Research On Methods Of Solving Linear System With Multiple Right-hand Sides

With the rapid development of science and technology and the continuous progress of the project,the solution of the linear equations becomes the core problem of the project.Especially,in many practical applications it needs to solve the large sparse linear system.In this thesis,we mainly study the solving method of linear systems with multiple right-hand sides and shifts.This problems often appear in quantum chromodynamics and image processing.The development of the Krylov subspace method is that the linear systems with multiple right-hand sides and shifts,and to solve the linear systems with multiple right-hand sides and shifts.The Krylov-subspace method is one of the common methods of solving this linear systems.Commonly used to solve such problems Krylov subspace method has a block method and one by one to solve the method.In this thesis,we mainly study the method of solving the linear system of symmetric positive definite matrix in solving method one by one.Because it solves the system is symmetric positive definite,the algorithm is based on the short recurrence-CG algorithm research.For the linear system with multiple right-hand sides,the information sharing of the deflation method accelerates the convergence to make it a common method.Andreas and Orginos propose eigCG method to quickly solve the linear system with multiple right-hand sides by sharing the information and deflating the eigenvector of the smallest eigenvalue.For the solving method of linear system with different shifts is the most commonly used method of solving multiple shift systems simultaneously——shifted CG algorithm,the other method is that Tony.Chan proposed Galerkin projection method by the use of projection technology through information sharing to its different linear system.The multiple linear system is deflated to speed up the convergence.In this thesis,the research on the linear system of multiple right-hand sides is presented.The advantages and disadvantages of the algorithm of solving the linear system with multiple right-hand sides are analyzed,and the advantages and disadvantages of the common methods of solving the linear system with different shifts are analyzed.Based on eigCG algorithm and the Galerkin projection algorithm is extended by adding the eigCG algorithm to linear system with the multiple right-hand sides and shifts.New Krylov Subspace Algorithm eigCG-Proj Algorithm is proposed.Our new algorithms efficiently solve the multiple right-hand sides while yet keeping costs for all of the multiply shifted systems close to those for one shift.The numerical results show that our algorithm is feasible.