Searcharxiv⌕ Search

arXiv subjects

Marc Mezzarobba

Publications and source records attributed to Marc Mezzarobba.

18 recordsLinked to original sources
arXiv2026

Rigorous numerical computation of the Stokes multipliers for linear differential equations with single level one

We describe a practical algorithm for computing the Stokes multipliers of a linear differential equation with polynomial coefficients at an irregular singular point of single level one. The algorithm follows a classical approach based on Borel summation and numerical ODE solving, but avoids a large amount of redundant work compared to a direct implementation. It applies to differential equations of arbitrary order, with no genericity assumption, and is suited to high-precision computations. In addition, we present an open-source implementation of this algorithm in the SageMath computer algebra system and illustrate its use with several examples. Our implementation supports arbitrary-precision computations and automatically provides rigorous error bounds. The article assumes minimal prior knowledge of the asymptotic theory of meromorphic differential equations and provides an elementary introduction to the linear Stokes phenomenon that may be of independent interest.

cs.MS↗
arXiv2024

First-order factors of linear Mahler operators

We develop and compare two algorithms for computing first-order right-hand factors in the ring of linear Mahler operators$\ell_r M^r + \dots + \ell_1 M + \ell_0$where $\ell_0, \dots, \ell_r$ are polynomials in~$x$ and $Mx = x^b M$ for some integer $b \geq 2$. In other words, we give algorithms for finding all formal infinite product solutions of linear functional equations$\ell_r(x) f(x^{b^r}) + \dots + \ell_1(x) f(x^b) + \ell_0(x) f(x) = 0$. The first of our algorithms is adapted from Petkov\v{s}ek's classical algorithm forthe analogous problem in the case of linear recurrences. The second one proceeds by computing a basis of generalized power series solutions of the functional equation and by using Hermite-Pad{\'e} approximants to detect those linear combinations of the solutions that correspond to first-order factors. We present implementations of both algorithms and discuss their use in combination with criteria from the literature to prove the differential transcendence of power series solutions of Mahler equations.

cs.SC↗
arXiv2023

Computing error bounds for asymptotic expansions of regular P-recursive sequences

Over the last several decades, improvements in the fields of analytic combinatorics and computer algebra have made determining the asymptotic behaviour of sequences satisfying linear recurrence relations with polynomial coefficients largely a matter of routine, under assumptions that hold often in practice. The algorithms involved typically take a sequence, encoded by a recurrence relation and initial terms, and return the leading terms in an asymptotic expansion up to a big-O error term. Less studied, however, are effective techniques giving an explicit bound on asymptotic error terms. Among other things, such explicit bounds typically allow the user to automatically prove sequence positivity (an active area of enumerative and algebraic combinatorics) by exhibiting an index when positive leading asymptotic behaviour dominates any error terms. In this article, we present a practical algorithm for computing such asymptotic approximations with rigorous error bounds, under the assumption that the generating series of the sequence is a solution of a differential equation with regular (Fuchsian) dominant singularities. Our algorithm approximately follows the singularity analysis method of Flajolet and Odlyzko, except that all big-O terms involved in the derivation of the asymptotic expansion are replaced by explicit error terms. The computation of the error terms combines analytic bounds from the literature with effective techniques from rigorous numerics and computer algebra. We implement our algorithm in the SageMath computer algebra system and exhibit its use on a variety of applications (including our original motivating example, solution uniqueness in the Canham model for the shape of genus one biomembranes).

cs.SC↗
arXiv2023

Rounding Error Analysis of Linear Recurrences Using Generating Series

We develop a toolbox for the error analysis of linear recurrences with constant or polynomial coefficients, based on generating series, Cauchy's method of majorants, and simple results from analytic combinatorics. We illustrate the power of the approach by several nontrivial application examples. Among these examples are a new worst-case analysis of an algorithm for computing Bernoulli numbers, and a new algorithm for evaluating differentially finite functions in interval arithmetic while avoiding interval blow-up.

