  Phase retrieval aims at recovering a complex-valued signal from
magnitude-only measurements, which attracts much attention since it has
numerous applications in many disciplines. However, phase recovery involves
solving a system of quadratic equations, indicating that it is a challenging
nonconvex optimization problem. To tackle phase retrieval in an effective and
efficient manner, we apply coordinate descent (CD) such that a single unknown
is solved at each iteration while all other variables are kept fixed. As a
result, only minimization of a univariate quartic polynomial is needed which is
easily achieved by finding the closed-form roots of a cubic equation. Three
computationally simple algorithms referred to as cyclic, randomized and greedy
CDs, based on different updating rules, are devised. It is proved that the
three CDs globally converge to a stationary point of the nonconvex problem, and
specifically, the randomized CD locally converges to the global minimum and
attains exact recovery at a geometric rate with high probability if the sample
size is large enough. The cyclic and randomized CDs are also modified via
minimization of the $\ell_1$-regularized quartic polynomial for phase retrieval
of sparse signals. Furthermore, a novel application of the three CDs, namely,
blind equalization in digital communications, is proposed. It is demonstrated
that the CD methodology is superior to the state-of-the-art techniques in terms
of computational efficiency and/or recovery performance.
