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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 $f: {\cal D}\mapsto\bbfR$ be defined on a set containing all diadic numbers in $[0,1]$ and $f(0)< 0< f(1)$. From the starting values $a\sb 0= 0$, $b\sb 0= 1$ the bisection method converges linearly to its limit $r$ if, and only if, either $r$ is a diadic point of discontinuity where $f(r)\ne 0$, or $r= (2a\sb n+ b\sb n)/3$ 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.
65H05Single nonlinear equations (numerical methods)
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