Heads up for a hot new root seeking algorithm!!

[Namir]

If you download the ZIP file from my web site, you get the report and an Excel file that has worksheets for the dozen test functions. What I observed is that in the majority of the cases, the methods of Halley and Ostrowski give the same results or don't differ by more than one iteration (both require three function calls per iteration). The results DO SHOW a consistent improvement over Newton's method. You can say that the methods of Halley and Ostrowski have a higher convergence rate than that of Newton. My new algorithm (which uses the Ostrowski's approach to enhance Halley) shows several cases where either the number of iterations or both the number of iterations AND number of function calls is less than that of Halley and Ostrowski. Since I developed the new algorithm using an empirical/heuristic approach I did not yield a long set of mathematical derivations that indicate the convergence rate. It may well be at least one order more than that of Ostrowski's method. It's hard to measure ... compared to, say determining the order of array sorting methods where you can widely vary the array size and calculate the number of array element comparisons and number of element swaps. See my article about enhancing the CombSort method where I was able to calculate the sort order of this method in Tables 3 and 4 of the article.