A proof of convergence and an error bound for the method of bisection in

Author:
Baker Kearfott

Journal:
Math. Comp. **32** (1978), 1147-1153

MSC:
Primary 65H10

DOI:
https://doi.org/10.1090/S0025-5718-1978-0494897-3

MathSciNet review:
0494897

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Abstract | References | Similar Articles | Additional Information

Abstract: Let be an *m*-simplex in . We define "bisection" of *S* as follows. We find the longest edge of *S*, calculate its midpoint , and define two new *m*-simplexes and by replacing by *M* or by *M*.

Suppose we bisect and , and continue the process for *p* iterations. It is shown that the diameters of the resulting Simplexes are no greater then times the diameter of the original simplex, where is the largest integer less than or equal to .

**[1]**P. ALEXANDROFF & H. HOPF,*Topologie*, Chelsea, New York, 1935; reprinted 1973.**[2]**Marvin J. Greenberg,*Lectures on algebraic topology*, W. A. Benjamin, Inc., New York-Amsterdam, 1967. MR**0215295****[3]**R. B. KEARFOTT,*Computing the Degree of Maps and a Generalized Method of Bisection*, Ph. D. dissertation, Univ. of Utah, 1977.**[4]**Baker Kearfott,*An efficient degree-computation method for a generalized method of bisection*, Numer. Math.**32**(1979), no. 2, 109–127. MR**529902**, https://doi.org/10.1007/BF01404868**[5]**Ivo G. Rosenberg and Frank Stenger,*A lower bound on the angles of triangles constructed by bisecting the longest side*, Math. Comp.**29**(1975), 390–395. MR**0375068**, https://doi.org/10.1090/S0025-5718-1975-0375068-5**[6]**MARTIN STYNES,*An Algorithm for the Numerical Calculation of the Degree of a Mapping*, Ph. D. dissertation, Oregon State Univ., 1977.**[7]**FRANK STENGER, "An algorithm for the topological degree of a mapping in ,"*Numer. Math.*, v. 25, 1976, pp. 23-28.

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DOI:
https://doi.org/10.1090/S0025-5718-1978-0494897-3

Article copyright:
© Copyright 1978
American Mathematical Society