Any non-decreasing sequence of non-negative numbers can be the convergence curve of the DGMRES method

Subject Areas : Statistics

Malihe Safarzadeh ¹ , Hossein Sadeghi Goughery ²

1 - Department of Mathematics, Kerman Branch, Islamic Azad University, Kerman, Iran
2 - Department of Mathematics, Kerman Branch, Islamic Azad University, Kerman, Iran

Received: 2020-02-06 Accepted : 2020-08-25 Published : 2021-09-23

Keywords: بردار مانده, معکوس درازین, شاخص, دستگاه معادلات خطی منفرد,

Abstract :

Investigating the convergence of the Krylov subspace methods has been considered as one of the favorite subjects in the field of numerical linear algebra. Given that the DGMRES method is a Krylov subspace methods, there is not much work to be done on its convergence. In this article we will discuss parts of the convergence of this method. We show that for any non-decreasing sequence of negative numbers f(0)≥f(1)≥⋯≥f(m-1)>f(m)=⋯=f(n)=0, he set of {λ_1,…,λ_m,0,…,0} of the complex numbers and the arbitrary number α≤n-m can be a set of singular linear equations n×n, Ax=b with the index α and the set {λ_1,…,λ_m,0,…,0} as the spectrum of matrix A such that if the DGMRES method is used to solve this system, assuming ‖A^α r_0 ‖_2=f(0)‖_2 = f (0), for k=1,…,m-1, the k^th residual vector is ‖A^α 〖r 〗_k ‖_2=f(k), wherer_0=b-Ax_0 is the initial residual vector. We attempt to do this by constructing the complete coefficient matrix of the system (s).

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