6,808 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
Symmetric Subresultants and Applications
Schur's transforms of a polynomial are used to count its roots in the unit
disk. These are generalized them by introducing the sequence of symmetric
sub-resultants of two polynomials. Although they do have a determinantal
definition, we show that they satisfy a structure theorem which allows us to
compute them with a type of Euclidean division. As a consequence, a fast
algorithm based on a dichotomic process and FFT is designed. We prove also that
these symmetric sub-resultants have a deep link with Toeplitz matrices.
Finally, we propose a new algorithm of inversion for such matrices. It has the
same cost as those already known, however it is fraction-free and consequently
well adapted to computer algebra
Computing Real Roots of Real Polynomials ... and now For Real!
Very recent work introduces an asymptotically fast subdivision algorithm,
denoted ANewDsc, for isolating the real roots of a univariate real polynomial.
The method combines Descartes' Rule of Signs to test intervals for the
existence of roots, Newton iteration to speed up convergence against clusters
of roots, and approximate computation to decrease the required precision. It
achieves record bounds on the worst-case complexity for the considered problem,
matching the complexity of Pan's method for computing all complex roots and
improving upon the complexity of other subdivision methods by several
magnitudes.
In the article at hand, we report on an implementation of ANewDsc on top of
the RS root isolator. RS is a highly efficient realization of the classical
Descartes method and currently serves as the default real root solver in Maple.
We describe crucial design changes within ANewDsc and RS that led to a
high-performance implementation without harming the theoretical complexity of
the underlying algorithm.
With an excerpt of our extensive collection of benchmarks, available online
at http://anewdsc.mpi-inf.mpg.de/, we illustrate that the theoretical gain in
performance of ANewDsc over other subdivision methods also transfers into
practice. These experiments also show that our new implementation outperforms
both RS and mature competitors by magnitudes for notoriously hard instances
with clustered roots. For all other instances, we avoid almost any overhead by
integrating additional optimizations and heuristics.Comment: Accepted for presentation at the 41st International Symposium on
Symbolic and Algebraic Computation (ISSAC), July 19--22, 2016, Waterloo,
Ontario, Canad
Modular polynomials via isogeny volcanoes
We present a new algorithm to compute the classical modular polynomial Phi_n
in the rings Z[X,Y] and (Z/mZ)[X,Y], for a prime n and any positive integer m.
Our approach uses the graph of n-isogenies to efficiently compute Phi_n mod p
for many primes p of a suitable form, and then applies the Chinese Remainder
Theorem (CRT). Under the Generalized Riemann Hypothesis (GRH), we achieve an
expected running time of O(n^3 (log n)^3 log log n), and compute Phi_n mod m
using O(n^2 (log n)^2 + n^2 log m) space. We have used the new algorithm to
compute Phi_n with n over 5000, and Phi_n mod m with n over 20000. We also
consider several modular functions g for which Phi_n^g is smaller than Phi_n,
allowing us to handle n over 60000.Comment: corrected a typo in equation (14), 31 page
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