Efficient Evaluation of Large Polynomials

In scientific computing, it is often required to evaluate a polynomial expression (or a matrix depending on some variables) at many points which are not known in advance or with coordinates containing “symbolic expressions”. In these circumstances, standard evaluation schemes, such as those based on Fast Fourier Transforms do not apply. Given a polynomial f expressed as the sum of its terms, we propose an algorithm which generates a representation of f optimizing the process of evaluating f at some points. In addition, this evaluation of f can be done efficiently in terms of data locality and parallelism. We have implemented our algorithm in the Cilk++ concurrency platform and our implementation achieves nearly linear speedup on 16 cores with large enough input. For some large polynomials, the generated schedule can be evaluated at least 10 times faster than the schedules produced by other available software solutions. Moreover, our code can handle much larger input polynomials

On the complexity of Generalized Chromatic Polynomials

J. Makowsky and B. Zilber (2004) showed that many variations of graph colorings, called CP-colorings in the sequel, give rise to graph polynomials. This is true in particular for harmonious colorings, convex colorings, mcct-colorings, and rainbow colorings, and many more. N. Linial (1986) showed that the chromatic polynomial �(G;X) is #P-hard to evaluate for all but three values X = 0, 1, 2, where evaluation is in P. This dichotomy includes evaluation at real or complex values, and has the further property that the set of points for which evaluation is in P is finite. We investigate how the complexity of evaluating univariate graph polynomials that arise from CPcolorings varies for different evaluation points. We show that for some CP-colorings (harmonious, convex) the complexity of evaluation follows a similar pattern to the chromatic polynomial. However, in other cases (proper edge colorings, mcct-colorings, H-free colorings) we could only obtain a dichotomy for evaluations at non-negative integer points. We also discuss some CP-colorings where we only have very partial results

Fast Mesh Refinement in Pseudospectral Optimal Control

Mesh refinement in pseudospectral (PS) optimal control is embarrassingly easy --- simply increase the order $N$ of the Lagrange interpolating polynomial and the mathematics of convergence automates the distribution of the grid points. Unfortunately, as $N$ increases, the condition number of the resulting linear algebra increases as $N^2$; hence, spectral efficiency and accuracy are lost in practice. In this paper, we advance Birkhoff interpolation concepts over an arbitrary grid to generate well-conditioned PS optimal control discretizations. We show that the condition number increases only as $\sqrt{N}$ in general, but is independent of $N$ for the special case of one of the boundary points being fixed. Hence, spectral accuracy and efficiency are maintained as $N$ increases. The effectiveness of the resulting fast mesh refinement strategy is demonstrated by using \underline{polynomials of over a thousandth order} to solve a low-thrust, long-duration orbit transfer problem.Comment: 27 pages, 12 figures, JGCD April 201

The complexity of class polynomial computation via floating point approximations

We analyse the complexity of computing class polynomials, that are an important ingredient for CM constructions of elliptic curves, via complex floating point approximations of their roots. The heart of the algorithm is the evaluation of modular functions in several arguments. The fastest one of the presented approaches uses a technique devised by Dupont to evaluate modular functions by Newton iterations on an expression involving the arithmetic-geometric mean. It runs in time $O (|D| \log^5 |D| \log \log |D|) = O (|D|^{1 + \epsilon}) = O (h^{2 + \epsilon})$ for any $\epsilon > 0$, where $D$ is the CM discriminant and $h$ is the degree of the class polynomial. Another fast algorithm uses multipoint evaluation techniques known from symbolic computation; its asymptotic complexity is worse by a factor of $\log |D|$. Up to logarithmic factors, this running time matches the size of the constructed polynomials. The estimate also relies on a new result concerning the complexity of enumerating the class group of an imaginary-quadratic order and on a rigorously proven upper bound for the height of class polynomials

STRINGVACUA: A Mathematica Package for Studying Vacuum Configurations in String Phenomenology

We give a simple tutorial introduction to the Mathematica package STRINGVACUA, which is designed to find vacua of string-derived or inspired four-dimensional N=1 supergravities. The package uses powerful algebro-geometric methods, as implemented in the free computer algebra system Singular, but requires no knowledge of the mathematics upon which it is based. A series of easy-to-use Mathematica modules are provided which can be used both in string theory and in more general applications requiring fast polynomial computations. The use of these modules is illustrated throughout with simple examples.Comment: 21 pages, 9 figure

Symbolic-Numeric Tools for Analytic Combinatorics in Several Variables

Analytic combinatorics studies the asymptotic behaviour of sequences through the analytic properties of their generating functions. This article provides effective algorithms required for the study of analytic combinatorics in several variables, together with their complexity analyses. Given a multivariate rational function we show how to compute its smooth isolated critical points, with respect to a polynomial map encoding asymptotic behaviour, in complexity singly exponential in the degree of its denominator. We introduce a numerical Kronecker representation for solutions of polynomial systems with rational coefficients and show that it can be used to decide several properties (0 coordinate, equal coordinates, sign conditions for real solutions, and vanishing of a polynomial) in good bit complexity. Among the critical points, those that are minimal---a property governed by inequalities on the moduli of the coordinates---typically determine the dominant asymptotics of the diagonal coefficient sequence. When the Taylor expansion at the origin has all non-negative coefficients (known as the `combinatorial case') and under regularity conditions, we utilize this Kronecker representation to determine probabilistically the minimal critical points in complexity singly exponential in the degree of the denominator, with good control over the exponent in the bit complexity estimate. Generically in the combinatorial case, this allows one to automatically and rigorously determine asymptotics for the diagonal coefficient sequence. Examples obtained with a preliminary implementation show the wide applicability of this approach.Comment: As accepted to proceedings of ISSAC 201