The Arnoldi method is a commonly used technique for finding a few eigenpairs of large, sparse and nonsymmetric matrices. Recently, a new variant of Arnoldi method (NVRA) was proposed. In NVRA, the modified Ritz vector is used to take the place of the Ritz vector by solving a minimization problem. Moreover, it was shown that if the refined Arnoldi method converges, then the NVRA method also converges. The contribution of this work is as follows. First, we point out that the convergence theory of the NVRA method is incomplete. More precisely, the cosine of the angle between the refined Ritz vector and the Ritz vector may not be uniformly lower-bounded, and it can be arbitrarily close to zero in theory. Consequently, the modified Ritz vector may fail to converge even when the search subspace is good enough. A remedy to the convergence of the NVRA method is given. Second, we show that the linear system for solving the modified Ritz vector in the NVRA method will become more and more ill-conditioned as the refined Ritz vector converges. If the Ritz vector also tends to converge as the refined Ritz vector does so, the ill-conditioning of the linear system will have little influence on the convergence of the modified Ritz vector, and the modified Ritz vector can improve the Ritz vector substantially. Otherwise, the ill-conditioning may have significant influence on the convergence of the modified Ritz vector. Third, to fix the NVRA method, we propose an improved refined Arnoldi method that uses improved refined Ritz vector to take the place of the modified Ritz vector. Theoretical results indicate that the improved refined Ritz method is often better than the refined Ritz method. Numerical experiments illustrate the numerical behavior of the improved refined Ritz method, and demonstrate the effectiveness of our theoretical analysis.

Arnoldi 方法是一种常用的技术，用于查找大型、稀疏和非对称矩阵的几个特征对。最近，提出了 Arnoldi 方法 (NVRA) 的新变体。在 NVRA 中，修改的 Ritz 向量用于通过解决最小化问题来代替 Ritz 向量。此外，结果表明，如果改进的 Arnoldi 方法收敛，则 NVRA 方法也收敛。这项工作的贡献如下。首先，我们指出NVRA方法的收敛理论是不完整的。更准确地说，细化的 Ritz 向量与 Ritz 向量的夹角的余弦可能不是均匀下界的，可以任意取理论上接近于零。因此，即使搜索子空间足够好，修改后的 Ritz 向量也可能无法收敛。给出了 NVRA 方法收敛的补救措施。其次，我们表明，随着改进的 Ritz 向量收敛，NVRA 方法中用于求解修正 Ritz 向量的线性系统将变得越来越病态。如果 Ritz 向量也像精化 Ritz 向量那样趋于收敛，则线性系统的病态对修正 Ritz 向量的收敛影响不大，而修正 Ritz 向量可以显着改善 Ritz 向量。否则，病态可能会对修改后的 Ritz 向量的收敛产生重大影响。三、修复NVRA方法，我们提出了一种改进的细化 Arnoldi 方法，该方法使用改进的细化 Ritz 向量来代替修改后的 Ritz 向量。理论结果表明，改进的细化 Ritz 方法往往优于细化 Ritz 方法。数值实验说明了改进的细化 Ritz 方法的数值行为，并证明了我们理论分析的有效性。