4,010 research outputs found
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
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
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
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
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
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
Using the distribution of cells by dimension in a cylindrical algebraic decomposition
We investigate the distribution of cells by dimension in cylindrical
algebraic decompositions (CADs). We find that they follow a standard
distribution which seems largely independent of the underlying problem or CAD
algorithm used. Rather, the distribution is inherent to the cylindrical
structure and determined mostly by the number of variables.
This insight is then combined with an algorithm that produces only
full-dimensional cells to give an accurate method of predicting the number of
cells in a complete CAD. Since constructing only full-dimensional cells is
relatively inexpensive (involving no costly algebraic number calculations) this
leads to heuristics for helping with various questions of problem formulation
for CAD, such as choosing an optimal variable ordering. Our experiments
demonstrate that this approach can be highly effective.Comment: 8 page
Program Verification in the presence of complex numbers, functions with branch cuts etc
In considering the reliability of numerical programs, it is normal to "limit
our study to the semantics dealing with numerical precision" (Martel, 2005). On
the other hand, there is a great deal of work on the reliability of programs
that essentially ignores the numerics. The thesis of this paper is that there
is a class of problems that fall between these two, which could be described as
"does the low-level arithmetic implement the high-level mathematics". Many of
these problems arise because mathematics, particularly the mathematics of the
complex numbers, is more difficult than expected: for example the complex
function log is not continuous, writing down a program to compute an inverse
function is more complicated than just solving an equation, and many algebraic
simplification rules are not universally valid.
The good news is that these problems are theoretically capable of being
solved, and are practically close to being solved, but not yet solved, in
several real-world examples. However, there is still a long way to go before
implementations match the theoretical possibilities
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
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
- …