Formally Verified Approximations of Definite Integrals

@article{Mahboubi2018FormallyVA,
  title={Formally Verified Approximations of Definite Integrals},
  author={Assia Mahboubi and Guillaume Melquiond and Thomas Sibut-Pinote},
  journal={Journal of Automated Reasoning},
  year={2018},
  volume={62},
  pages={281-300}
}
Finding an elementary form for an antiderivative is often a difficult task, so numerical integration has become a common tool when it comes to making sense of a definite integral. Some of the numerical integration methods can even be made rigorous: not only do they compute an approximation of the integral value but they also bound its inaccuracy. Yet numerical integration is still missing from the toolbox when performing formal proofs in analysis. This paper presents an efficient method for… 

Approximations of Definite Integrals

This paper presents an efficient method for automatically computing and proving bounds on some definite integrals inside the Coq formal system, based on computing and evaluating antiderivatives of rigorous polynomial approximations, combined with an adaptive domain splitting.

A Certificate-Based Approach to Formally Verified Approximations

A library to verify rigorous approximations of univariate functions on real numbers, with the Coq proof assistant, and implements a technique of validation a posteriori based on the Banach fixed-point theorem.

Numerical integration in arbitrary-precision ball arithmetic

This work presents an implementation of arbitrary-precision numerical integration with rigorous error bounds in the Arb library, which combines adaptive bisection with adaptive Gaussian quadrature where error bounds are determined via complex magnitudes without evaluating derivatives.

Investigations in Computer-Aided Mathematics : Experimentation, Computation, and Certification. (Investigations en Mathématiques Assistées par Ordinateur : Expérimentation, Calcul et Certification)

This thesis presents three contributions to the topic of computer-assisted proofs: both in the case of proofs relying on computations and of formal proofs produced and verified using a piece of

A Verified ODE Solver and Smale's 14th Problem

  • Fabian Immler
  • Computer Science
    Ausgezeichnete Informatikdissertationen
  • 2018
This dissertation presents a formalization of ordinary differential equations (ODEs) and the verification of rigorous numerical algorithms in the interactive theorem prover Isabelle/HOL using Runge-Kutta methods and affine arithmetic.

Plotting in a Formally Verified Way

This paper investigates what it means for a plot to be correct and how to formally verify this property, and the Coq proof assistant is turned into a tool for plotting function graphs that are guaranteed to becorrect, by using reliable polynomial approximations.

Extensional constructive real analysis via locators

  • A. Booij
  • Mathematics, Computer Science
    Mathematical Structures in Computer Science
  • 2020
This work considers real numbers equipped with additional structure, which it is called a locator, and considers a certain locatedness structure on real numbers, which is reminiscent of computable analysis.

Towards justifying computer algebra algorithms in Isabelle/HOL

  • Wenda Li
  • Computer Science, Mathematics
  • 2019
This thesis proposes a library of real algebraic numbers, whose distinguishing features include a modular architecture and a sign determination algorithm requiring only rational arithmetic, and formalised various analytical results including Cauchy’s residue theorem and the bivariate case of the projection theorem of CAD.

A Verified ODE Solver and the Lorenz Attractor

A rigorous numerical algorithm is used to certify the computations that Tucker used to prove chaos for the Lorenz attractor, based on a formalization of a diverse variety of mathematics and algorithms.

Formal Verification for Numerical Computations, and the Other Way Around

It is shown here how to solve the differential equation ordinary problem using Euler-Mascheroni’s inequality principle.

References

SHOWING 1-10 OF 22 REFERENCES

Adaptive, self-validating numerical quadrature

Integrals of a function of a single variable can be expressed as the sum of a numerical quadrature rule and a remainder term. The quadrature rule is a linear combination of function values and

Proving Tight Bounds on Univariate Expressions with Elementary Functions in Coq

This paper presents a tactic for the Coq proof assistant that is designed to automatically and formally prove bounds on univariate expressions based on a formalization of floating-point and interval arithmetic, associated with an on-the-fly computation of Taylor expansions.

Formally Verified Computation of Enclosures of Solutions of Ordinary Differential Equations

A functional algorithm is implemented that computes enclosures of solutions of ODEs in the interactive theorem prover Isabelle/HOL and is based on the well-known Euler method, which abstracts discretization and round-off errors in the domain of affine forms.

Verification methods: rigorous results using floating-point arithmetic

  • S. Rump
  • Computer Science
    Acta Numerica
  • 2010
A main goal of this talk is to introduce the principles of how to design verification algorithms, and how these principles differ from those for traditional numerical algorithms.

Validated Numerics: A Short Introduction to Rigorous Computations

This book is an essential resource for those entering this fast-developing field, and it is also the ideal textbook for graduate students and advanced undergraduates needing an accessible introduction to the subject.

Coquelicot: A User-Friendly Library of Real Analysis for Coq

A user-friendly library that comes with a comprehensive set of theorems that cover not only these notions, but also some extensions such as parametric integrals, two-dimensional differentiability, asymptotic behaviors, and more.

Interval Tools for ODEs and DAEs

  • N. Nedialkov
  • Computer Science
    12th GAMM - IMACS International Symposium on Scientific Computing, Computer Arithmetic and Validated Numerics (SCAN 2006)
  • 2006
A major goal of the VNODE-LP work is to produce an interval solver such that its correctness can be verified by a human expert, similar to how mathematical results are certified for correctness.

A computer-verified monadic functional implementation of the integral

Minor arcs for Goldbach's problem

The ternary Goldbach conjecture states that every odd number n>=7 is the sum of three primes. The estimation of sums of the form \sum_{p\leq x} e(\alpha p), \alpha = a/q + O(1/q^2), has been a

Formally Verified Certificate Checkers for Hardest-to-Round Computation

This paper shows how certificates based on Hensel’s lemma can be added to an algorithm using lattice basis reduction so that the result of a computation can be formally checked in the Coq proof assistant.