Rayleigh quotient with bolzano booster for faster convergence of dominant eigenvalues

Computation ranking algorithms are widely used in several informatics fields. One of them is the PageRank algorithm, recognized as the most popular search engine globally. Many researchers have improvised the ranking algorithm in order to get better results. Recent research using Rayleigh Quotient to speed up PageRank can guarantee the convergence of the dominant eigenvalues as a key value for stopping computation. Bolzano's method has a convergence character on a linear function by dividing an interval into two intervals for better convergence. This research aims to implant the Bolzano algorithm into Rayleigh for faster computation. This research produces an algorithm that has been tested and validated by mathematicians, which shows an optimization speed of a maximum 7.08% compared to the sole Rayleigh approach. Analysis of computation results using statistics software shows that the degree of the curve of the new algorithm, which is Rayleigh with Bolzano booster (RB), is positive and more significant than the original method. In other words, the linear function will always be faster in the subsequent computation than the previous method

Off-diagonal low-rank preconditioner for difficult PageRank problems

PageRank problem is the cornerstone of Google search engine and is usually stated as solving a huge linear system. Moreover, when the damping factor approaches 1, the spectrum properties of this system deteriorate rapidly and this system becomes difficult to solve. In this paper, we demonstrate that the coefficient matrix of this system can be transferred into a block form by partitioning its rows into special sets. In particular, the off-diagonal part of the block coefficient matrix can be compressed by a simple low-rank factorization, which can be beneficial for solving the PageRank problem. Hence, a matrix partition method is proposed to discover the special sets of rows for supporting the low rank factorization. Then a preconditioner based on the low-rank factorization is proposed for solving difficult PageRank problems. Numerical experiments are presented to support the discussions and to illustrate the effectiveness of the proposed methods. (C) 2018 Elsevier B.V. All rights reserved