Numerische Mathematik

, Volume 70, Issue 1, pp 111–118 | Cite as

Bisection hardly ever converges linearly

  • Yves Nievergelt


For real functions that cross the unit interval, the method of bisection converges linearly if, but only if, the point of crossing is a diadic number where the function does not vanish, or, except for finitely many digits, its binary expansion coincides with that of one third or two thirds. Otherwise, the order of convergence remains undefined. If the point of crossing is one of Borel's normal real numbers (Lebesgue's measure of all of which equals one), then the sequence of ratios of two consecutive errors accumulates simultaneously at zero, one half, and negative infinity. Thus, in every finite sequence of estimates from the bisection, the last estimate need not be more accurate than the first one.

Mathematics Subject Classification (1991):65H05, 11K16 


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Copyright information

© Springer-Verlag Berlin Heidelberg 1995

Authors and Affiliations

  • Yves Nievergelt
    • 1
  1. 1.Department of Mathematics, MS 32, Eastern Washington University, Cheney, WA 99004-2415, USA US

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