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

Speeding up Cylindrical Algebraic Decomposition by Gr\"obner Bases

Gr\"obner Bases and Cylindrical Algebraic Decomposition are generally thought of as two, rather different, methods of looking at systems of equations and, in the case of Cylindrical Algebraic Decomposition, inequalities. However, even for a mixed system of equalities and inequalities, it is possible to apply Gr\"obner bases to the (conjoined) equalities before invoking CAD. We see that this is, quite often but not always, a beneficial preconditioning of the CAD problem. It is also possible to precondition the (conjoined) inequalities with respect to the equalities, and this can also be useful in many cases.Comment: To appear in Proc. CICM 2012, LNCS 736

A comparison of three heuristics to choose the variable ordering for CAD

Cylindrical algebraic decomposition (CAD) is a key tool for problems in real algebraic geometry and beyond. When using CAD there is often a choice over the variable ordering to use, with some problems infeasible in one ordering but simple in another. Here we discuss a recent experiment comparing three heuristics for making this choice on thousands of examples

Cylindrical Algebraic Decomposition With Frontier Condition

We propose a novel algorithm for computing Cylindrical Algebraic Decomposition satisfying the frontier condition without preliminary change of coordinates, i.e., in the potential presence of blow-ups. Frontier condition means that the frontier of each cell is made up of a union of some other cells in the decoposition. This construction can be useful for computing topological properties of semialgebraic sets defined by first-order formulas with quantifiers. The algorithm uses a recursion on lexicographic order of cells in the initial decomposition

Using cylindrical algebraic decomposition and local Fourier analysis to study numerical methods: two examples

Local Fourier analysis is a strong and well-established tool for analyzing the convergence of numerical methods for partial differential equations. The key idea of local Fourier analysis is to represent the occurring functions in terms of a Fourier series and to use this representation to study certain properties of the particular numerical method, like the convergence rate or an error estimate. In the process of applying a local Fourier analysis, it is typically necessary to determine the supremum of a more or less complicated term with respect to all frequencies and, potentially, other variables. The problem of computing such a supremum can be rewritten as a quantifier elimination problem, which can be solved with cylindrical algebraic decomposition, a well-known tool from symbolic computation. The combination of local Fourier analysis and cylindrical algebraic decomposition is a machinery that can be applied to a wide class of problems. In the present paper, we will discuss two examples. The first example is to compute the convergence rate of a multigrid method. As second example we will see that the machinery can also be used to do something rather different: We will compare approximation error estimates for different kinds of discretizations.Comment: The research was funded by the Austrian Science Fund (FWF): J3362-N2

A cluster-based cylindrical algebraic decomposition algorithm

Let A ⊂ Z [x1,000, xr be a finite set. An A-invariant cylindrical algebraic decomposition (cad) is a certain partitlon of r-dlmensional euclidean space Er into semi-algebralc cells such that the value of each Ai ∈ A has constant sign (positive, negative, or zero) throug|umt each cell. Two cells are adjacent if their union is connected. Recently a number of mathoda have been given for augmenting Colllns' cad construction algorithm (1975), so that in addition to specifying the cell~ that comprise a cad, it identifies the pairs of adjacent cells. Assuming the availability of such an adjacency algorithm, in this paper we give a modified cad construction algorithm based on the utillzatloa of clusters of cells in a cad (a cluster is a collection of cells whose union is connected). Preliminary observations indicate that the 11ew algorithm can be significantly more efficient in some cases than the original, although in other examples it is somewhat less efficient