  In this paper, we develop a new computational approach which is based on
minimizing the difference of two convex functionals (DC) to solve a broader
class of phase retrieval problems. The approach splits a standard nonlinear
least squares minimizing function associated with the phase retrieval problem
into the difference of two convex functions and then solves a sequence of
convex minimization sub-problems. For each subproblem, the Nesterov's
accelerated gradient descent algorithm or the Barzilai-Borwein (BB) algorithm
is used. In the setting of sparse phase retrieval, a standard $\ell_1$ norm
term is added into the minimization mentioned above. The subproblem is
approximated by a proximal gradient method which is solved by the
shrinkage-threshold technique directly without iterations. In addition, a
modified Attouch-Peypouquet technique is used to accelerate the iterative
computation. These lead to more effective algorithms than the Wirtinger flow
(WF) algorithm and the Gauss-Newton (GN) algorithm and etc.. A convergence
analysis of both DC based algorithms shows that the iterative solutions is
convergent linearly to a critical point and will be closer to a global
minimizer than the given initial starting point. Our study is a deterministic
analysis while the study for the Wirtinger flow (WF) algorithm and its
variants, the Gauss-Newton (GN) algorithm, the trust region algorithm is based
on the probability analysis. In particular, the DC based algorithms are able to
retrieve solutions using a number $m$ of measurements which is about twice of
the number $n$ of entries in the solution with high frequency of successes.
When $m\approx n$, the $\ell_1$ DC based algorithm is able to retrieve sparse
signals.
