Jasmina Mansour: Att lösa femtegradsekvationen: En jämförelse av Newton-Raphsons och Bring-Jerrards metoder

Abstract:

This essay consists of a literary study of two methods for solving quintic equations: the Newton-Raphson method and the Bring-Jerrard method. In the latter, we will also use Tschirnhaus transformation, the Lagrange inversion theorem, as well as Newton’s binominal theorem. Each method will be presented with a corresponding equation that exemplifies their respective procedure in application. The essay concludes with a comparative discussion of difficulties in applying the respective methods. The Newton-Raphson method suffers from problems of how to find viable initial starting points so that the process converges fast enough. In contrast, the Bring-Jerrard method has the issue of though producing an exact expression, this expression may still converge too slowly, and have to be continued by analytic continuation.