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Zbl 0822.65024
Nievergelt, Yves
Bisection hardly ever converges linearly.
(English)
[J] Numer. Math. 70, No.1, 111-118 (1995). ISSN 0029-599X; ISSN 0945-3245/e

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.
[W.C.Rheinboldt (Pittsburgh)]
MSC 2000:
*65H05 Single nonlinear equations (numerical methods)

Keywords: bisection method; order of convergence

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