Study On The Krylov Subspace Methods For Systems With Multiple Right-hand Sides

Many scientific and engineering applications require one to solve large sparse linear systems with multiple right-hand sides,for example: scattering calculation,simulation of semiconductor,image restoration,quantum chromodynamics and so on.In recent years,with the rapid development of information and computer technology,people demand higher computational accuracy,and how to solve the large sparse linear equations has become one of the important direction of scientific computing research.At present,Krylov subspace is the most effective projection method to solve the systems with multiple right-hand sides,because it has less storage,less amount of calculation and so on,Krylov subspace method has been studied by many people.Many methods based on Krylov subspace have been proposed to solve linear systems with different right-hand sides and a symmetric positive definite matrix,for example,block methods,deflating eigenvalues,the seed projection methods and so on.The thesis mainly studies seed projection methods to solve symmetric positive definite systems of linear equations with multiple right-hand sides.Smith,Peterson and Mittra argued that a seed projection algorithm based on CG algorithm is the most effective,namely the standard seed conjugate gradient algorithm.But,rounding error in the CG method limits the extent to which the seeding can improve the rate of convergence.Abdel-Rehim and Morgan etal improved the seed CG algorithm and proposed seeding once algorithm,only the first right-hand side is used for seeding,do not need to be repeated seeding.Numerical experiments show that the seeding once has better convergence than multiple seeding algorithm.This thesis proposes improved seeding once algorithm based on seeding once algorithm by improving the initial guess after it has been seeded in order to reduce error.Numerical experiments show the efficacy.The thesis gives an actuality of the study on solving the linear systems with multiple right-hand sides,introducing relevant algorithm for solving symmetric positive definite linear systems with multiple right-hand sides,including InitCG algorithm,AugCG algorithm,multiple seeding,seeding once algorithm,and the differences and relations among different types of algorithms are analyzed.Finally,this paper proposes improved seeding once algorithm based on seeding once algorithm,combined with technology of improving the initial guess proposed byErhel Guyomarc'h in 2000.Improved seeding once algorithm improves the convergence rate and reduces the computing time.Numerical experiments show the efficacy of improved seeding once algorithm.