{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,14]],"date-time":"2025-10-14T00:45:59Z","timestamp":1760402759638,"version":"build-2065373602"},"reference-count":21,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2021,4,10]],"date-time":"2021-04-10T00:00:00Z","timestamp":1618012800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Anderson(m0) extrapolation, an accelerator to a fixed-point iteration, stores m0+1 prior evaluations of the fixed-point iteration and computes a linear combination of those evaluations as a new iteration. The computational cost of the Anderson(m0) acceleration becomes expensive with the parameter m0 increasing, thus m0 is a common choice in most practice. In this paper, with the aim of improving the computations of PageRank problems, a new method was developed by applying Anderson(1) extrapolation at periodic intervals within the Arnoldi-Inout method. The new method is called the AIOA method. Convergence analysis of the AIOA method is discussed in detail. Numerical results on several PageRank problems are presented to illustrate the effectiveness of our proposed method.<\/jats:p>","DOI":"10.3390\/sym13040636","type":"journal-article","created":{"date-parts":[[2021,4,12]],"date-time":"2021-04-12T11:05:06Z","timestamp":1618225506000},"page":"636","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Anderson Acceleration of the Arnoldi-Inout Method for Computing PageRank"],"prefix":"10.3390","volume":"13","author":[{"given":"Xia","family":"Tang","sequence":"first","affiliation":[{"name":"School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu 610054, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Chun","family":"Wen","sequence":"additional","affiliation":[{"name":"School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu 610054, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7895-2050","authenticated-orcid":false,"given":"Xian-Ming","family":"Gu","sequence":"additional","affiliation":[{"name":"School of Economic Mathematics, Southwestern University of Finance and Economics, Chengdu 611130, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Zhao-Li","family":"Shen","sequence":"additional","affiliation":[{"name":"College of Science, Sichuan Agricultural University, Ya\u2019an 625000, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2021,4,10]]},"reference":[{"key":"ref_1","unstructured":"Page, L., Brin, S., and Motwani, R. (1999). The PageRank Citation Ranking: Bringing Order to the Web, Stanford InfoLab."},{"key":"ref_2","unstructured":"Haveliwala, T., and Kamvar, S. (2003). The Second Eigenvalue of the Google Matrix, Stanford InfoLab."},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Kamvar, S.D., Haveliwala, T.H., Manning, C.D., and Golub, G.H. (2003, January 20\u201324). Extrapolation methods for accelerating PageRank computations. Proceedings of the 12th International Conference on World Wide Web, Budapest, Hungary.","DOI":"10.1145\/775189.775190"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"551","DOI":"10.1137\/050626612","article-title":"The PageRank vector: Properties, computation, approximation, and acceleration","volume":"28","author":"Brezinski","year":"2006","journal-title":"SIAM J. Matrix Anal. Appl."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"349","DOI":"10.1137\/080727397","article-title":"An inner-outer iteration for computing PageRank","volume":"32","author":"Gleich","year":"2010","journal-title":"SIAM J. Sci. Comput."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"19","DOI":"10.1016\/j.cam.2014.09.022","article-title":"A two-step matrix splitting iteration for computing PageRank","volume":"278","author":"Gu","year":"2015","journal-title":"J. Comput. Appl. Math."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"759","DOI":"10.1007\/s10543-006-0091-y","article-title":"An Arnoldi-type algorithm for computing page rank","volume":"46","author":"Golub","year":"2006","journal-title":"BIT"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"219","DOI":"10.1016\/j.cam.2016.05.026","article-title":"An Arnoldi-Inout algorithm for computing PageRank problems","volume":"309","author":"Gu","year":"2017","journal-title":"J. Comput. Appl. Math."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"96","DOI":"10.1016\/j.laa.2005.07.024","article-title":"A harmonic restarted Arnoldi algorithm for calculating eigenvalues and determining multiplicity","volume":"415","author":"Morgan","year":"2006","journal-title":"Linear Algebra Appl."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"113034","DOI":"10.1016\/j.cam.2020.113034","article-title":"A variant of the Power\u2013Arnoldi algorithm for computing PageRank","volume":"381","author":"Hu","year":"2021","journal-title":"J. Comput. Appl. Math."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"521","DOI":"10.1002\/nla.531","article-title":"A Power\u2014Arnoldi algorithm for computing PageRank","volume":"14","author":"Wu","year":"2007","journal-title":"Numer. Linear Algebra Appl."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"383","DOI":"10.1016\/j.cam.2016.08.034","article-title":"A new extrapolation method for PageRank computations","volume":"313","author":"Tan","year":"2017","journal-title":"J. Comput. Appl. Math."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"547","DOI":"10.1145\/321296.321305","article-title":"Iterative procedures for nonlinear integral equations","volume":"12","author":"Anderson","year":"1965","journal-title":"JACM"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"1715","DOI":"10.1137\/10078356X","article-title":"Anderson acceleration for fixed-point iterations","volume":"49","author":"Walker","year":"2011","journal-title":"SIAM J. Numer. Anal."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"805","DOI":"10.1137\/130919398","article-title":"Convergence analysis for Anderson acceleration","volume":"53","author":"Toth","year":"2015","journal-title":"SIAM J. Numer. Anal."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"43","DOI":"10.1016\/j.jcp.2015.11.018","article-title":"Anderson acceleration of the Jacobi iterative method: An efficient alternative to Krylov methods for large, sparse linear systems","volume":"306","author":"Pratapa","year":"2016","journal-title":"J. Comput. Phys."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"1854","DOI":"10.1137\/060661442","article-title":"A trust region direct constrained minimization algorithm for the kohn\u2013sham equation","volume":"29","author":"Yang","year":"2007","journal-title":"SIAM J. Sci. Comput."},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Allaire, G., Kaber, S.M., Trabelsi, K., and Allaire, G. (2008). Numerical Linear Algebra, Springer.","DOI":"10.1007\/978-0-387-68918-0"},{"key":"ref_19","unstructured":"Walker, H.F. (2011). Anderson Acceleration: Algorithms and Implementations, WPI Math. Sciences Dept.. Report MS-6-15-50."},{"key":"ref_20","unstructured":"Saad, Y. (1992). Numerical Methods for Large Eigenvalue Problems, Manchester University Press."},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Langville, A., and Meyer, C. (2006). Google\u2019s PageRank and Beyond: The Science of the Search Engine Rankings, Princeton University Press.","DOI":"10.1515\/9781400830329"}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/13\/4\/636\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,13]],"date-time":"2025-10-13T13:59:18Z","timestamp":1760363958000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/13\/4\/636"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,4,10]]},"references-count":21,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2021,4]]}},"alternative-id":["sym13040636"],"URL":"https:\/\/doi.org\/10.3390\/sym13040636","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2021,4,10]]}}}