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.