Based on the paper Sometimes Newton's Method Cycles, we first asked ourselves if there were any Newtonian Method Cycle functions which have non-trivial guesses. We encountered a way to create functions that cycle between a set number of points with any initial, non-trivial guesses when Newton's Method is applied. We exercised these possibilities through the methods of 2-cycles, 3-cycles and 4-cycles. We then generalized these cycles into k-cycles. After generalizing Newton's Method, we found the conditions that skew the cycles into a spiral pattern which will either converge, diverge or become a near-cycle. Once we obtained all this information, we explored additional questions that rose up from our initial exploration of Newton's Method.
Particle Accelerator Society of Japan (PASJ)
Template of proceedings of annual meeting of PASJ
&coyp; Particle Accelerator Society of Japan. All Rights Reserved.
Dr Driver's standard reading quiz template, downloaded from the github gist on 20th August 2014.
The version on writeLaTeX has the commands for choosing the Meta and Meta Serif fonts commented out as these are not currently installed on the system. The template compiles with XeLaTeX.