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Finding the roots of a non-linear equation: history and reliability

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Research in History and Philosophy of Mathematics

Abstract

Finding the roots of a non-linear equation \( f(x)=0 \) is one of the most commonly occurring problems of applied mathematics. This work concerns the nineteenth century history of the fixed point and the bisection methods of root finding. We present the linear convergence properties of the fixed point technique as explained by Sancery in 1862 and Schröder in 1870. The bisection method does not have the prestigious past of other methods of root finding, however because the bisection method is linked to the intermediate value theorem, we examine the Bolzano (1817), Cauchy (1821), and Sarrus (1841) approaches. We conclude by looking at some contemporary approaches to the problem.

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Notes

  1. 1.

    Note that Picard’s fixed point theorem concerns differential equations.

  2. 2.

    For example, this property M can correspond to \( f(x)<0 \), \( \forall x<U \).

  3. 3.

    Gaston Darboux, in his Mémoire sur les fonctions discontinues (1975), follows Cauchy’s method by dividing an interval (A, B) in n parts. It is clear that neither Cauchy nor Darboux was aware of Bolzano’s work.

  4. 4.

    Bradley said that Cauchy gave a first proof of the Intermediate Value Theorem in Chapter II, but it was rather unsatisfactory.

  5. 5.

    Already, in 1833, Sarrus published an article “Nouvelle méthode pour la résolution des équations numériques.”

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Authors and Affiliations

  1. Department of Mathematics and Computer Science, Royal Military College of Canada, 92 Florence Street, Kingston, ON, Canada, K7M 1Y6

    Roger Godard

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  1. Roger Godard
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Correspondence to Roger Godard .

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Editors and Affiliations

  1. Mathematical Information and Computer Sciences, Point Loma Nazarene University, San Diego, California, USA

    Maria Zack

  2. Department of Philosophy, University of California, Davis, Davis, California, USA

    Elaine Landry

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Godard, R. (2016). Finding the roots of a non-linear equation: history and reliability. In: Zack, M., Landry, E. (eds) Research in History and Philosophy of Mathematics. Proceedings of the Canadian Society for History and Philosophy of Mathematics/La Société Canadienne d’Histoire et de Philosophie des Mathématiques. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-46615-6_5

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