Understanding Brent's root finding method

[phantomvommand]

TL;DR Summary

I am unsure about the conditions for rejecting the secant or IQI method in favour of bisection

I am unsure about why in the case where bisect_flag == False, we should check b-1 - b-2. Is the objective not to check that we are halving the interval between our best guesses b, so it should be abs(b - b-1), regardless of whether the previous step was a bisection or not?

 

[pbuk]

I am unsure about why in the case where bisect_flag == False, we should check b-1 - b-2. Is the objective not to check that we are halving the interval between our best guesses b, so it should be abs(b - b-1), regardless of whether the previous step was a bisection or not?

According to p.50 of Brent's original publication (it is available on archive.org): "practical tests show that this [would slow down] convergence for well-behaved functions by performing unnecessary bisections".

Like many practical algorithms this is a compromise between optimizing performance in the majority of situations whilst avoiding poor performance (or even failure) in pathological cases.

Note that this question probably belongs in the Programming and Computer Science forum. I'll get it moved.