Using the Regular Chains Library to build cylindrical algebraic decompositions by projecting and lifting

Cylindrical algebraic decomposition (CAD) is an important tool, both for quantifier elimination over the reals and a range of other applications. Traditionally, a CAD is built through a process of projection and lifting to move the problem within Euclidean spaces of changing dimension. Recently, an alternative approach which first decomposes complex space using triangular decomposition before refining to real space has been introduced and implemented within the RegularChains Library of Maple. We here describe a freely available package ProjectionCAD which utilises the routines within the RegularChains Library to build CADs by projection and lifting. We detail how the projection and lifting algorithms were modified to allow this, discuss the motivation and survey the functionality of the package

An implementation of Sub-CAD in Maple

Cylindrical algebraic decomposition (CAD) is an important tool for the investigation of semi-algebraic sets, with applications in algebraic geometry and beyond. We have previously reported on an implementation of CAD in Maple which offers the original projection and lifting algorithm of Collins along with subsequent improvements. Here we report on new functionality: specifically the ability to build cylindrical algebraic sub-decompositions (sub-CADs) where only certain cells are returned. We have implemented algorithms to return cells of a prescribed dimensions or higher (layered {\scad}s), and an algorithm to return only those cells on which given polynomials are zero (variety {\scad}s). These offer substantial savings in output size and computation time. The code described and an introductory Maple worksheet / pdf demonstrating the full functionality of the package are freely available online at http://opus.bath.ac.uk/43911/.Comment: 9 page

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

Choosing a variable ordering for truth-table invariant cylindrical algebraic decomposition by incremental triangular decomposition

Cylindrical algebraic decomposition (CAD) is a key tool for solving problems in real algebraic geometry and beyond. In recent years a new approach has been developed, where regular chains technology is used to first build a decomposition in complex space. We consider the latest variant of this which builds the complex decomposition incrementally by polynomial and produces CADs on whose cells a sequence of formulae are truth-invariant. Like all CAD algorithms the user must provide a variable ordering which can have a profound impact on the tractability of a problem. We evaluate existing heuristics to help with the choice for this algorithm, suggest improvements and then derive a new heuristic more closely aligned with the mechanics of the new algorithm

An implementation of CAD in Maple utilising problem formulation, equational constraints and truth-table invariance

Cylindrical algebraic decomposition (CAD) is an important tool for the investigation of semi-algebraic sets, with applications within algebraic geometry and beyond. We recently reported on a new implementation of CAD in Maple which implemented the original algorithm of Collins and the subsequent improvement to projection by McCallum. Our implementation was in contrast to Maple's in-built CAD command, based on a quite separate theory. Although initially developed as an investigative tool to compare the algorithms, we found and reported that our code offered functionality not currently available in any other existing implementations. One particularly important piece of functionality is the ability to produce order-invariant CADs. This has allowed us to extend the implementation to produce CADs invariant with respect to either equational constraints (ECCADs) or the truth-tables of sequences of formulae (TTICADs). This new functionality is contained in the second release of our code, along with commands to consider problem formulation which can be a major factor in the tractability of a CAD. In the report we describe the new functionality and some theoretical discoveries it prompted. We describe how the CADs produced using equational constraints are able to take advantage of not just improved projection but also improvements in the lifting phase. We also present an extension to the original TTICAD algorithm which increases both the applicability of TTICAD and its relative benefit over other algorithms. The code and an introductory Maple worksheet / pdf demonstrating the full functionality of the package are freely available online.Comment: 12 pages; University of Bath, Dept. Computer Science Technical Report Series, 2013-02, 201

An Incremental Algorithm for Computing Cylindrical Algebraic Decompositions

In this paper, we propose an incremental algorithm for computing cylindrical algebraic decompositions. The algorithm consists of two parts: computing a complex cylindrical tree and refining this complex tree into a cylindrical tree in real space. The incrementality comes from the first part of the algorithm, where a complex cylindrical tree is constructed by refining a previous complex cylindrical tree with a polynomial constraint. We have implemented our algorithm in Maple. The experimentation shows that the proposed algorithm outperforms existing ones for many examples taken from the literature

Real Root Isolation of Regular Chains

We present an algorithm RealRootIsolate for isolating the real roots of a system of multivariate polynomials given by a zerodimensional squarefree regular chain. The output of the algorithm is guaranteed in the sense that all real roots are obtained and are described by boxes of arbitrary precision. Real roots are encoded with a hybrid representation, combining a symbolic object, namely a regular chain, and a numerical approximation given by intervals. Our isolation algorithm is a generalization, for regular chains, of the algorithm proposed by Collins and Akritas. We have implemented RealRootIsolate as a command of the module SemiAlgebraicSetTools of the RegularChains library in Maple. Benchmarks are reported.

Determinantal sets, singularities and application to optimal control in medical imagery

Control theory has recently been involved in the field of nuclear magnetic resonance imagery. The goal is to control the magnetic field optimally in order to improve the contrast between two biological matters on the pictures. Geometric optimal control leads us here to analyze mero-morphic vector fields depending upon physical parameters , and having their singularities defined by a deter-minantal variety. The involved matrix has polynomial entries with respect to both the state variables and the parameters. Taking into account the physical constraints of the problem, one needs to classify, with respect to the parameters, the number of real singularities lying in some prescribed semi-algebraic set. We develop a dedicated algorithm for real root classification of the singularities of the rank defects of a polynomial matrix, cut with a given semi-algebraic set. The algorithm works under some genericity assumptions which are easy to check. These assumptions are not so restrictive and are satisfied in the aforementioned application. As more general strategies for real root classification do, our algorithm needs to compute the critical loci of some maps, intersections with the boundary of the semi-algebraic domain, etc. In order to compute these objects, the determinantal structure is exploited through a stratifi-cation by the rank of the polynomial matrix. This speeds up the computations by a factor 100. Furthermore, our implementation is able to solve the application in medical imagery, which was out of reach of more general algorithms for real root classification. For instance, computational results show that the contrast problem where one of the matters is water is partitioned into three distinct classes