Chapter 11
Solving Nonlinear Equations
Nonlinear equations in physics may have no closed-form solution, several roots, or mathematical roots outside the physical regime. This chapter develops bisection and false position for roots that can be bracketed, then Newton’s method and the secant method, which can converge more rapidly from suitable starting guesses. It considers how to locate several solutions, choose a method, and judge whether a computed root is physically meaningful. Worked examples include Wien’s displacement law and the bound states of a finite square well.
In this chapter
- 11.1 Nonlinear equations in physics
- 11.2 Mathematical formulation and basic ideas
- 11.3 Bracketing methods: bisection and false position
- 11.4 Open methods: Newton and secant
- 11.5 Multiple roots and multiple solutions
- 11.6 Practical guidelines for root finding
- 11.7 Worked examples from physics
- 11.8 Computational exercises and projects
Companion resources
Codes
MATLAB, Python, C++, and Fortran programs for bisection, false position, Newton’s method, the secant method, and interval scanning for several roots.
View Chapter 11 codes →Exercises & Projects
Additional exercises and physics projects for choosing intervals and starting guesses, checking solutions, and interpreting their physical meaning.
View Chapter 11 exercises & projects →Resources
Benchmark equations, method-selection guidance, practical notes on convergence and failure, and selected references.
View Chapter 11 resources →Updates & Errata
Corrections, clarifications, and post-publication updates associated with Chapter 11.
View Chapter 11 updates & errata →