Refined projection methods proposed by the author have received attention internationally. We are concerned with a conventional projection method and its refined counterpart for computing approximations to a simple ei...Refined projection methods proposed by the author have received attention internationally. We are concerned with a conventional projection method and its refined counterpart for computing approximations to a simple eigenpair (λ, x) of a large matrix A. Given a subspace ω that contains an approximation to x, these two methods compute approximations (μ,x) and μ,x) to (λ,x), respectively. We establish three results. First, the refined eigenvector approximation or simply the refined Ritz vector x is unique as the deviation of x from ω approaches zero if A is simple. Second, in terms of residual norm of the refined approximate eigenpair (μ, x), we derive lower and upper bounds for the sine of the angle between the Ritz vector x and the refined eigenvector approximation x, and we prove that x≠x unless x = x. Third, we establish relationships between the residual norm ||AX -μx|| of the conventional methods and the residual norm ||Ax -μx|| of the refined methods, and we show that the latter is always smaller than the former if (μ, x) is not an exact eigenpair of A, indicating that the refined projection method is superior to the corresponding conventional counterpart.展开更多
We discuss a variant of restarted GMRES method that allows changes of the restarting vector at each cycle of iterations.The merit of the variant is that previously generated information can be utilized to select a new...We discuss a variant of restarted GMRES method that allows changes of the restarting vector at each cycle of iterations.The merit of the variant is that previously generated information can be utilized to select a new starting vector,such that the occurrence of stagnation be mitigated or the convergence be accelerated.The more appealing utilization of the new method is in conjunction with a harmonic Ritz vector as the starting vector,which is discussed in detail.Numerical experiments are carried out to demonstrate that the proposed procedure can effectively mitigate the occurrence of stagnation due to the presence of small eigenvalues in modulus.展开更多
文摘Refined projection methods proposed by the author have received attention internationally. We are concerned with a conventional projection method and its refined counterpart for computing approximations to a simple eigenpair (λ, x) of a large matrix A. Given a subspace ω that contains an approximation to x, these two methods compute approximations (μ,x) and μ,x) to (λ,x), respectively. We establish three results. First, the refined eigenvector approximation or simply the refined Ritz vector x is unique as the deviation of x from ω approaches zero if A is simple. Second, in terms of residual norm of the refined approximate eigenpair (μ, x), we derive lower and upper bounds for the sine of the angle between the Ritz vector x and the refined eigenvector approximation x, and we prove that x≠x unless x = x. Third, we establish relationships between the residual norm ||AX -μx|| of the conventional methods and the residual norm ||Ax -μx|| of the refined methods, and we show that the latter is always smaller than the former if (μ, x) is not an exact eigenpair of A, indicating that the refined projection method is superior to the corresponding conventional counterpart.
文摘We discuss a variant of restarted GMRES method that allows changes of the restarting vector at each cycle of iterations.The merit of the variant is that previously generated information can be utilized to select a new starting vector,such that the occurrence of stagnation be mitigated or the convergence be accelerated.The more appealing utilization of the new method is in conjunction with a harmonic Ritz vector as the starting vector,which is discussed in detail.Numerical experiments are carried out to demonstrate that the proposed procedure can effectively mitigate the occurrence of stagnation due to the presence of small eigenvalues in modulus.
