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ArXiv:191006709v2 [mathHO] 16 Dec 2019

December 16 2019 Abstract This article provides a simple proof of the quadratic formula which also produces an efficient and natural method for solving general quadratic equations The derivation is computationally light and conceptually natural and has the potential to demystify quadratic equations for stu-dents worldwide 1 Introduction

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Fast and Correct SIMD Algorithms for Interval

4 Fr´ed´eric Goualard presence of inﬁnities and zeros we have to test the bounds in advance which leads to algorithms with 9 cases for the multiplication and up to 36 cases for the division [7] Wolﬀ von Gudenberg [2021] implemented interval arithmetic with SSE in-structions by using the “opposite trick” above

PDF	Gaol: NOT Just Another Interval Arithmetic Library Gaol 4 2 0 NOT Just Another Interval Arithmetic Library Edition 4 4 Last updated 21 May 2015 Frédéric Goualard Laboratoire d’Informatique de Nantes-Atlantique France

PDF	The Ins and Outs of Solving Quadratic Equations with Floating Floating-point arithmetic numerical errors robustness quadratic equation

Est-ce que les terminales auront maths en 2023 ?
Retour des maths obligatoires pour tous les lycéens dès la première à la rentrée 2023. À la rentrée 2023, tous les élèves de première et terminale retrouveront les mathématiques dans leur programme scolaire en tant que matière obligatoire et non plus comme une option.
Est-ce que les terminales auront maths en 2024 ?
les épreuves écrites dans les deux enseignements de spécialités conservés en terminale.
Ces épreuves auront lieu en juin 2024 (préparez-vous au mieux )
C'est quoi maths complémentaire ?
L'option "mathématiques complémentaires" est destinée prioritairement aux élèves qui, ayant suivi la spécialité mathématiques en première et ne souhaitant pas poursuivre cet enseignement en terminale, ont cependant besoin de compléter leurs connaissances mathématiques par un enseignement adapté à leur poursuite d'étude
La spécialité abandonnée en fin de Première, dans votre cas les Mathématiques, représente le plus fort coefficient du contrôle continu en voie générale.
En effet, si vous candidatez pour le Bac 2022, les Mathématiques seront coefficient 5.
Si vous candidatez pour le Bac 2023, les Mathématiques seront coefficient 8.

blog de jean-Paul GOUALARD · Cours de spécialité en Première (année 2023-2024) · Feuille d'exercices et corrigés donnés en mathématiques complémentaires (année Autres questions

KEYWORDS:

Floating-point arithmetic, numerical errors, robustness, quadratic equation d197for5662m48.cloudfront.net

3.10 Einarsson, 2005

Despite its name, the book “Accuracy and reliability in scientific computing,”29by Einarsson, seems only concerned, when considering the solving of a quadratic equation, by the potential cancellation when computing the solutions. His method is the citardauq Even a simple same as Menzel’s or Nievergelt’s, relying on the for one of the solutions. As

3.11 Press et al., 2007

The often-cited book “Numerical recipes: the art of scientific computing,”30 et al. by Press , has something to say about many numerical problems, the solving of quadratic equations being one of them. Sadly, it is disappointingly light on the matter, simply considering the cancellation problem when computing the solutions. Its solution is the one a

4 A REVIEW OF THE IMPLEMENTATIONS IN SOFTWARE

An article may dismissively consider some of the problems that can plague a quadratic equation solver, referring the reader to other publications or to “common sense” for details. A piece of software does not have that luxury, and must implement each and every strategy that is necessary to handle correctly all inputs. Not all programming languages

4.1 C++ Boost libraries

C++ The Boost libraries are a huge set of code that is oftentimes the antechamber for code that will be included into a future C++ standard of . As such, it is usually of high quality. boost::math::tools::quadratic_roots() Boost implements the function to solve quadratic equations with radicals. The comment in the header of the source code for that

(Δ = 2 − 4 errors

) with no extra precision. The documentation for the function acknowledges that by stating that “[the] may cause a discrete change in the number of roots. ”. Cancellation in computing the solutions is avoided by using the modified Fagnano’s formulas, as the Boost libraries do. d197for5662m48.cloudfront.net

4.5 Rust “roots” module

roots roots::find_roots_quadratic() The module of the Rust language ofers the function to solve quadratic equations. According to its source code, the algorithm used handles separately the linear case. The discriminant is computed with the naive formula without any extra precision, and no provision is made to avoid underflows and overflows. On the

FindQuadraticPolynomialRoots()

in the report, nor present in the code of the C++ function of the Scilab source code on github. d197for5662m48.cloudfront.net

5.1 Pitfall 1: Exceptional parameters

Handling exceptional parameters simply requires to detect them at the beginning of the function. Programming languages usually isnan() isinf() C C++ provide methods to determine whether a float is an NaN or an infinity (e.g., and , in and ). We also have to decide what we will return in that case. In our Julia implementation, we have chosen to retu

QuadraticEquation

and more, are present in our Julia package available on github. There are five categories: Fifteen quadratics with no solution, not even complex ones (an NaN or an infinity is among the parameters); Four quadratics with no real solution; Seventeen degenerate quadratics (at least one parameter is zero); One quadratic with one solution; Eighteen quad

QuadraticEquation

The code for all the algorithms presented in this article has been assembled into a Julia package, , which is available on github. The package also contains challenging tests for the various solvers. d197for5662m48.cloudfront.net

ACKNOWLEDGMENTS

Discussions with Dr. Christophe Jermann helped shape this article as it currently is. An invitation to present computer arithmetic pitfalls to high school French mathematics teachers by Pr. Magali Hersant and Dr. Emmanuel Desmontils, co-directors of the d197for5662m48.cloudfront.net

Institute for Research on Teaching Mathematics

(IREM des Pays de la Loire), was the initial impetus for the research presented therein. d197for5662m48.cloudfront.net

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PDF	Fast and Correct SIMD Algorithms for Interval Arithmetic Fast and Correct SIMD Algorithms for Interval Arithmetic Frédéric Goualard Université de Nantes Nantes Atlantique Universités CNRS LINA UMR 6241

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