A numerical method for oscillatory integrals with coalescing saddle points

The value of a highly oscillatory integral is typically determined asymptotically by the behaviour of the integrand near a small number of critical points. These include the endpoints of the integration domain and the so-called stationary points or saddle points -- roots of the derivative of the phase of the integrand -- where the integrand is locally non-oscillatory. Modern methods for highly oscillatory quadrature exhibit numerical issues when two such saddle points coalesce. On the other hand, integrals with coalescing saddle points are a classical topic in asymptotic analysis, where they give rise to uniform asymptotic expansions in terms of the Airy function. In this paper we construct Gaussian quadrature rules that remain uniformly accurate when two saddle points coalesce. These rules are based on orthogonal polynomials in the complex plane. We analyze these polynomials, prove their existence for even degrees, and describe an accurate and efficient numerical scheme for the evaluation of oscillatory integrals with coalescing saddle points

Towards an exact adaptive algorithm for the determinant of a rational matrix

In this paper we propose several strategies for the exact computation of the determinant of a rational matrix. First, we use the Chinese Remaindering Theorem and the rational reconstruction to recover the rational determinant from its modular images. Then we show a preconditioning for the determinant which allows us to skip the rational reconstruction process and reconstruct an integer result. We compare those approaches with matrix preconditioning which allow us to treat integer instead of rational matrices. This allows us to introduce integer determinant algorithms to the rational determinant problem. In particular, we discuss the applicability of the adaptive determinant algorithm of [9] and compare it with the integer Chinese Remaindering scheme. We present an analysis of the complexity of the strategies and evaluate their experimental performance on numerous examples. This experience allows us to develop an adaptive strategy which would choose the best solution at the run time, depending on matrix properties. All strategies have been implemented in LinBox linear algebra library

A Generic Library for Floating-Point Numbers and Its Application to Exact Computing

International audienceIn this paper we present a general library to reason about floating-point numbers within the Coq system. Most of the results of the library are proved for an arbitrary floating-point format and an arbitrary base. A special emphasis has been put on proving properties for exact computing, i.e. computing without rounding errors

Combinatorial summation of Feynman diagrams: Equation of state of the 2D SU(N) Hubbard model

We introduce a universal framework for efficient summation of connected or skeleton Feynman diagrams for generic quantum many-body systems. It is based on explicit combinatorial construction of the sum of the integrands by dynamic programming, at a computational cost that can be made only exponential in the diagram order. We illustrate the technique by an unbiased diagrammatic Monte Carlo calculation of the equation of state of the $2D$ $SU(N)$ Hubbard model in an experimentally relevant regime, which has remained challenging for state-of-the-art numerical methods.Comment: 8 pages, 4 figure

Some applications of computing to number theory

This thesis describes the application of a computer to certain problems in number theory. Chapter 1 is a general description of the use of computers in this field. Chapter 2 contains the results of computations relating to certain non - congruence subgroups of the full modular group, while chapter 3 describes the use of variable precision arithmetic, mainly in connection with continued fraction expansions. The last chapter describes a small library of subroutines, the majority of which are written in Fortran. A computer - printed listing of these subroutines is given at the end of the thesis