5,940 research outputs found
On the asymptotic and practical complexity of solving bivariate systems over the reals
This paper is concerned with exact real solving of well-constrained,
bivariate polynomial systems. The main problem is to isolate all common real
roots in rational rectangles, and to determine their intersection
multiplicities. We present three algorithms and analyze their asymptotic bit
complexity, obtaining a bound of \sOB(N^{14}) for the purely projection-based
method, and \sOB(N^{12}) for two subresultant-based methods: this notation
ignores polylogarithmic factors, where bounds the degree and the bitsize of
the polynomials. The previous record bound was \sOB(N^{14}).
Our main tool is signed subresultant sequences. We exploit recent advances on
the complexity of univariate root isolation, and extend them to sign evaluation
of bivariate polynomials over two algebraic numbers, and real root counting for
polynomials over an extension field. Our algorithms apply to the problem of
simultaneous inequalities; they also compute the topology of real plane
algebraic curves in \sOB(N^{12}), whereas the previous bound was
\sOB(N^{14}).
All algorithms have been implemented in MAPLE, in conjunction with numeric
filtering. We compare them against FGB/RS, system solvers from SYNAPS, and
MAPLE libraries INSULATE and TOP, which compute curve topology. Our software is
among the most robust, and its runtimes are comparable, or within a small
constant factor, with respect to the C/C++ libraries.
Key words: real solving, polynomial systems, complexity, MAPLE softwareComment: 17 pages, 4 algorithms, 1 table, and 1 figure with 2 sub-figure
Bases for bivariate spline spaces : a constructive approach
We give a basis for bivariate spline spaces on crosscut partitions which can be computed without solving systems of linear equations. In particular, we develop a recursion formula for these basis functions that cannot be written as polynomials or truncated power functions
A Generic Position Based Method for Real Root Isolation of Zero-Dimensional Polynomial Systems
We improve the local generic position method for isolating the real roots of
a zero-dimensional bivariate polynomial system with two polynomials and extend
the method to general zero-dimensional polynomial systems. The method mainly
involves resultant computation and real root isolation of univariate polynomial
equations. The roots of the system have a linear univariate representation. The
complexity of the method is for the bivariate case, where
, resp., is an upper bound on the degree, resp., the
maximal coefficient bitsize of the input polynomials. The algorithm is
certified with probability 1 in the multivariate case. The implementation shows
that the method is efficient, especially for bivariate polynomial systems.Comment: 24 pages, 5 figure
Roots of bivariate polynomial systems via determinantal representations
We give two determinantal representations for a bivariate polynomial. They
may be used to compute the zeros of a system of two of these polynomials via
the eigenvalues of a two-parameter eigenvalue problem. The first determinantal
representation is suitable for polynomials with scalar or matrix coefficients,
and consists of matrices with asymptotic order , where is the degree
of the polynomial. The second representation is useful for scalar polynomials
and has asymptotic order . The resulting method to compute the roots of
a system of two bivariate polynomials is competitive with some existing methods
for polynomials up to degree 10, as well as for polynomials with a small number
of terms.Comment: 22 pages, 9 figure
An Elimination Method for Solving Bivariate Polynomial Systems: Eliminating the Usual Drawbacks
We present an exact and complete algorithm to isolate the real solutions of a
zero-dimensional bivariate polynomial system. The proposed algorithm
constitutes an elimination method which improves upon existing approaches in a
number of points. First, the amount of purely symbolic operations is
significantly reduced, that is, only resultant computation and square-free
factorization is still needed. Second, our algorithm neither assumes generic
position of the input system nor demands for any change of the coordinate
system. The latter is due to a novel inclusion predicate to certify that a
certain region is isolating for a solution. Our implementation exploits
graphics hardware to expedite the resultant computation. Furthermore, we
integrate a number of filtering techniques to improve the overall performance.
Efficiency of the proposed method is proven by a comparison of our
implementation with two state-of-the-art implementations, that is, LPG and
Maple's isolate. For a series of challenging benchmark instances, experiments
show that our implementation outperforms both contestants.Comment: 16 pages with appendix, 1 figure, submitted to ALENEX 201
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