The Research And Application Of E-Transform GMRES(m) Algorithm

Today, many algorithms are applied to solve a large-scale sparse linear system,such as GMRES(m) algorithm. Amount of problems from some fields: systemengineering, science and so on, can be transformed into a linear equations to be solvedby the “difference”,“discrete” and “linear” process ultimately. The direct methodapplies to the low-order linear equations effectively, but if the coefficient matrix isadvanced as tridiagonal matrix, diagonally dominant matrix and symmetric positivedefinite matrices, the iterative method should be more effective. However, thepractical problems of life tend to higher order large matrices and the higher orderlinear equations are often not only sparse but also irregular. The GMRES(m) algorithmcomes into being with shortcomings in the computational efficiency and in the processof solving. In recent years, these shortcomings have been paid attention to study bymore researchers. In terms of the GMRES(m) algorithm, presents a E-transformGMRES(m) algorithm based on Krylov subspace. The new algorithm transformscoefficient matrix of GMRES(m) algorithm equations to be a diagonal matrix by usingthe matrix E and makes the solution of problems greatly simplified.The content of this paper is like this: at first, a brief introduction of thedevelopment history, research situation of linear equations and some solution ideas,specific iterative steps were introduced. Then, we summarizes the source of theGMRES(m) algorithm, introduces the GMRES(m) algorithm research status and theiterative thoughts of the GMRES(m) algorithm, gave the practical significance of theGMRES(m) algorithm. The feasibility and convergence of the algorithms were giventhrough the theoretical analysis and numerical experiments. we assess the influencebetween the restart m and the GMRES(m) algorithm in the numerical examples.Secondly, the basic idea and detailed calculation steps of E-transform GMRES(m)algorithm were given which is a variant of the GMRES(m) algorithm. Theconvergence and the feasibility of the E-transform GMRES(m) algorithm has beenproved form the theoretical analysis. Compared with the GMRES(m) algorithm, the calculation results of the E-transform GMRES(m) algorithm reflect the high efficiencyand high precision. Finally, we introduced the basic theory of the five-point differencemethod and Crank-Nicholson method, studied the benefit of E-transform GMRES(m)algorithm in actual application and meaning in the problems of partial differentialequations. Thus, the E-transform GMRES(m) algorithm that this paper puts forwardplays an important role in practical problems can solve and simplify the practicalproblems.