Convergence Rate of GMRES on Tridiagonal Toeplitz Linear System

This paper aimed to review paper [01] titled The rate of convergence of GMRES on a tridiagonal toeplitz linear system II by Ren-cang Li & Wei Zhang and its references, in which linear system Ax=b is solved using Toeplitz matrices by generalized minimal residual method. Based on this thesis,this method is the best and most complete way to solve asymmetric matrices. Since the analysis of theconvergence of such systems is difficult and complex, 2nd order orthogonal Chebyshev polynomials have been used to obtain more accurate limits and expressions for residual. In this thesis, convergence rate of GMRES was increased for linear toepltiz matrices using certain formulas. Simpler and faster bounds will be provided for linear system of Ax=b while all entries of A except for first and last ones are zero. Further, comparison of this method with others will show its superiority. While providingdifferent methods, they will be compared. In the end, algorithm of the above method was implemented using MATLAB software and in section on numerical results, diagrams of convergence rate and eigenvalues matrix were provided. By comparing them, a logical and experimental relationbetween convergence rate and eigenvalue will be obtained. Toeplitz systems We want to study toeplitz linear matrix equation system, which is type of band matrices. First, we provide a series of preliminary definitions.