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Improving Hybrid GMRES Algorithm

Lan He Zhang

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Abstract

The Hybrid GMRES of N.M.Nachtigal,L.Reichel,and L.N.Trefethen for solving large nonsymmetric systems of linear equations is a very effective algorithm,which can saves the computing time,however the theory of this algorithm is imperfect.To some extent,Hybrid GMRES is an empirical algorithm and may be slowly or not converge.In order to enhance the usability of Hybrid GMRES,a polynomial preconditioner is described based on GMRES,and proposed a new improved algorithm called ImprovedHybrid GMRES as follows.The results of numerical experiments show that the new algorithm is quite simple to be implemented and can converge to a prescribed accuracy quickly with a small steps m,which reduces computation and overcome the flaw of Hybrid GMRES successfully.

About this research paper

What this paper is about

The Hybrid GMRES of N.M.Nachtigal,L.Reichel,and L.N.Trefethen for solving large nonsymmetric systems of linear equations is a very effective algorithm,which can saves the computing time,however the theory of this algorithm is imperfect.To some extent,Hybrid GMRES is an empirical algorithm and may be slowly or not converge.In order to enhance the usability of Hybrid GMRES,a polynomial preconditioner is described based on GMRES,and proposed a new improved algorithm called ImprovedHybrid GMRES as follows.The results of numerical experiments show that the new algorithm is quite simple to be implemented and can converge to a prescribed accuracy quickly with a small steps m,which reduces computation and overcome the flaw of Hybrid GMRES successfully.

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Available abstract

The Hybrid GMRES of N.M.Nachtigal,L.Reichel,and L.N.Trefethen for solving large nonsymmetric systems of linear equations is a very effective algorithm,which can saves the computing time,however the theory of this algorithm is imperfect.To some extent,Hybrid GMRES is an empirical algorithm and may be slowly or not converge.In order to enhance the usability of Hybrid GMRES,a polynomial preconditioner is described based on GMRES,and proposed a new improved algorithm called ImprovedHybrid GMRES as follows.The results of numerical experiments show that the new algorithm is quite simple to be implemented and can converge to a prescribed accuracy quickly with a small steps m,which reduces computation and overcome the flaw of Hybrid GMRES successfully.

Key concepts: Generalized minimal residual method, Preconditioner, Computation, Polynomial, Algorithm, Computer science, Linear system, Simple (philosophy)

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