On Convergence Rates of Root-Seeking Methods

[emece67]

(03-08-2018 01:21 AM)emece67 Wrote:  Now I will check if the Taylor method used by Gerson can be used to get a 4th order method with only one division. Such a method, if not too complex, may be even faster than Halley.

I've tested this 4th order Taylor with only 1 division. The results are a little disappointing. At 100000 digits it behaves as the other 4th order method I've tested (Ostrowsky-Traub), the effect of avoiding a division is balanced by the more complex algorithm and also by needing one iteration more. At 1 million digits it gets faster than Ostrowsky-Traub and Newton-Raphson & comparable to Grau+Díaz-Barrero, but still slower than Halley.

The test at 10 million digits is now in progress (but it may take some days to complete).

Regards.