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Bisection hardly ever converges linearly. (English) Zbl 0822.65024
A real number is called diadic if it is the sum of finitely many integral powers of two. The following theorem is proved: Let f:𝒟 be defined on a set containing all diadic numbers in [0,1] and f(0)<0<f(1). From the starting values a 0 =0, b 0 =1 the bisection method converges linearly to its limit r if, and only if, either r is a diadic point of discontinuity where f(r)0, or r=(2a n +b n )/3 for some positive integer n. For all other limit points the method still converges but the order of convergence remains undefined. Note that for bisection to converge it suffices that f is defined at all diadic numbers in [0,1] and that it need neither be continuous nor measurable.
MSC:
65H05Single nonlinear equations (numerical methods)