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: 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 GREENBERG,*Lectures on Algebraic Topology*, Benjamin, New York, 1967. MR**0215295 (35:6137)****[3]**R. B. KEARFOTT,*Computing the Degree of Maps and a Generalized Method of Bisection*, Ph. D. dissertation, Univ. of Utah, 1977.**[4]**R. B. KEARFOTT, "An efficient degree-computation method for a generalized method of bisection,"*Numer. Math.*(To appear.) MR**529902 (80g:65062)****[5]**J. ROSENBERG & F. STENGER, "A lower bound on the angles of triangles constructed by bisecting the longest side,"*Math. Comp.*, v. 29, 1975, pp. 390-395. MR**0375068 (51:11264)****[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