Variant of the Secant Method

[Albert Chan]

(12-12-2020 02:41 PM)ttw Wrote:  Methods with high convergence can have problems numerically; they also tend to have a smaller domain of convergence.

This might not always true.
In fact, recently we come across opposite case, LambertW function implementation.

Solving f(x) = x*e^x - a, with Newton's method is very unstable.

Higher convergence method, say Halley, it becomes "safe"

P.S. the next post, I had suggested another way, to suppress the exponential.
This version, Newton's method is fast, and safe.

>>> from mpmath import *
>>> a = -1e99
>>> findroot(lambda x: x + ln(x/a), 200+3j) # bad guess
mpc(real='222.55067066150301', imag='3.1275404175517207')
>>> lambertw(a)
mpc(real='222.55067066150301', imag='3.127540417551721')