36,047 research outputs found
Involutive Division Technique: Some Generalizations and Optimizations
In this paper, in addition to the earlier introduced involutive divisions, we
consider a new class of divisions induced by admissible monomial orderings. We
prove that these divisions are noetherian and constructive. Thereby each of
them allows one to compute an involutive Groebner basis of a polynomial ideal
by sequentially examining multiplicative reductions of nonmultiplicative
prolongations. We study dependence of involutive algorithms on the completion
ordering. Based on properties of particular involutive divisions two
computational optimizations are suggested. One of them consists in a special
choice of the completion ordering. Another optimization is related to
recomputing multiplicative and nonmultiplicative variables in the course of the
algorithm.Comment: 19 page
Thomas Decomposition of Algebraic and Differential Systems
In this paper we consider disjoint decomposition of algebraic and non-linear
partial differential systems of equations and inequations into so-called simple
subsystems. We exploit Thomas decomposition ideas and develop them into a new
algorithm. For algebraic systems simplicity means triangularity, squarefreeness
and non-vanishing initials. For differential systems the algorithm provides not
only algebraic simplicity but also involutivity. The algorithm has been
implemented in Maple
Tensor Computation: A New Framework for High-Dimensional Problems in EDA
Many critical EDA problems suffer from the curse of dimensionality, i.e. the
very fast-scaling computational burden produced by large number of parameters
and/or unknown variables. This phenomenon may be caused by multiple spatial or
temporal factors (e.g. 3-D field solvers discretizations and multi-rate circuit
simulation), nonlinearity of devices and circuits, large number of design or
optimization parameters (e.g. full-chip routing/placement and circuit sizing),
or extensive process variations (e.g. variability/reliability analysis and
design for manufacturability). The computational challenges generated by such
high dimensional problems are generally hard to handle efficiently with
traditional EDA core algorithms that are based on matrix and vector
computation. This paper presents "tensor computation" as an alternative general
framework for the development of efficient EDA algorithms and tools. A tensor
is a high-dimensional generalization of a matrix and a vector, and is a natural
choice for both storing and solving efficiently high-dimensional EDA problems.
This paper gives a basic tutorial on tensors, demonstrates some recent examples
of EDA applications (e.g., nonlinear circuit modeling and high-dimensional
uncertainty quantification), and suggests further open EDA problems where the
use of tensor computation could be of advantage.Comment: 14 figures. Accepted by IEEE Trans. CAD of Integrated Circuits and
System
Involutive Bases Algorithm Incorporating F5 Criterion
Faugere's F5 algorithm is the fastest known algorithm to compute Groebner
bases. It has a signature-based and an incremental structure that allow to
apply the F5 criterion for deletion of unnecessary reductions. In this paper,
we present an involutive completion algorithm which outputs a minimal
involutive basis. Our completion algorithm has a nonincremental structure and
in addition to the involutive form of Buchberger's criteria it applies the F5
criterion whenever this criterion is applicable in the course of completion to
involution. In doing so, we use the G2V form of the F5 criterion developed by
Gao, Guan and Volny IV. To compare the proposed algorithm, via a set of
benchmarks, with the Gerdt-Blinkov involutive algorithm (which does not apply
the F5 criterion) we use implementations of both algorithms done on the same
platform in Maple.Comment: 24 pages, 2 figure
Minimal Involutive Bases
In this paper we present an algorithm for construction of minimal involutive
polynomial bases which are Groebner bases of the special form. The most general
involutive algorithms are based on the concept of involutive monomial division
which leads to partition of variables into multiplicative and
non-multiplicative. This partition gives thereby the self-consistent
computational procedure for constructing an involutive basis by performing
non-multiplicative prolongations and multiplicative reductions. Every specific
involutive division generates a particular form of involutive computational
procedure. In addition to three involutive divisions used by Thomas, Janet and
Pommaret for analysis of partial differential equations we define two new ones.
These two divisions, as well as Thomas division, do not depend on the order of
variables. We prove noetherity, continuity and constructivity of the new
divisions that provides correctness and termination of involutive algorithms
for any finite set of input polynomials and any admissible monomial ordering.
We show that, given an admissible monomial ordering, a monic minimal involutive
basis is uniquely defined and thereby can be considered as canonical much like
the reduced Groebner basis.Comment: 22 page
Algorithmic Thomas Decomposition of Algebraic and Differential Systems
In this paper, we consider systems of algebraic and non-linear partial
differential equations and inequations. We decompose these systems into
so-called simple subsystems and thereby partition the set of solutions. For
algebraic systems, simplicity means triangularity, square-freeness and
non-vanishing initials. Differential simplicity extends algebraic simplicity
with involutivity. We build upon the constructive ideas of J. M. Thomas and
develop them into a new algorithm for disjoint decomposition. The given paper
is a revised version of a previous paper and includes the proofs of correctness
and termination of our decomposition algorithm. In addition, we illustrate the
algorithm with further instructive examples and describe its Maple
implementation together with an experimental comparison to some other
triangular decomposition algorithms.Comment: arXiv admin note: substantial text overlap with arXiv:1008.376
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