15 research outputs found
Nearly Optimal Refinement of Real Roots of a Univariate Polynomial
International audienceWe assume that a real square-free polynomial has a degree , a maximum coefficient bitsize and a real root lying in an isolating interval and having no nonreal roots nearby (we quantify this assumption). Then we combine the {\em Double Exponential Sieve} algorithm (also called the {\em Bisection of the Exponents}), the bisection, and Newton iteration to decrease the width of this inclusion interval by a factor of . The algorithm has Boolean complexity . This substantially decreases the known bound and is optimal up to a polylogarithmic factor. Furthermore we readily extend our algorithm to support the same upper bound on the complexity of the refinement of real roots, for any , by incorporating the known efficient algorithms for multipoint polynomial evaluation. The main ingredient for the latter is an efficient algorithm for (approximate) polynomial division; we present a variation based on structured matrix computation with quasi-optimal Boolean complexity
Solving rank-constrained semidefinite programs in exact arithmetic
We consider the problem of minimizing a linear function over an affine
section of the cone of positive semidefinite matrices, with the additional
constraint that the feasible matrix has prescribed rank. When the rank
constraint is active, this is a non-convex optimization problem, otherwise it
is a semidefinite program. Both find numerous applications especially in
systems control theory and combinatorial optimization, but even in more general
contexts such as polynomial optimization or real algebra. While numerical
algorithms exist for solving this problem, such as interior-point or
Newton-like algorithms, in this paper we propose an approach based on symbolic
computation. We design an exact algorithm for solving rank-constrained
semidefinite programs, whose complexity is essentially quadratic on natural
degree bounds associated to the given optimization problem: for subfamilies of
the problem where the size of the feasible matrix is fixed, the complexity is
polynomial in the number of variables. The algorithm works under assumptions on
the input data: we prove that these assumptions are generically satisfied. We
also implement it in Maple and discuss practical experiments.Comment: Published at ISSAC 2016. Extended version submitted to the Journal of
Symbolic Computatio
Accelerated Approximation of the Complex Roots of a Univariate Polynomial (Extended Abstract)
International audienceHighly efficient and even nearly optimal algorithms have been developed for the classical problem of univariate polynomial root-finding (see, e.g., \cite{P95}, \cite{P02}, \cite{MNP13}, and the bibliography therein), but this is still an area of active research. By combining some powerful techniques developed in this area we devise new nearly optimal algorithms, whose substantial merit is their simplicity, important for the implementation
Nearly Optimal Computations with Structured Matrices
We estimate the Boolean complexity of multiplication of structured matrices
by a vector and the solution of nonsingular linear systems of equations with
these matrices. We study four basic most popular classes, that is, Toeplitz,
Hankel, Cauchy and Van-der-monde matrices, for which the cited computational
problems are equivalent to the task of polynomial multiplication and division
and polynomial and rational multipoint evaluation and interpolation. The
Boolean cost estimates for the latter problems have been obtained by Kirrinnis
in \cite{kirrinnis-joc-1998}, except for rational interpolation, which we
supply now. All known Boolean cost estimates for these problems rely on using
Kronecker product. This implies the -fold precision increase for the -th
degree output, but we avoid such an increase by relying on distinct techniques
based on employing FFT. Furthermore we simplify the analysis and make it more
transparent by combining the representation of our tasks and algorithms in
terms of both structured matrices and polynomials and rational functions. This
also enables further extensions of our estimates to cover Trummer's important
problem and computations with the popular classes of structured matrices that
generalize the four cited basic matrix classes.Comment: (2014-04-10
New Acceleration of Nearly Optimal Univariate Polynomial Root-findERS
Univariate polynomial root-finding has been studied for four millennia and is
still the subject of intensive research. Hundreds of efficient algorithms for
this task have been proposed. Two of them are nearly optimal. The first one,
proposed in 1995, relies on recursive factorization of a polynomial, is quite
involved, and has never been implemented. The second one, proposed in 2016,
relies on subdivision iterations, was implemented in 2018, and promises to be
practically competitive, although user's current choice for univariate
polynomial root-finding is the package MPSolve, proposed in 2000, revised in
2014, and based on Ehrlich's functional iterations. By proposing and
incorporating some novel techniques we significantly accelerate both
subdivision and Ehrlich's iterations. Moreover our acceleration of the known
subdivision root-finders is dramatic in the case of sparse input polynomials.
Our techniques can be of some independent interest for the design and analysis
of polynomial root-finders.Comment: 89 pages, 5 figures, 2 table
Efficient Sampling from Feasible Sets of SDPs and Volume Approximation
We present algorithmic, complexity, and implementation results on the problem
of sampling points from a spectrahedron, that is the feasible region of a
semidefinite program. Our main tool is geometric random walks. We analyze the
arithmetic and bit complexity of certain primitive geometric operations that
are based on the algebraic properties of spectrahedra and the polynomial
eigenvalue problem. This study leads to the implementation of a broad
collection of random walks for sampling from spectrahedra that experimentally
show faster mixing times than methods currently employed either in theoretical
studies or in applications, including the popular family of Hit-and-Run walks.
The different random walks offer a variety of advantages , thus allowing us to
efficiently sample from general probability distributions, for example the
family of log-concave distributions which arise in numerous applications. We
focus on two major applications of independent interest: (i) approximate the
volume of a spectrahedron, and (ii) compute the expectation of functions coming
from robust optimal control. We exploit efficient linear algebra algorithms and
implementations to address the aforemen-tioned computations in very high
dimension. In particular, we provide a C++ open source implementation of our
methods that scales efficiently, for the first time, up to dimension 200. We
illustrate its efficiency on various data sets