  Finding a Z-eigenpair of a symmetric tensor is equivalent to finding a KKT
point of a sphere constrained minimization problem. Based on this equivalency,
in this paper, we first propose a class of iterative methods to get a
Z-eigenpair of a symmetric tensor. Each method can generate a sequence of
feasible points such that the sequence of function evaluations is decreasing.
These methods can be regarded as extensions of the descent methods for
unconstrained optimization problems. We pay particular attention to the
Newton's method. We show that under appropriate conditions, the Newton's method
is globally and quadratically convergent. Moreover, after finitely many
iterations, the unit steplength will always be accepted. We also propose a
nonlinear equations based Newton's method and establish its global and
quadratic convergence. In the end, we do several numerical experiments to test
the proposed Newton's methods. The results show that both Newton's methods are
very efficient.
