Question

Advantage of false position method over bisection method

Answer 1

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Regula falsi might be interpreted as computing a convex combination of the interval endpoints at each iteration. The pure method uses the inverse function values as weights,

c=|f(a)|−1a+|f(b)|−1b|f(a)|−1+|f(b)|−1

so that the endpoint with the smaller function value gets more weight. Because of f(a)f(b)<0, this can be rewritten in the usual way.

The Illinois and other variants modify these weights according to the iteration history

c=u|f(a)|−1a+v|f(b)|−1bu|f(a)|−1+v|f(b)|−1

so that the weight of the unmodified side is increased in each step, drawing the midpoint in its direction and thus eventually over the root that is (too) slowly approximated by the changing side of the interval.

Brents method, in contrast and among other things, performs explicit detection of stalling and inserts a bisection step to move also the other interval end.