In recent two decades, palindromic eigenvalue problems (PEP) have great relevance in control applications associated with the continuous-time descriptor systems and the vibration of fast trains. In this talk, we investigate the an accelerated technique for solving an eigenpair of PEP. Our contribution is twofold. First, we introduce a method to transform the eigenpair of linear PEP into a solution of a kind of algebraic Riccati equations (ARE). Second, we propose an accelerated iteration to obtain a solution of ARE with high order convergence. We point out that traditional approaches for finding a numerical solution of a nonlinear equation are based on the fixed-point iteration, and the speed of the convergence is usually linear. With our method, once the existence of ``semigroup property''. holds for a given iteration, It can immediately construct an iterative method which can converge to the solution with the speed of convergence of any desired order.