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Abstract:
We study polynomials of degree up to 4 over the rationals or a computable real subfield. Our motivation comes from the need to evaluate predicates in nonlinear computational geometry efficiently and exactly. We show a new method to compare real algebraic numbers by precomputing generalized Sturm sequences, thus avoiding iterative methods; the method, moreover handles all degenerate cases. Our first contribution is the determination of rational isolating points, as functions of the coefficients, between any pair of real roots. Our second contribution is to exploit invariants and Bezoutian subexpressions in writing the sequences, in order to reduce bit complexity. The degree of the tested quantities in the input coefficients is optimal for degree up to 3, and for degree 4 in certain cases. Our methods readily apply to real solving of pairs of quadratic equations, and to sign determination of polynomials over algebraic numbers of degree up to 4. Our third contribution is an implementation in a new module of library synaps v2.1. It improves significantly upon the efficiency of certain publicly available implementations: Rioboo's approach on axiom, the package of Guibas-Karavelas-Russel [11], and core v1.6, maple v9, and synaps v2.0. Some existing limited tests had shown that it is faster than commercial library leda v4.5 for quadratic algebraic numbers.
BibTeX:
@InProceedings{et-esa-04,
Author = {Ioannis~Z. Emiris and Elias~P. Tsigaridas},
Title = {Computing with real algebraic numbers of small
degree},
BookTitle = {Proc. 12th European Symp. of Algorithms (ESA)},
Series = "LNCS",
Editor = {S. Albers and T. Radzik},
Volume = 3221,
Pages = "652--663",
address = {Bergen, Norway},
month = {Sep 14--17},
Publisher = "Springer Verlag",
year = 2004,
abstract = "We study polynomials of degree up to 4 over the
rationals or a computable real subfield. Our
motivation comes from the need to evaluate
predicates in nonlinear computational geometry
efficiently and exactly. We show a new method to
compare real algebraic numbers by precomputing
generalized Sturm sequences, thus avoiding iterative
methods; the method, moreover handles all degenerate
cases. Our first contribution is the determination
of rational isolating points, as functions of the
coefficients, between any pair of real roots. Our
second contribution is to exploit invariants and
Bezoutian subexpressions in writing the sequences,
in order to reduce bit complexity. The degree of the
tested quantities in the input coefficients is
optimal for degree up to 3, and for degree 4 in
certain cases. Our methods readily apply to real
solving of pairs of quadratic equations, and to sign
determination of polynomials over algebraic numbers
of degree up to 4. Our third contribution is an
implementation in a new module of library synaps
v2.1. It improves significantly upon the efficiency
of certain publicly available implementations:
Rioboo's approach on axiom, the package of
Guibas-Karavelas-Russel [11], and core v1.6, maple
v9, and synaps v2.0. Some existing limited tests had
shown that it is faster than commercial library leda
v4.5 for quadratic algebraic numbers.",
}
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