<p>This paper addresses the solution of periodic tridiagonal Toeplitz linear systems. Most existing approaches decompose the coefficient matrix into a tridiagonal matrix plus low-rank perturbations and then apply the Sherman-Morrison-Woodbury formula to obtain the solution. However, this necessitates solving multiple auxiliary linear systems. To reduce this computational overhead, we propose an asymptotic approximation algorithm. This algorithm involves solving only one tridiagonal Toeplitz linear system, followed by a correction step using an asymptotic explicit formula to yield the final solution. Compared to existing algorithms, our proposed algorithm not only requires fewer floating-point operations but also consumes less memory storage. Finally, several numerical experiments implemented in MATLAB confirm the efficiency and accuracy of the proposed algorithm.</p>