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

• DOI: 10.2307/2006307
• Publication Date: 2006-04-22T13:38:00Z
• Authors: Krzysztof Sikorski
• Keywords: 0101 mathematics; 01 natural sciences

This paper deals with a three-dimensional analogue to the method of bisections for solving a nonlinear system of equations F ( X ) = θ = ( 0 , 0 , 0 ) T 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 ∈ R 3 Z \in {R^3} such that F ( Z ) = θ F(Z) = \theta . The process of bisecting a tetrahedron > | > E 1 E 2 E 3 E 4 > | > {E_1}{E_2}{E_3}{E_4} with vertices E i {E_i} is defined as follows. We first locate the longest edge E i E j , i ≠ j {E_i}{E_j},i \ne j , set D = ( E i + E j ) / 2 D = ({E_i} + {E_j})/2 , and then define two new tetrahedra > | > E i D E k E l > | > {E_i}D{E_k}{E_l} and > | > D E j E k E l > | > D{E_j}{E_k}{E_l} , where j ≠ l , l ≠ i , i ≠ k , k ≠ j j \ne l,l \ne i,i \ne k,k \ne j and k ≠ l 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.