  We study a family of high order Ehrlich-type methods for approximating all
zeros of a polynomial simultaneously. Let us denote by $T^{(1)}$ the famous
Ehrlich method (1967). Starting from $T^{(1)}$, Kjurkchiev and Andreev (1987)
have introduced recursively a sequence ${(T^{(N)})_{N=1}^\infty}$ of iterative
methods for simultaneous finding polynomial zeros. For given $N \ge 1$, the
Ehrlich-type method $T^{(N)}$ has the order of convergence ${2 N + 1}$. In this
paper, we establish two new local convergence theorems as well as a semilocal
convergence theorem (under computationally verifiable initial conditions and
with a posteriori error estimate) for the Ehrlich-type methods $T^{(N)}$. Our
first local convergence theorem generalizes a result of Proinov (2015) and
improves the result of Kjurkchiev and Andreev (1987). The second local
convergence theorem generalizes another recent result of Proinov (2015), but
only in the case of maximum-norm. Our semilocal convergence theorem is the
first result in this direction.
