NUMERICAL LINEAR ALGEBRA WITH APPLICATIONS Numer. Linear Algebra Appl. 2007; 14:521–546 Published online 7 March 2007 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/nla.531 A Power–Arnoldi algorithm for computing PageRank Gang Wu 1, ‡ and Yimin Wei 2, 3, ∗, † 1 School of Mathematical Sciences, Xuzhou Normal University, Xuzhou, Jiangsu 221116, People’s Republic of China 2 School of Mathematical Sciences, Fudan University, Shanghai 200433, People’s Republic of China 3 Key Laboratory of Mathematics for Nonlinear Sciences, Fudan University, Ministry of Education, Shanghai 200433, People’s Republic of China SUMMARY The PageRank algorithm plays an important role in modern search engine technology. It involves using the classical power method to compute the principle eigenvector of the Google matrix representing the web link graph. However, when the largest eigenvalue is not well separated from the second one, the power method may perform poorly. This happens when the damping factor is sufficiently close to 1. Therefore, it is worth developing new techniques that are more sophisticated than the power method. The approach presented here, called Power–Arnoldi, is based on a periodic combination of the power method with the thick restarted Arnoldi algorithm. The justification for this new approach is presented. Numerical tests illustrate the efficiency and convergence behaviour of the new algorithm. Copyright 2007 John Wiley & Sons, Ltd. Received 30 April 2006; Revised 17 December 2006; Accepted 12 January 2007 KEY WORDS: PageRank; Google matrix; thick restarted Arnoldi; Krylov subspace; Power–Arnoldi Dedicated to Professor Zhihao Cao on the occasion of his 70th birthday 1. INTRODUCTION Google is one of the most successful search engines in recent years, due to its comprehensive and accurate search results. The key algorithm, called PageRank, plays a major role in Google’s ∗ Correspondence to: Yimin Wei, School of Mathematical Sciences, Fudan University, Shanghai 200433, People’s Republic of China. † E-mail: ymwei@fudan.edu.cn, ymwei cn@yahoo.com ‡ E-mail: gangwu76@yahoo.com.cn Contract/grant sponsor: National Natural Science Foundation of China; contract/grant numbers: 10626044, 10471027 Contract/grant sponsor: Doctoral Foundation of Xuzhou Normal University; contract/grant number: KY2004121 Contract/grant sponsor: Shanghai Education Committee Contract/grant sponsor: Qing-Lan Project of Jiangsu Province Copyright 2007 John Wiley & Sons, Ltd.