| Autor: |
Thomas Arildsen, Peter R. Turner, Kathleen Kavanagh |
| Rok vydání: |
2018 |
| Předmět: |
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| Zdroj: |
Texts in Computer Science ISBN: 9783319895741 |
| Popis: |
This chapter addresses one of the most fundamental mathematical problems: the solution of a scalar nonlinear equation, \(f(x)=0\). All the methods presented are iterative in nature. We begin with perhaps the simplest idea – using bisection to reduce an interval which we know contains a solution to an acceptable tolerance. Next, we then present Newton’s method which is based on where the tangent line at a particular point would cross the axis. Provided we can get a “good enough” starting point Newton’s method will converge very quickly to the desired solution of the equation. Building on both the bisection method and Newton’s method gives rise to the secant method. Now instead of simply halving the interval we look at the point where the chord of the graph between the two ends of the interval would cross the axis. The secant method can have significantly faster convergence than mere bisection. It is both an improvement on bisection and a difference approximation to Newton’s method that does not require knowledge of the derivative. Finally, we present the setting for systems of nonlinear equations. Two of our standard modeling examples are extended from one unknown to two. The details for implementing Newton’s method for two equations and two unknowns are provided (in addition to the scalar algorithms) so that you have the solvers all ready to go and tackle the applied problems in this chapter – and beyond. |
| Databáze: |
OpenAIRE |
| Externí odkaz: |
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