  We consider the tensor equation whose coefficient tensor is a nonsingular
M-tensor and whose right side vector is nonnegative. Such a tensor equation may
have a large number of nonnegative solutions. It is already known that the
tensor equation has a maximal nonnegative solution and a minimal nonnegative
solution (called extremal solutions collectively). However, the existing proofs
do not show how the extremal solutions can be computed. The existing numerical
methods can find one of the nonnegative solutions, without knowing whether the
computed solution is an extremal solution. In this paper, we present new proofs
for the existence of extremal solutions. Our proofs are much shorter than
existing ones and more importantly they give numerical methods that can compute
the extremal solutions. Linear convergence of these numerical methods is also
proved under mild assumptions. Some of our discussions also allow the
coefficient tensor to be a Z-tensor or allow the right side vector to have some
negative elements.
