{"id":153088,"date":"2007-04-01T00:00:00","date_gmt":"2007-04-01T00:00:00","guid":{"rendered":"https:\/\/www.microsoft.com\/en-us\/research\/msr-research-item\/iteratively-constructing-preconditioners-via-the-conjugate-gradient-method\/"},"modified":"2018-10-16T20:47:11","modified_gmt":"2018-10-17T03:47:11","slug":"iteratively-constructing-preconditioners-via-the-conjugate-gradient-method","status":"publish","type":"msr-research-item","link":"https:\/\/www.microsoft.com\/en-us\/research\/publication\/iteratively-constructing-preconditioners-via-the-conjugate-gradient-method\/","title":{"rendered":"Iteratively Constructing Preconditioners via the Conjugate Gradient Method"},"content":{"rendered":"<div class=\"asset-content\">\n<p>We consider the problem of solving a symmetric, positive definite system of linear equations. The most well-known and widely-used method for solving such systems is the preconditioned Conjugate Gradient method. The performance of this method depends crucially on knowing a good preconditioner matrix. We show that the Conjugate Gradient method itself can produce good preconditioners as a by-product. These preconditioners allow us to derive new asymptotic bounds on the time to solve multiple related linear systems.<\/p>\n<\/div>\n<p><!-- .asset-content --><\/p>\n","protected":false},"excerpt":{"rendered":"<p>We consider the problem of solving a symmetric, positive definite system of linear equations. The most well-known and widely-used method for solving such systems is the preconditioned Conjugate Gradient method. The performance of this method depends crucially on knowing a good preconditioner matrix. We show that the Conjugate Gradient method itself can produce good preconditioners [&hellip;]<\/p>\n","protected":false},"featured_media":0,"template":"","meta":{"msr-url-field":"","msr-podcast-episode":"","msrModifiedDate":"","msrModifiedDateEnabled":false,"ep_exclude_from_search":false,"_classifai_error":"","msr-author-ordering":null,"msr_publishername":"Association for Computing Machinery, Inc.","msr_publisher_other":"","msr_booktitle":"","msr_chapter":"","msr_edition":"","msr_editors":"","msr_how_published":"","msr_isbn":"","msr_issue":"","msr_journal":"","msr_number":"MSR-TR-2007-42","msr_organization":"","msr_pages_string":"10","msr_page_range_start":"10","msr_page_range_end":"","msr_series":"","msr_volume":"","msr_copyright":"Copyright \u00a9 2004 by the Association for Computing Machinery, Inc. Permission to make digital or hard copies of part or all of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, to republish, to post on servers, or to redistribute to lists, requires prior specific permission and\/or a fee. Request permissions from Publications Dept, ACM Inc., fax +1 (212) 869-0481, or permissions@acm.org. The definitive version of this paper can be found at ACM's Digital Library -http:\/\/www.acm.org\/dl\/.","msr_conference_name":"","msr_doi":"","msr_arxiv_id":"","msr_s2_paper_id":"","msr_mag_id":"","msr_pubmed_id":"","msr_other_authors":"","msr_other_contributors":"","msr_speaker":"","msr_award":"","msr_affiliation":"","msr_institution":"Microsoft Research","msr_host":"","msr_version":"","msr_duration":"","msr_original_fields_of_study":"","msr_release_tracker_id":"","msr_s2_match_type":"","msr_citation_count_updated":"","msr_published_date":"2007-04-01","msr_highlight_text":"","msr_notes":"","msr_longbiography":"","msr_publicationurl":"http:\/\/www.acm.org\/","msr_external_url":"","msr_secondary_video_url":"","msr_conference_url":"","msr_journal_url":"","msr_s2_pdf_url":"","msr_year":2007,"msr_citation_count":0,"msr_influential_citations":0,"msr_reference_count":0,"msr_s2_match_confidence":0,"msr_microsoftintellectualproperty":true,"msr_s2_open_access":false,"msr_s2_author_ids":[],"msr_pub_ids":[],"msr_hide_image_in_river":0,"footnotes":""},"msr-research-highlight":[],"research-area":[13560],"msr-publication-type":[193718],"msr-publisher":[],"msr-focus-area":[],"msr-locale":[268875],"msr-post-option":[],"msr-field-of-study":[],"msr-conference":[],"msr-journal":[],"msr-impact-theme":[],"msr-pillar":[],"class_list":["post-153088","msr-research-item","type-msr-research-item","status-publish","hentry","msr-research-area-programming-languages-software-engineering","msr-locale-en_us"],"msr_publishername":"Association for Computing Machinery, Inc.","msr_edition":"","msr_affiliation":"","msr_published_date":"2007-04-01","msr_host":"","msr_duration":"","msr_version":"","msr_speaker":"","msr_other_contributors":"","msr_booktitle":"","msr_pages_string":"10","msr_chapter":"","msr_isbn":"","msr_journal":"","msr_volume":"","msr_number":"MSR-TR-2007-42","msr_editors":"","msr_series":"","msr_issue":"","msr_organization":"","msr_how_published":"","msr_notes":"","msr_highlight_text":"","msr_release_tracker_id":"","msr_original_fields_of_study":"","msr_download_urls":"","msr_external_url":"","msr_secondary_video_url":"","msr_longbiography":"","msr_microsoftintellectualproperty":1,"msr_main_download":"226684","msr_publicationurl":"http:\/\/www.acm.org\/","msr_doi":"","msr_publication_uploader":[{"type":"file","title":"tr-2007-42.pdf","viewUrl":"https:\/\/www.microsoft.com\/en-us\/research\/wp-content\/uploads\/2007\/04\/tr-2007-42.pdf","id":226684,"label_id":0},{"type":"url","title":"http:\/\/www.acm.org\/","viewUrl":false,"id":false,"label_id":0}],"msr_related_uploader":"","msr_citation_count":0,"msr_citation_count_updated":"","msr_s2_paper_id":"","msr_influential_citations":0,"msr_reference_count":0,"msr_arxiv_id":"","msr_s2_author_ids":[],"msr_s2_open_access":false,"msr_s2_pdf_url":null,"msr_attachments":[{"id":0,"url":"http:\/\/www.acm.org\/"},{"id":226684,"url":"https:\/\/www.microsoft.com\/en-us\/research\/wp-content\/uploads\/2007\/04\/tr-2007-42.pdf"}],"msr-author-ordering":[{"type":"user_nicename","value":"jdunagan","user_id":32212,"rest_url":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/microsoft-research\/v1\/researchers?person=jdunagan"},{"type":"text","value":"Nicholas J.A. 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