An asymptotic approximation of the solution for nearly tridiagonal quasi-Toeplitz linear systems

We introduce an asymptotic approximate algorithm for solving nearly tridiagonal quasi-Toeplitz linear systems. When addressing low-rank perturbations of a tridiagonal Toeplitz matrix system based on the Sherman–Morrison–Woodbury formula (or Woodbury identity), conventional methods require solving at least two simpler systems. The proposed algorithm overcomes this limitation by providing an explicit asymptotic formula for one of these systems. This asymptotic approximation enables a rapid resolution of the original system with minimal additional computation. To validate the accuracy and efficiency of the proposed algorithm, we conduct numerical experiments on two cases, comparing the results with those of existing methods. The results demonstrate that the proposed algorithm significantly reduces computation time while maintaining accuracy compared to the existing methods.