{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/multi-way-monte-carlo-method-for-linear","title":"Multi-way Monte Carlo Method for Linear Systems","arxiv_id":"1608.04361","date":"2016-08-15","proceeding":null,"authors":["Tao Wu","David F. Gleich"],"abstract":"We study the Monte Carlo method for solving a linear system of the form $x =\nH x + b$. A sufficient condition for the method to work is $\\| H \\| < 1$, which\ngreatly limits the usability of this method. We improve this condition by\nproposing a new multi-way Markov random walk, which is a generalization of the\nstandard Markov random walk. Under our new framework we prove that the\nnecessary and sufficient condition for our method to work is the spectral\nradius $\\rho(H^{+}) < 1$, which is a weaker requirement than $\\| H \\| < 1$. In\naddition to solving more problems, our new method can work faster than the\nstandard algorithm. In numerical experiments on both synthetic and real world\nmatrices, we demonstrate the effectiveness of our new method.","url_abs":"http://arxiv.org/abs/1608.04361v1","url_pdf":"http://arxiv.org/pdf/1608.04361v1.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"abstracts"},"code_links":[{"paper_slug":"multi-way-monte-carlo-method-for-linear","repo_url":"https://github.com/wutao27/multi-way-MC","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":0,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}