Krylov Subspace Algorithms Based On Projection

Krylov subspace method is one of the ten famous algorithms,which is very useful to deal with large sparse linear system,so it is of great significance to study it in depth.The convergence of Krylov subspace method is closely related to the distribution of character-istic information of solving linear system.The convergence of Krylov subspace method is closely related to the distribution of characteristic information of solving linear system.Singular value is an important characteristic information of solving linear system,so it is of great significance to study the singular value of solving linear system.The generalized inverse is closely related to the singular value.In the second chapter,starting from the minimum polynomial,we find the solution space containing the minimum norm solution of singular linear system Ax=b,and get the formula of solving the generalized inverse only by calculating the different singular value.The minimum norm solution can be rep-resented by singular value,A,b,and the convergence of least square QR decomposition algorithm(LSQR)is analyzed theoretically.The results show that the convergence of LSQR is closely related to the nonzero singular value of matrix A:The LSQR algorithm is used to solve the singular linear system Ax=b,If there is no error,the number of non-zero singular values of A is represented by k_?,the LSQR algorithm will converge to the least square solution by iterating at most k_?.As we all know,many Krylov subspace variational methods have been proposed based on the idea of iterative solution,but the iterative method is easy to produce error accumulation effect,resulting in stagnation and irregular oscillation.Taking restart mea-sures can break some”undesirable”iteration periods,release a large amount of storage space,and iterate from the new residual reduction direction,which is conducive to solv-ing stagnation and irregular oscillation.But at the same time,it also discards a lot of useful information generated by previous iterations,which affects the convergence speed.In the third chapter of this paper,GMRES(m)method,which is a typical representative of Krylov subspace method with restart strategy,is studied.The algorithm is very effective for solv-ing asymmetric linear system.The research focuses on the change of GMRES(m)iterative solution in each restart,which is also the useful information discarded after restart.Ac-celerating Krylov subspace algorithm with abandoned information and projection method,we propose an extension of GMRES(m)method based on error equation,which is named LGMRES(m).The LBGMRES(m)algorithm is obtained by introducing backtracking restart technology into GMRES(m)algorithm,and the LLBGMRES(m)algorithm is ob-tained by adding backtracking restart technology into LGMRES(m)algorithm.The results of numerical experiments on LLBGMRES(m),LBGMRES(m),LGMRES(m)and GM-RES(m)show that the convergence of LGMRES(m)and LLBGMRES(m)is better than the traditional restart Krylov subspace algorithm in time and accuracy,and LGMRES(m)is better than LBGMRES(m),Good acceleration effect is obtained by using projection method.