13 Root finding by trial-and-error

25 Mar 2021

An elementary numerical problem is to find the root \(x_0\) of an equation

\[f(x_0) = 0.\]

The applications are manifold. For instance, if the function \(f\) is a derivative of another function \(f(x) = \frac{dg}{dx}\) then finding a root allows us to find extrema (maxima or mimima) or saddle points (depending on the values of the higher derivatives) of the function \(g\).

Trial-and-error root finding methods move along the function graph and, depending on current values, investigate new regions of the function. They continue until they either converge on an answer to a pre-set precision or fail (perhaps after a maximum number of steps). Trial-and-error search is a common technique for cases where analytic solutions are lacking or not practical.

We will study two algorithms: bisection and Newton-Raphson searching.

The Jupyter notebook 13-Root-finding.ipynb contains the lecture notes.¹ The derivations of the bisection and the Newton-Raphson algorithms (PDF) can be found in the slides. Skeleton code for in-class problem exercises can be found in the notebook 13-Root-finding-students.ipynb.²

Additional resources:

• Computational Modelling: Chapter 3.3
• Computational Physics: Chapter 7

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Footnotes