ENH: Only use Halley's adjustment to Newton if close enough.

[pvanmulbregt]

Closes gh-8881.

Halley's enhancement to the Newton-Raphson rule divides the step by (1 - 0.5 * f/f' * f''/f')
If (1 - 0.5 * f/f' * f''/f') is not close to 1, then the iteration could easily go in the wrong direction.
This only happens when the approximation is not that close. See gh-8881.

Algorithm change: If the denominator (1 - 0.5 * f/f' * f''/f') in the adjustment is not close to 1, don't use it, as the current approximation is not close enough to the root to guarantee that the necessary assumptions hold. Just fall-back to a Newton step.

Expected changes in behavior in existing usage:

1. Some convergence failures will be replaced with successes.
2. For an approximation now considered not close enough, the algorithm may take one or more Newton steps until close enough. This may lead to an increase in the number of iterations, or convergence to a different root.
3. In some cases, newton(f,...,fprime2=fpp, ) may no longer converge when previously it did. This may occur if the derivative has a zero between the current approximation and the root, or if the sign of f'(x0) and f'(root) are different. [E.g f(x) = 16*x*(x-1)*(x+1)+7, which has a single real root -1.17 and is situated on an upward sloping part of the graph. Starting at x0 in a positive but downward sloping part of the graph (say between -0.5 and 0.5), the algorithm may or may not converge.]
   I.e. If Newton's algorithm would be on shaky ground due to the presence of stationary points, then using the modification here may change some previous Halley convergences to failures. And vice-versa.

[rgommers]

Sounds like a reasonable change to me. Could you add the example of gh-8881 as a test?

[pvanmulbregt]

Added an example of gh-8881 as a test. Rebased on a recent master.

[rgommers]

Merged, thanks Paul!