231 research outputs found
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
A "Piano Movers" Problem Reformulated
It has long been known that cylindrical algebraic decompositions (CADs) can
in theory be used for robot motion planning. However, in practice even the
simplest examples can be too complicated to tackle. We consider in detail a
"Piano Mover's Problem" which considers moving an infinitesimally thin piano
(or ladder) through a right-angled corridor.
Producing a CAD for the original formulation of this problem is still
infeasible after 25 years of improvements in both CAD theory and computer
hardware. We review some alternative formulations in the literature which use
differing levels of geometric analysis before input to a CAD algorithm. Simpler
formulations allow CAD to easily address the question of the existence of a
path. We provide a new formulation for which both a CAD can be constructed and
from which an actual path could be determined if one exists, and analyse the
CADs produced using this approach for variations of the problem.
This emphasises the importance of the precise formulation of such problems
for CAD. We analyse the formulations and their CADs considering a variety of
heuristics and general criteria, leading to conclusions about tackling other
problems of this form.Comment: 8 pages. Copyright IEEE 201
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
Experience with Heuristics, Benchmarks & Standards for Cylindrical Algebraic Decomposition
The Complexity of Cylindrical Algebraic Decomposition with Respect to Polynomial Degree
Cylindrical algebraic decomposition (CAD) is an important tool for working
with polynomial systems, particularly quantifier elimination. However, it has
complexity doubly exponential in the number of variables. The base algorithm
can be improved by adapting to take advantage of any equational constraints
(ECs): equations logically implied by the input. Intuitively, we expect the
double exponent in the complexity to decrease by one for each EC. In ISSAC 2015
the present authors proved this for the factor in the complexity bound
dependent on the number of polynomials in the input. However, the other term,
that dependent on the degree of the input polynomials, remained unchanged.
In the present paper the authors investigate how CAD in the presence of ECs
could be further refined using the technology of Groebner Bases to move towards
the intuitive bound for polynomial degree
Iterated Resultants in CAD
Cylindrical Algebraic Decomposition (CAD) by projection and lifting requires
many iterated univariate resultants. It has been observed that these often
factor, but to date this has not been used to optimise implementations of CAD.
We continue the investigation into such factorisations, writing in the specific
context of SC-Square.Comment: Presented at the 2023 SC-Square Worksho
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
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