# Solving Non-linear Equations

• Simon Širca
• Martin Horvat
Part of the Graduate Texts in Physics book series (GTP)

## Abstract

Solving non-linear equations and systems of such equations is one of the daily chores of a physicist or engineer. In this chapter basic techniques for scalar equations are explained: bisection, Newton–Raphson with optional convergence improvements, the secant and Müller’s method. Vector non-linear equations are treated by Newton–Raphson and Broyden’s method, illustrated by a robot engineering example. Solving polynomial equations of a single variable is described next, along with the efficient means of counting and locating the zeros. Special attention is given to the sensitivity of zeros to perturbations. The chapter ends with a discussion on how to solve algebraic equations of several variables (frequently occurring in automated assembly of mechanical systems, robot control, coding and cryptography), for which powerful algorithms based on Gröbner bases have been developed. The Examples and Problems include Kepler’s equation, Wien’s law of black-body radiation, Heisenberg’s model in the mean-field approximation, energy levels of one-dimensional quantum-mechanical systems, fluid flow through systems of pipes, and automated assembly of three-dimensional structures.

## Keywords

Jacobi Matrix Real Zero Raphson Method Quadratic Convergence Secant Method
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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