. 1989a; Holthusen 1974; Hong 1989, 1990a, 1990b, 1991a, 1991b, 1991c; Johnson 1991; Langemyr 1990; Lazard 1990; McCallum 1984; Renegar 1992a, 1992b, 1992c; Seidenberg 1954).

Safe starting regions by fixed points and tightening

In this paper, we present a method for finding safe starting regions for a given system of non-linear equations. The method is an improvement of the usual method which is based on the fixed point theorem. The improvement is obtained by enclosing the components of the equation system by univariante interval polynomials whose zero sets are found. This operation is called “tightening”. Preliminary experiments show that the tightening operation usually reduces the number of bisections, and thus the computing time. The reduction seems to become more dramatic when the number of variables increases.ZusammenfassungIn dieser Arbeit wird eine Methode zur Bestimmung von Startintervallen mit garantierter Konvergenz für ein gegebenes nichtlineares Gleichungssystem vorgestellt. Die Methode ist eine verbesserung der gebräuchlichen, auf dem Fixpunkt Theorem basierenden Methode. Die Verbesserung wird durch Einschließen der Komponenten des Gleichungssystems durch univariate Intervallpolynome, deren Lösungsmengen berechnet werden, erzielt. Diese Operation wird “Einengung” genannt. Erste experimentelle Untersuchungen zeigen, daß Einengung im allgemeinen die Anzahl der Intervallhalbierungen und somit die Rechenzeit reduziert. Die Reduktion scheint umso signifikanter, je höher die Anzahl der Variablen ist.

Bounds for Absolute Positiveness of Multivariate Polynomials

A multivariate polynomialP(x1, �,xn) with real coefficients is said to beabsolutely positivefrom a real numberBiff it and all of its non-zero partial derivatives of every order are positive forx1,�,xn�B. We call suchBaboundfor the absolute positiveness ofP. This paper provides a simple formula for computing such bounds. We also prove that the resulting bounds are guaranteed to be close to the optimal ones.

Groebner Basis Under Composition I

Composition is the operation of replacing variables in a polynomial with other polynomials. The main question of this paper is:When does composition commute with Groebner basis computation?We prove that this happens iff the composition is `compatible? with the term ordering and the nondivisibility. This has a natural application in the computation of Groebner bases of composed polynomials which often arises in real-life problems.

Quantifier Elimination for Formulas Constrained by Quadratic Equations via Slope Resultants

An algorithm is given for eliminating the quantifier from a formula: (∃x∈R)[α 2 x 2 +α 1 z+α 0 =0ΛF], where F is a quantifier free formula in x 1 ,..., x r , x, and α 2 , α 1 , α 0 are polynomials in x 1 ,..., x r with real coefficients such that the system {α 2 =0, α 1 =0, α 0 =0} has no solution in R r . The output formulas are made of resultants and their variants, which we call slope resultants. The slope resultants can be, like the resultants, expressed as determinants of certain matrices

The design of the SACLIB/PACLIB kernels

This paper describes the design of the kernels of two variants of the SAC-2 computer algebra library: Saclib and Paclib. Saclib is a C version of SAC-2, supporting automatic garbage collection and embeddability. Paclib is a parallel version of Saclib, supporting lightweight concurrency, non-determinism, and parallel garbage collection.

Non-linear Real Constraints in Constraint Logic Programming

Dealing with non-linear constraints over real numbers is one of the most important and non-trivial problems in constraint logic programming. We report our initial effort in tackling the problem with two methods developed in computer algebra during last three decades: Partial Cylindrical Algebraic Decomposition and Grobner basis. We have implemented a prototype called RISC-CLP(Real). Experience with the prototype suggests that it is desirable and in fact feasible to provide a full support of non-linear constraints.

Quantifier elimination for formulas constrained by quadratic equations

An algorithm is given for constructing a quantifier free formula (a boolean expression of polynomial equations and inequalities) equivalent to a given formula of the form: (% c R)[azzz + alz + a. = O A F], where F is a quantifier free formula in Z1, . . . . z~, z, and az, al, ao are polynomials in z 1, . . . . Xr with real coefficients such that the system {az = O, al = O, a. = O} has no solution in Rr. Formulas of this form frequently occur in the context of constraint logic programming over the real numbers. The output formulas are made of resultants and two variants, which we call trace and slope resultants. Both of these variant resultants can be expressed as determinants of certain matrices.