A three-dimensional analogue to the method of bisections for solving nonlinear equations
Abstract: This paper deals with a three-dimensional analogue to the method of bisections for solving a nonlinear system of equations , 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 such that . The process of bisecting a tetrahedron with vertices is defined as follows. We first locate the longest edge , set , and then define two new tetrahedra and , where and .
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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-  Frank Stenger, Computing the topological degree of a mapping in 𝑅ⁿ, Numer. Math. 25 (1975/76), no. 1, 23–38. MR 0394639, https://doi.org/10.1007/BF01419526
- CH. HARVEY & F. STENGER, "A two dimensional analogue to the method of bisections for solving nonlinear equations," Quart. Appl. Math., v. 33, 1976, pp. 351-368. MR 0455361 (56:13600)
- J. M. ORTEGA & W. C. RHEINBOLDT, Iterative Solutions of Nonlinear Equations in Several Variables, Academic Press, New York, 1970. MR 0273810 (42:8686)
- F. STENGER, "Computing the topological degree of a mapping in n-space," Numer. Math., v. 25, 1975, pp. 23-38. MR 0394639 (52:15440)
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