  We propose in this paper New Q-Newton's method. The update rule is very
simple conceptually, for example $x_{n+1}=x_n-w_n$ where
$w_n=pr_{A_n,+}(v_n)-pr_{A_n,-}(v_n)$, with $A_n=\nabla ^2f(x_n)+\delta
_n||\nabla f(x_n)||^2.Id$ and $v_n=A_n^{-1}.\nabla f(x_n)$. Here $\delta _n$ is
an appropriate real number so that $A_n$ is invertible, and $pr_{A_n,\pm}$ are
projections to the vector subspaces generated by eigenvectors of positive
(correspondingly negative) eigenvalues of $A_n$.
  The main result of this paper roughly says that if $f$ is $C^3$ (can be
unbounded from below) and a sequence $\{x_n\}$, constructed by the New
Q-Newton's method from a random initial point $x_0$, {\bf converges}, then the
limit point is a critical point and is not a saddle point, and the convergence
rate is the same as that of Newton's method. The first author has recently been
successful incorporating Backtracking line search to New Q-Newton's method,
thus resolving the convergence guarantee issue observed for some (non-smooth)
cost functions. An application to quickly finding zeros of a univariate
meromorphic function will be discussed. Various experiments are performed,
against well known algorithms such as BFGS and Adaptive Cubic Regularization
are presented.
