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 $N$ 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

Separating linear forms and Rational Univariate Representations of bivariate systems

International audienceWe address the problem of solving systems of bivariate polynomials with integer coefficients. We first present an algorithm for computing a separating linear form of such systems, that is a linear combination of the variables that takes different values when evaluated at distinct (complex) solutions of the system. In other words, a separating linear form defines a shear of the coordinate system that sends the algebraic system in generic position, in the sense that no two distinct solutions are vertically aligned. The computation of such linear forms is at the core of most algorithms that solve algebraic systems by computing rational parameterizations of the solutions and, moreover, the computation of a separating linear form is the bottleneck of these algorithms, in terms of worst-case bit complexity. Given two bivariate polynomials of total degree at most $d$ with integer coefficients of bitsize at most~$\tau$, our algorithm computes a separating linear form {of bitsize $O(\log d)$} in \comp\ bit operations in the worst case, which decreases by a factor $d^2$ the best known complexity for this problem (where \sO refers to the complexity where polylogarithmic factors are omitted and $O_B$ refers to the bit complexity). We then present simple polynomial formulas for the Rational Univariate Representations (RURs) of such systems. {This yields that, given a separating linear form of bitsize $O(\log d)$, the corresponding RUR can be computed in worst-case bit complexity \sOB(d^7+d^6\tau) and that its coefficients have bitsize \sO(d^2+d\tau).} We show in addition that isolating boxes of the solutions of the system can be computed from the RUR with \sOB(d^{8}+d^7\tau) bit operations in the worst case. Finally, we show how a RUR can be used to evaluate the sign of a bivariate polynomial (of degree at most $d$ and bitsize at most $\tau$) at one real solution of the system in \sOB(d^{8}+d^7\tau) bit operations and at all the $\Theta(d^2)$ real solutions in only $O(d)$ times that for one solution

Cache-Friendly, Modular and Parallel Schemes For Computing Subresultant Chains

The RegularChains library in Maple offers a collection of commands for solving polynomial systems symbolically with taking advantage of the theory of regular chains. The primary goal of this thesis is algorithmic contributions, in particular, to high-performance computational schemes for subresultant chains and underlying routines to extend that of RegularChains in a C/C++ open-source library. Subresultants are one of the most fundamental tools in computer algebra. They are at the core of numerous algorithms including, but not limited to, polynomial GCD computations, polynomial system solving, and symbolic integration. When the subresultant chain of two polynomials is involved in a client procedure, not all polynomials of the chain, or not all coefficients of a given subresultant, may be needed. Based on that observation, we design so-called speculative and caching strategies which yield great performance improvements within our polynomial system solver. Our implementation of these techniques has been highly optimized. We have implemented optimized core arithmetic routines and multithreaded subresultant algorithms for univariate, bivariate and multivariate polynomials. We further examine memory access patterns and data locality for computing subresultants of multivariate polynomials, and study different optimization techniques for the fraction-free LU decomposition algorithm to compute subresultants based on determinant of Bezout matrices. Our code is publicly available at www.bpaslib.org as part of the Basic Polynomial Algebra Subprograms (BPAS) library that is mainly written in C, with concurrency support and user interfaces written in C++

Bivariate triangular decompositions in the presence of asymptotes

International audienceGiven two coprime polynomials $P$ and $Q$ in $\mathbb{Z}[x,y]$ of degree at most $d$ and coefficients of bitsize at most $\tau$, we address the problem of computing a triangular decomposition $\{(U_i(x),V_i(x,y))\}_{i\in\cal I}$ of the system $\{P,Q\}$.The state-of-the-art worst-case complexities for computing such triangular decompositions when thecurves defined by the input polynomials do not have common vertical asymptotes are $\widetilde{O}(d^4)$ for the arithmetic complexity and $\widetilde{O}_B(d^{6} +d^{5}\tau)$ for thebit complexity, where $\widetilde{O}$ refers to thecomplexity where polylogarithmic factors are omitted and $O_B$ refers to the bit complexity.We show that the same worst-case complexities can be achieved even when the curves defined by the input polynomials may have common vertical asymptotes.We actually present refined complexities, $\widetilde{O}(d_xd_y^3+d_x^2d_y^2)$ for the arithmetic complexity and $\widetilde{O}_B(d_x^3d_y^3 +(d_x^2d_y^3+d_xd_y^4)\tau)$ for the bit complexity, where $d_x$ and $d_y$ bound the degrees of $P$ and $Q$ in $x$ and $y$, respectively. We also prove that the total bitsize of the decomposition is in $\widetilde{O}((d_x^2d_y^3+d_xd_y^4)\tau)$

Bivariate Triangular Decompositions in the Presence of Asymptotes

Given two coprime polynomials $P$ and $Q$ in $\mathbb{Z}[x,y]$ of degree at most $d$ and coefficients of bitsize at most $\tau$, we address the problem of computing a triangular decomposition $\{(U_i(x),V_i(x,y))\}_{i\in\cal I}$ of the system $\{P,Q\}$. The state-of-the-art worst-case bit complexity for computing such triangular decompositions when the curves defined by the input polynomials do not have common vertical asymptotes is in $\tilde{O}_B(d^6+d^5\tau)$ [Bouzidi et al. 2015, Prop. 16], where $\tilde{O}$ refers to the complexity where polylogarithmic factors are omitted and $O_B$ refers to the bit complexity. We show that the same worst-case bit complexity can be achieved even when the curves defined by the input polynomials may have common vertical asymptotes.We actually present a refined bit complexity in $\tilde{O}_B(d_x^3d_y^3 +(d_x^2d_y^3+d_xd_y^4)\tau)$ where $d_x$ and $d_y$ bound the degrees of $P$ and $Q$ in $x$ and $y$, respectively. We also prove that the total bitsize of the decomposition is in $\tilde{O}((d_x^2d_y^3+d_xd_y^4)\tau)$

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

Towards faster real algebraic numbers

AbstractThis paper presents a new encoding scheme for real algebraic number manipulations which enhances current Axiom’s real closure. Algebraic manipulations are performed using different instantiations of sub-resultant-like algorithms instead of Euclidean-like algorithms. We use these algorithms to compute polynomial gcds and Bezout relations, to compute the roots and the signs of algebraic numbers. This allows us to work in the ring of real algebraic integers instead of the field of real algebraic numbers avoiding many denominators