Adam W. Strzebonski

Cylindrical Algebraic Decomposition using validated numerics

Abstract We present a version of the Cylindrical Algebraic Decomposition (CAD) algorithm which uses interval sample points in the lifting phase, whenever the results can be validated. This gives substantial time savings by avoiding computations with exact algebraic numbers. We use bounds based on Rouche’s theorem combined with information collected during the projection phase and during construction of the current cell to validate the singularity structure of roots. We compare empirically our implementation of this variant of CAD with implementations of CAD using exact algebraic sample points (our and QEPCAD) and with our implementation of CAD using interval sample points with validation based solely on interval data.

Solving Systems of Strict Polynomial Inequalities

We present an algorithm for finding an explicit description of solution sets of systems of strict polynomial inequalities, correct up to lower dimensional algebraic sets. Such a description is sufficient for many practical purposes, such as volume integration, graphical representation of solution sets, or global optimization over open sets given by polynomial inequality constraints. Our algorithm is based on the cylindrical algebraic decomposition algorithm. It uses a simplified projection operator, and constructs only rational sample points.

Computing in the Field of Complex Algebraic Numbers

In this paper we present two methods of computing with complex algebraic numbers. The first uses isolating rectangles to distinguish between the roots of the minimal polynomial, the second method uses validated numeric approximations. We present algorithms for arithmetic and for solving polynomial equations, and compare implementations of both methods inMathematica.

Computation with semialgebraic sets represented by cylindrical algebraic formulas

Cylindrical algebraic formulas are an explicit representation of semialgebraic sets as finite unions of cylindrically arranged disjoint cells bounded by graphs of algebraic functions. We present a version of the Cylindrical Algebraic Decomposition (CAD) algorithm customized for efficient computation of arbitrary combinations of unions, intersections and complements of semialgebraic sets given in this representation. The algorithm can also be used to eliminate quantifiers from Boolean combinations of cylindrical algebraic formulas. We show application examples and an empirical comparison with direct CAD computation for unions and intersections of semialgebraic sets.

Real root isolation for exp-log functions

We present a real root isolation procedure for univariate functions obtained by composition and rational operations from exp, log, and real constants. We discuss implementation of the procedure and give empirical results. The procedure requires the ability to determine signs of exp-log functions at simple roots of other exp-log functions. The currently known method to do this depends on Schanuels conjecture [6].

Cylindrical algebraic decomposition using local projections

We present an algorithm which computes a cylindrical algebraic decomposition of a semialgebraic set using projection sets computed for each cell separately. Such local projection sets can be significantly smaller than the global projection set used by the Cylindrical Algebraic Decomposition (CAD) algorithm. This leads to reduction in the number of cells the algorithm needs to construct. We give an empirical comparison of our algorithm and the classical CAD algorithm.

Solving polynomial systems over semialgebraic sets represented by cylindrical algebraic formulas

Cylindrical algebraic formulas are an explicit representation of semialgebraic sets as finite unions of cylindrically arranged disjoint cells bounded by graphs of algebraic functions. We present a version of the Cylindrical Algebraic Decomposition (CAD) algorithm customized for solving systems of polynomial equations and inequalities over semialgebraic sets given in this representation. The algorithm can also be used to solve conjunctions of polynomial conditions in an incremental manner. We show application examples and give an empirical comparison of incremental and direct CAD computation.

A Real Polynomial Decision Algorithm Using Arbitrary-Precision Floating Point Arithmetic

We study the problem of deciding whether a system of real polynomial equations and inequalities has solutions, and if yes finding a sample solution. For polynomials with exact rational number coefficients the problem can be solved using a variant of the cylindrical algebraic decomposition (CAD) algorithm. We investigate how the CAD algorithm can be adapted to the situation when the coefficients are inexact, or, more precisely, Mathematica arbitrary-precision floating point numbers. We investigate what changes need to be made in algorithms used by CAD, and how reliable are the results we get.

Real root isolation for tame elementary functions

We present a real root isolation procedure for univariate elementary functions. The procedure finds the domain and the zero set of a function f in an arbitrary, possibly unbounded, interval as long as f is represented by a tame expression. An elementary expression is tame if the arguments of its trigonometric subexpressions are bounded. We discuss implementation of the procedure and give empirical results. The procedure requires the ability to determine signs of elementary functions at simple roots of other elementary functions. The currently known method to do this depends on Schanuels conjecture [7].

Advances on the continued fractions method using better estimations of positive root bounds

We present an implementation of the Continued Fractions (CF) real root isolation method using a recently developed upper bound on the positive values of the roots of polynomials. Empirical results presented in this paper verify that this implementation makes the CF method always faster than the Vincent-Collins-Akritas bisection method, or any of its variants.