Modulo partial mapping of fractal potential and field lines

Abstract

Faster methods for topological categorization and field line calculations are developed by using decomposition regions together with the self-winding techniques first developed in a prior patent application. A point iteration technique provides direct calculation of low order digits of winding counts without use of complex intervals. Easy to calculate derivatives define decomposition interval boundaries which substitute for methods using the slower complex interval processing of the prior patent. Methods common to this and the prior patent are developed for visualizing conformal mappings of iterated functions.

What is claimed is: 
     
         1 . A machine executed full modulo iterator method for creating a modulo point from a predetermined point iterate data, a predetermined point iterate goal, a predetermined winding digit adjustment, and a predetermined complex iteration expression, comprising:
 (A) providing a branchcut adjuster means to return a winding count adjustment for a predetermined receiver point by counting directional transversal of the negative axis branchcut arising from the shortest rotation angular adjustment to rotate from a predetermined donor point to said predetermined receiver point,   (B) providing a complex dominant splitter means for separating said predetermined complex iteration expression into a dominant expression as the term having the highest degree of an iteration variable and setting an inferior expression to the remainder of the expression,   (C) providing a self winding operator pool means for returning a winding operator from a pool of the winding operators based on a predetermined winding operator request,   (D) providing a self winding dominant stepper means for returning a winding point resulting from iterating said dominate expression in a predetermined iteration direction, using a predetermined point iterate data, by utilizing the winding operators from said self winding operator pool means,   (E) providing a modulo banding creator means to construct a modulo point from a predetermined modulo banding parameters,   (F) providing a full modulo resolver means to create a modulo point by calling said modulo banding creator means with said predetermined modulo banding parameters of (E) constructed by iterating said predetermined point iterate data to achieve said predetermined point iterate goal.   
     
     
         2 . The machine executed full modulo iterator method of  claim 1 , enhanced to eliminate self winding calls, with the new method called an optimized modulo iterator method, further including:
 (A) providing a traditional iterator means of returning a traditional result by evaluating a predetermined domain start point with said predetermined complex iteration expression using traditional evaluation of iterated functions until a predetermined escape distance is achieved,   (B) providing an optimized modulo resolver means to both call the traditional iterator means to obtain a partial iteration result called the traditional result and then to use said traditional result as part of a modified point iterate data with a winding point iterate goal constructed from the predetermined point iterate goal, as parameters to the full modulo iterator method to produce a modulo point as the final result.   
     
     
         3 . A programmed machine comprising:
 one or more processors;   machine storage that stores instructions for performing the method of  claim 1 .   
     
     
         4 . A programmed machine comprising:
 one or more processors;   machine storage that stores instructions for performing the method of  claim 2 .