A Three-Dimensional Analogue to the Method of Bisections for Solving Nonlinear Equations

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.