math.NA↗
arXiv2022

Symbolic-Numeric Factorization of Differential Operators

We present a symbolic-numeric Las Vegas algorithm for factoring Fuchsian ordinary differential operators with rational function coefficients. The new algorithm combines ideas of van Hoeij's "local-to-global" method and of the ''analytic'' approach proposed by van der Hoeven. It essentially reduces to the former in ''easy'' cases where the local-to-global method succeeds, and to an optimized variant of the latter in the "hardest" cases, while handling intermediate cases more efficiently than both.

cs.SC↗
arXiv2021

Fast evaluation of some p-adic transcendental functions

We design algorithms for computing values of many p-adic elementary and special functions, including logarithms, exponentials, polylogarithms, and hypergeometric functions. All our algorithms feature a quasi-linear complexity with respect to the target precision and most of them are based on an adaptation to the-adic setting of the binary splitting and bit-burst strategies.

cs.SC↗
arXiv2020

Sequence Positivity Through Numeric Analytic Continuation: Uniqueness of the Canham Model for Biomembranes

We prove solution uniqueness for the genus one Canham variational problem arising in the shape prediction of biomembranes. The proof builds on a result of Yu and Chen that reduces the variational problem to proving non-negativity of a sequence defined by a linear recurrence relation with polynomial coefficients. We combine rigorous numeric analytic continuation of D-finite functions with classic bounds from singularity analysis to derive an effective index where the asymptotic behaviour of the sequence, which is positive, dominates the sequence behaviour. Positivity of the finite number of remaining terms is then checked computationally.

math.CO↗
arXiv2020

The expected signature of Brownian motion stopped on the boundary of a circle has finite radius of convergence

The expected signature is an analogue of the Laplace transform for rough paths. Chevyrev and Lyons showed that, under certain moment conditions, the expected signature determines the laws of signatures. Lyons and Ni posed the question of whether the expected signature of Brownian motion up to the exit time of a domain satisfies Chevyrev and Lyons' moment condition. We provide the first example where the answer is negative.

math.PR↗
arXiv2019

Computing the volume of compact semi-algebraic sets

Let $S\subset R^n$ be a compact basic semi-algebraic set defined as the real solution set of multivariate polynomial inequalities with rational coefficients. We design an algorithm which takes as input a polynomial system defining $S$ and an integer $p\geq 0$ and returns the $n$-dimensional volume of $S$ at absolute precision $2^{-p}$.Our algorithm relies on the relationship between volumes of semi-algebraic sets and periods of rational integrals. It makes use of algorithms computing the Picard-Fuchs differential equation of appropriate periods, properties of critical points, and high-precision numerical integration of differential equations.The algorithm runs in essentially linear time with respect to~$p$. This improves upon the previous exponential bounds obtained by Monte-Carlo or moment-based methods. Assuming a conjecture of Dimca, the arithmetic cost of the algebraic subroutines for computing Picard-Fuchs equations and critical points is singly exponential in $n$ and polynomial in the maximum degree of the input.

cs.SC↗
arXiv2018

Computing solutions of linear Mahler equations

Mahler equations relate evaluations of the same function $f$ at iterated $b$th powers of the variable. They arise in particular in the study of automatic sequences and in the complexity analysis of divide-and-conquer algorithms. Recently, the problem of solving Mahler equations in closed form has occurred in connection with number-theoretic questions. A difficulty in the manipulation of Mahler equations is the exponential blow-up of degrees when applying a Mahler operator to a polynomial. In this work, we present algorithms for solving linear Mahler equations for series, polynomials, and rational functions, and get polynomial-time complexity under a mild assumption. Incidentally, we develop an algorithm for computing the gcrd of a family of linear Mahler operators.

cs.SC↗
arXiv2018

Fast and rigorous arbitrary-precision computation of Gauss-Legendre quadrature nodes and weights

We describe a strategy for rigorous arbitrary-precision evaluation of Legendre polynomials on the unit interval and its application in the generation of Gauss-Legendre quadrature rules. Our focus is on making the evaluation practical for a wide range of realistic parameters, corresponding to the requirements of numerical integration to an accuracy of about 100 to 100 000 bits. Our algorithm combines the summation by rectangular splitting of several types of expansions in terms of hypergeometric series with a fixed-point implementation of Bonnet's three-term recurrence relation. We then compute rigorous enclosures of the Gauss-Legendre nodes and weights using the interval Newton method. We provide rigorous error bounds for all steps of the algorithm. The approach is validated by an implementation in the Arb library, which achieves order-of-magnitude speedups over previous code for computing Gauss-Legendre rules with simultaneous high degree and precision.

