# Two fast algorithms for finding the solution of the lower Hessenberg quasi-Toeplitz linear system from Markov chain

## TL;DR

This paper introduces two efficient algorithms for solving a specific type of linear system that arises in Markov chains.

## Contribution

The novelty lies in the development of two O(n log n) algorithms for lower Hessenberg quasi-Toeplitz systems.

## Key findings

- The algorithms leverage the structure of the matrix as a sum of a Toeplitz and a rank-one matrix.
- Numerical experiments show the algorithms outperform existing methods in terms of accuracy and speed.

## Abstract

We present two fast algorithms for finding the solution of the nonsingular lower Hessenberg quasi-Toeplitz linear system stem from Markov chain. And we confirm the complexity of these two algorithms is both O\documentclass[12pt]{minimal}
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\begin{document}$$(n\log n)$$\end{document} based on the fact that a lower Hessenberg quasi-Toeplitz matrix can be written as the sum of a Toeplitz matrix and a rank-one matrix, such that the fast solver involves O\documentclass[12pt]{minimal}
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\begin{document}$$(n\log n)$$\end{document} operators for solving the Toeplitz linear system can be adopted. Finally, numerical results prove the superiority and accuracy of our algorithms by comparing the values of residual and CPU time with existing algorithms.