{"id":{"repo_id":"vt","oai_identifier":"oai:vtechworks.lib.vt.edu:10919/50922"},"canonical_url":"https://search.dev.ndltd.org/etd/vt/oai:vtechworks.lib.vt.edu:10919/50922","repository":{"repo_id":"vt","name":"Virginia Tech","base_url":"https://vtechworks.lib.vt.edu/oai/request"},"display":{"title":"Analysis of the BiCG Method","abstract":"The Biconjugate Gradient (BiCG) method is an iterative Krylov subspace method that utilizes a 3-term recurrence. BiCG is the basis of several very popular methods, such as BiCGStab. The short recurrence makes BiCG preferable to other Krylov methods because of decreased memory usage and CPU time. However, BiCG does not satisfy any optimality conditions and it has been shown that for up to n/2-1 iterations, a special choice of the left starting vector can cause BiCG to follow {em any} 3-term recurrence. Despite this apparent sensitivity, BiCG often converges well in practice. This paper seeks to explain why BiCG converges so well, and what conditions can cause BiCG to behave poorly. We use tools such as the singular value decomposition and eigenvalue decomposition to establish bounds on the residuals of BiCG and make links between BiCG and optimal Krylov methods.","abstract_html":"The Biconjugate Gradient (BiCG) method is an iterative Krylov subspace method that utilizes a 3-term recurrence. BiCG is the basis of several very popular methods, such as BiCGStab. The short recurrence makes BiCG preferable to other Krylov methods because of decreased memory usage and CPU time. However, BiCG does not satisfy any optimality conditions and it has been shown that for up to n/2-1 iterations, a special choice of the left starting vector can cause BiCG to follow {em any} 3-term recurrence. Despite this apparent sensitivity, BiCG often converges well in practice. This paper seeks to explain why BiCG converges so well, and what conditions can cause BiCG to behave poorly. We use tools such as the singular value decomposition and eigenvalue decomposition to establish bounds on the residuals of BiCG and make links between BiCG and optimal Krylov methods.","abstract_has_math":false,"creators":["Renardy, Marissa"],"institution":"Virginia Tech","degree_name":"Master of Science","degree_level":"masters","degree_discipline":"Mathematics","degree_department":"Mathematics","school":null,"contributors":[],"advisors":[],"committee_chairs":["de Sturler, Eric"],"committee_members":["Rossi, John F.","Linnell, Peter A."],"year":2013,"date_issued":"2013-05-31","date_published":"2013-05-31","updated_at":"2026-07-24T05:56:01Z","subjects":["Krylov methods","BiCG","GMRES","FOM"],"languages":[],"rights":["In Copyright"],"rights_urls":["http://rightsstatements.org/vocab/InC/1.0/"],"identifier_entries":[{"key":"dc:identifier.other","label":"Dc Identifier Other","values":["vt_gsexam:934"],"render_values":[{"text":"vt_gsexam:934","href":null,"code":true}]}]},"links":{"outbound_url":"http://hdl.handle.net/10919/50922","outbound_label":"Handle","outbound_source":"dc:identifier.uri"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.committeechair","label":"Committee Chair","values":["de Sturler, Eric"]},{"key":"dc:contributor.committeemember","label":"Committee Member","values":["Rossi, John F.","Linnell, Peter A."]},{"key":"dc:contributor.department","label":"Department","values":["Mathematics"]},{"key":"dc:creator","label":"Author","values":["Renardy, Marissa"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.accessioned","label":"Dc Date Accessioned","values":["2014-11-23T07:00:09Z"]},{"key":"dc:date.available","label":"Dc Date Available","values":["2014-11-23T07:00:09Z"]},{"key":"dc:date.issued","label":"Date","values":["2013-05-31"]},{"key":"dc:publisher","label":"Institution","values":["Virginia Tech"]},{"key":"dc:type","label":"Dc Type","values":["Thesis"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["masters"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Master of Science"]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["Virginia Polytechnic Institute and State University"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Krylov methods","BiCG","GMRES","FOM"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:rights","label":"Dc Rights","values":["In Copyright"]},{"key":"dc:rights.uri","label":"Rights URI","values":["http://rightsstatements.org/vocab/InC/1.0/"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier.other","label":"Dc Identifier Other","values":["vt_gsexam:934"]},{"key":"dc:identifier.uri","label":"Identifier URI","values":["http://hdl.handle.net/10919/50922"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["The Biconjugate Gradient (BiCG) method is an iterative Krylov subspace method that utilizes a 3-term recurrence. BiCG is the basis of several very popular methods, such as BiCGStab. The short recurrence makes BiCG preferable to other Krylov methods because of decreased memory usage and CPU time. However, BiCG does not satisfy any optimality conditions and it has been shown that for up to n/2-1 iterations, a special choice of the left starting vector can cause BiCG to follow {em any} 3-term recurrence. Despite this apparent sensitivity, BiCG often converges well in practice. This paper seeks to explain why BiCG converges so well, and what conditions can cause BiCG to behave poorly. We use tools such as the singular value decomposition and eigenvalue decomposition to establish bounds on the residuals of BiCG and make links between BiCG and optimal Krylov methods."]},{"key":"dc:description.degree","label":"Dc Description Degree","values":["Master of Science"]},{"key":"dc:format.medium","label":"Dc Format Medium","values":["ETD"]},{"key":"dc:title","label":"Title","values":["Analysis of the BiCG Method"]}]}],"canonical_facts":{"dc:contributor.committeechair":["de Sturler, Eric"],"dc:contributor.committeemember":["Rossi, John F.","Linnell, Peter A."],"dc:contributor.department":["Mathematics"],"dc:creator":["Renardy, Marissa"],"dc:date.accessioned":["2014-11-23T07:00:09Z"],"dc:date.available":["2014-11-23T07:00:09Z"],"dc:date.issued":["2013-05-31"],"dc:description.abstract":["The Biconjugate Gradient (BiCG) method is an iterative Krylov subspace method that utilizes a 3-term recurrence. BiCG is the basis of several very popular methods, such as BiCGStab. The short recurrence makes BiCG preferable to other Krylov methods because of decreased memory usage and CPU time. However, BiCG does not satisfy any optimality conditions and it has been shown that for up to n/2-1 iterations, a special choice of the left starting vector can cause BiCG to follow {em any} 3-term recurrence. Despite this apparent sensitivity, BiCG often converges well in practice. This paper seeks to explain why BiCG converges so well, and what conditions can cause BiCG to behave poorly. We use tools such as the singular value decomposition and eigenvalue decomposition to establish bounds on the residuals of BiCG and make links between BiCG and optimal Krylov methods."],"dc:description.degree":["Master of Science"],"dc:format.medium":["ETD"],"dc:identifier.other":["vt_gsexam:934"],"dc:identifier.uri":["http://hdl.handle.net/10919/50922"],"dc:publisher":["Virginia Tech"],"dc:rights":["In Copyright"],"dc:rights.uri":["http://rightsstatements.org/vocab/InC/1.0/"],"dc:subject":["Krylov methods","BiCG","GMRES","FOM"],"dc:title":["Analysis of the BiCG Method"],"dc:type":["Thesis"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["masters"],"thesis:degree_name":["Master of Science"],"thesis:institution_name":["Virginia Polytechnic Institute and State University"]},"updated_at":"2026-07-24T05:56:01Z"}