A proof of convergence and an error bound for the method of bisection in $\textbf {R}^{n}$
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- by Baker Kearfott PDF
- Math. Comp. 32 (1978), 1147-1153 Request permission
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 $p/m$.References
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P. ALEXANDROFF & H. HOPF, Topologie, Chelsea, New York, 1935; reprinted 1973.
- Marvin J. Greenberg, Lectures on algebraic topology, W. A. Benjamin, Inc., New York-Amsterdam, 1967. MR 0215295 R. B. KEARFOTT, Computing the Degree of Maps and a Generalized Method of Bisection, Ph. D. dissertation, Univ. of Utah, 1977.
- Baker Kearfott, An efficient degree-computation method for a generalized method of bisection, Numer. Math. 32 (1979), no. 2, 109–127. MR 529902, DOI 10.1007/BF01404868
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Additional Information
- © Copyright 1978 American Mathematical Society
- Journal: Math. Comp. 32 (1978), 1147-1153
- MSC: Primary 65H10
- DOI: https://doi.org/10.1090/S0025-5718-1978-0494897-3
- MathSciNet review: 0494897