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Mathematics of Computation

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A proof of convergence and an error bound for the method of bisection in $ {\bf R}\sp{n}$

Author: Baker Kearfott
Journal: Math. Comp. 32 (1978), 1147-1153
MSC: Primary 65H10
MathSciNet review: 0494897
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Abstract: Let $ S = \langle {X_0},..,{X_m}\rangle $ be an m-simplex in $ {{\mathbf{R}}^n}$. We define "bisection" of S as follows. We find the longest edge $ \langle {X_i},{X_j}\rangle $ of S, calculate its midpoint $ M = ({X_i} + {X_j})/2$, and define two new m-simplexes $ {S_1}$ and $ {S_2}$ by replacing $ {X_i}$ by M or $ {X_j}$ by M.

Suppose we bisect $ {S_1}$ and $ {S_2}$, and continue the process for p iterations. It is shown that the diameters of the resulting Simplexes are no greater then $ {(\sqrt 3 /2)^{\left\lfloor {p/m} \right\rfloor }}$ times the diameter of the original simplex, where $ \left\lfloor {p/m} \right\rfloor $ is the largest integer less than or equal to $ p/m$.

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Article copyright: © Copyright 1978 American Mathematical Society

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