A three-dimensional analogue to the method of bisections for solving nonlinear equations
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- by Krzysztof Sikorski PDF
- Math. Comp. 33 (1979), 722-738 Request permission
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 $< | > {E_1}{E_2}{E_3}{E_4}$ 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 $< | > {E_i}D{E_k}{E_l}$ and $< | > D{E_j}{E_k}{E_l}$, 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.References
- Charles Harvey and Frank Stenger, A two-dimensional analogue to the method of bisections for solving nonlinear equations, Quart. Appl. Math. 33 (1975/76), no. 4, 351–368. MR 455361, DOI 10.1090/S0033-569X-1976-0455361-7
- J. M. Ortega and W. C. Rheinboldt, Iterative solution of nonlinear equations in several variables, Academic Press, New York-London, 1970. MR 0273810
- Frank Stenger, Computing the topological degree of a mapping in $\textbf {R}^{n}$, Numer. Math. 25 (1975/76), no. 1, 23–38. MR 394639, DOI 10.1007/BF01419526
Additional Information
- © Copyright 1979 American Mathematical Society
- Journal: Math. Comp. 33 (1979), 722-738
- MSC: Primary 65H10
- DOI: https://doi.org/10.1090/S0025-5718-1979-0521286-6
- MathSciNet review: 521286