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Mathematics of Computation

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

 

 

A three-dimensional analogue to the method of bisections for solving nonlinear equations


Author: Krzysztof Sikorski
Journal: Math. Comp. 33 (1979), 722-738
MSC: Primary 65H10
DOI: https://doi.org/10.1090/S0025-5718-1979-0521286-6
MathSciNet review: 521286
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Abstract: This paper deals with a three-dimensional analogue to the method of bisections for solving a nonlinear system of equations $ F(X) = \theta = {(0,0,0)^T}$, which does not require the evaluation of derivatives of F.

We divide the original parallelepiped (Figure 2.1) into 8 tetrahedra (Figure 2.2), and then bisect the tetrahedra to form an infinite sequence of tetrahedra, whose vertices converge to $ Z \in {R^3}$ such that $ F(Z) = \theta $. The process of bisecting a tetrahedron $ < \vert > {E_1}{E_2}{E_3}{E_4}$ with vertices $ {E_i}$ is defined as follows. We first locate the longest edge $ {E_i}{E_j},i \ne j$, set $ D = ({E_i} + {E_j})/2$, and then define two new tetrahedra $ < \vert > {E_i}D{E_k}{E_l}$ and $ < \vert > D{E_j}{E_k}{E_l}$, where $ j \ne l,l \ne i,i \ne k,k \ne j$ and $ k \ne l$.

We give sufficient conditions for convergence of the algorithm. The results of our numerical experiments show that the required storage may be large in some cases.


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Additional Information

DOI: https://doi.org/10.1090/S0025-5718-1979-0521286-6
Keywords: Bisection
Article copyright: © Copyright 1979 American Mathematical Society