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Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.98.

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On faster convergence of the bisection method for all triangles
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by Martin Stynes PDF
Math. Comp. 35 (1980), 1195-1201 Request permission

Abstract:

Let $\Delta ABC$ be a triangle with vertices A, B, and C. It is "bisected" as follows: choose a/the longest side (say AB) of $\Delta ABC$, let D be the midpoint of AB, then replace $\Delta ABC$ by two triangles $\Delta ADC$ and $\Delta DBC$. Let ${\Delta _{01}}$ be a given triangle. Bisect ${\Delta _{01}}$ into two triangles ${\Delta _{11}}$ and ${\Delta _{12}}$. Next bisect each ${\Delta _{1i}},\;i = 1,2$, forming four new triangles ${\Delta _{2i}},\;i = 1,2,3,4$. Continue thus, forming an infinite sequence ${T_j},\;j = 0,1,2, \ldots$, of sets of triangles, where ${T_j} = \left \{ {{\Delta _{ji}}:1 \leqslant i \leqslant {2^j}} \right \}$. Let ${m_j}$ denote the mesh of ${T_j}$. It is shown that there exists $N = N({\Delta _{01}})$ such that, for $j \geqslant N$, ${m_{2j}} \leqslant {(\sqrt 3 /2)^N}{(1/2)^{j - N}}{m_0}$, thus greatly improving the previous best known bound of ${m_{2j}} \leqslant {(\sqrt 3 /2)^j}{m_0}$. It is also shown that only a finite number of distinct shapes occur among the triangles produced, and that, as the method proceeds, ${\Delta _{01}}$ tends to become covered by triangles which are approximately equilateral in a certain sense.
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Additional Information
  • © Copyright 1980 American Mathematical Society
  • Journal: Math. Comp. 35 (1980), 1195-1201
  • MSC: Primary 51N99; Secondary 65B99, 65N30
  • DOI: https://doi.org/10.1090/S0025-5718-1980-0583497-1
  • MathSciNet review: 583497