  In recent years, the PageRank algorithm has garnered significant attention
due to its crucial role in search engine technologies and its applications
across various scientific fields. It is well-known that the power method is a
classical method for computing PageRank. However, there is a pressing demand
for alternative approaches that can address its limitations and enhance its
efficiency. Specifically, the power method converges very slowly when the
damping factor is close to 1. To address this challenge, this paper introduces
a new multi-step splitting iteration approach for accelerating PageRank
computations. Furthermore, we present two new approaches for computating
PageRank, which are modifications of the new multi-step splitting iteration
approach, specifically utilizing the thick restarted Arnoldi and generalized
Arnoldi methods. We provide detailed discussions on the construction and
theoretical convergence results of these two approaches. Extensive experiments
using large test matrices demonstrate the significant performance improvements
achieved by our proposed algorithms.