math.NA↗
arXiv2014

Rigorous uniform approximation of D-finite functions using Chebyshev expansions

A wide range of numerical methods exists for computing polynomial approximations of solutions of ordinary differential equations based on Chebyshev series expansions or Chebyshev interpolation polynomials. We consider the application of such methods in the context of rigorous computing (where we need guarantees on the accuracy of the result), and from the complexity point of view. It is well-known that the order-n truncation of the Chebyshev expansion of a function over a given interval is a near-best uniform polynomial approximation of the function on that interval. In the case of solutions of linear differential equations with polynomial coefficients, the coefficients of the expansions obey linear recurrence relations with polynomial coefficients. Unfortunately, these recurrences do not lend themselves to a direct recursive computation of the coefficients, owing among other things to a lack of initial conditions. We show how they can nevertheless be used, as part of a validated process, to compute good uniform approximations of D-finite functions together with rigorous error bounds, and we study the complexity of the resulting algorithms. Our approach is based on a new view of a classical numerical method going back to Clenshaw, combined with a functional enclosure method.

cs.SC↗
arXiv2013

Multiple precision evaluation of the Airy Ai function with reduced cancellation

The series expansion at the origin of the Airy function Ai(x) is alternating and hence problematic to evaluate for x > 0 due to cancellation. Based on a method recently proposed by Gawronski, Müller, and Reinhard, we exhibit two functions F and G, both with nonnegative Taylor expansions at the origin, such that Ai(x) = G(x)/F(x). The sums are now well-conditioned, but the Taylor coefficients of G turn out to obey an ill-conditioned three-term recurrence. We use the classical Miller algorithm to overcome this issue. We bound all errors and our implementation allows an arbitrary and certified accuracy, that can be used, e.g., for providing correct rounding in arbitrary precision.

cs.SC↗
arXiv2013

Finding Hyperexponential Solutions of Linear ODEs by Numerical Evaluation

We present a new algorithm for computing hyperexponential solutions of ordinary linear differential equations with polynomial coefficients. The algorithm relies on interpreting formal series solutions at the singular points as analytic functions and evaluating them numerically at some common ordinary point. The numerical data is used to determine a small number of combinations of the formal series that may give rise to hyperexponential solutions.

cs.SC↗
arXiv2012

A Note on the Space Complexity of Fast D-Finite Function Evaluation

We state and analyze a generalization of the "truncation trick" suggested by Gourdon and Sebah to improve the performance of power series evaluation by binary splitting. It follows from our analysis that the values of D-finite functions (i.e., functions described as solutions of linear differential equations with polynomial coefficients) may be computed with error bounded by 2^(-p) in time O(p*(lg p)^(3+o(1))) and space O(p). The standard fast algorithm for this task, due to Chudnovsky and Chudnovsky, achieves the same time complexity bound but requires Θ(p*lg p) bits of memory.

cs.SC↗
arXiv2010

NumGfun: a Package for Numerical and Analytic Computation with D-finite Functions

This article describes the implementation in the software package NumGfun of classical algorithms that operate on solutions of linear differential equations or recurrence relations with polynomial coefficients, including what seems to be the first general implementation of the fast high-precision numerical evaluation algorithms of Chudnovsky & Chudnovsky. In some cases, our descriptions contain improvements over existing algorithms. We also provide references to relevant ideas not currently used in NumGfun.

cs.SC↗
arXiv2009

Effective Bounds for P-Recursive Sequences

We describe an algorithm that takes as input a complex sequence $(u_n)$ given by a linear recurrence relation with polynomial coefficients along with initial values, and outputs a simple explicit upper bound $(v_n)$ such that $|u_n| \leq v_n$ for all $n$. Generically, the bound is tight, in the sense that its asymptotic behaviour matches that of $u_n$. We discuss applications to the evaluation of power series with guaranteed precision.

cs.SC↗