{"id":"https://openalex.org/W2997929922","doi":"https://doi.org/10.1137/19m1298263","title":"Exploiting Lower Precision Arithmetic in Solving Symmetric Positive Definite Linear Systems and Least Squares Problems","display_name":"Exploiting Lower Precision Arithmetic in Solving Symmetric Positive Definite Linear Systems and Least Squares Problems","publication_year":2021,"publication_date":"2021-01-01","ids":{"openalex":"https://openalex.org/W2997929922","doi":"https://doi.org/10.1137/19m1298263","mag":"2997929922"},"language":"en","primary_location":{"id":"doi:10.1137/19m1298263","is_oa":true,"landing_page_url":"https://doi.org/10.1137/19m1298263","pdf_url":"https://epubs.siam.org/doi/pdf/10.1137/19M1298263","source":{"id":"https://openalex.org/S165512578","display_name":"SIAM Journal on Scientific Computing","issn_l":"1064-8275","issn":["1064-8275","1095-7197"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320508","host_organization_name":"Society for Industrial and Applied Mathematics","host_organization_lineage":["https://openalex.org/P4310320508"],"host_organization_lineage_names":["Society for Industrial and Applied Mathematics"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"SIAM Journal on Scientific Computing","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":true,"oa_status":"hybrid","oa_url":"https://epubs.siam.org/doi/pdf/10.1137/19M1298263","any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5069339254","display_name":"Nicholas J. Higham","orcid":"https://orcid.org/0000-0001-5956-4976"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Nicholas J. Higham","raw_affiliation_strings":[],"raw_orcid":"https://orcid.org/0000-0001-5956-4976","affiliations":[]},{"author_position":"last","author":{"id":"https://openalex.org/A5070702896","display_name":"Srikara Pranesh","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Srikara Pranesh","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":2,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":6.25,"has_fulltext":false,"cited_by_count":33,"citation_normalized_percentile":{"value":0.97164546,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":95,"max":99},"biblio":{"volume":"43","issue":"1","first_page":"A258","last_page":"A277"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10963","display_name":"Advanced Optimization Algorithms Research","score":0.9979000091552734,"subfield":{"id":"https://openalex.org/subfields/2612","display_name":"Numerical Analysis"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T10963","display_name":"Advanced Optimization Algorithms Research","score":0.9979000091552734,"subfield":{"id":"https://openalex.org/subfields/2612","display_name":"Numerical Analysis"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10792","display_name":"Matrix Theory and Algorithms","score":0.996999979019165,"subfield":{"id":"https://openalex.org/subfields/1703","display_name":"Computational Theory and Mathematics"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11697","display_name":"Numerical Methods and Algorithms","score":0.9951000213623047,"subfield":{"id":"https://openalex.org/subfields/1703","display_name":"Computational Theory and Mathematics"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.8457095623016357},{"id":"https://openalex.org/keywords/positive-definite-matrix","display_name":"Positive-definite matrix","score":0.6576215028762817},{"id":"https://openalex.org/keywords/least-squares-function-approximation","display_name":"Least-squares function approximation","score":0.5998495817184448},{"id":"https://openalex.org/keywords/non-linear-least-squares","display_name":"Non-linear least squares","score":0.5469480156898499},{"id":"https://openalex.org/keywords/linear-least-squares","display_name":"Linear least squares","score":0.5447171926498413},{"id":"https://openalex.org/keywords/arithmetic","display_name":"Arithmetic","score":0.5183971524238586},{"id":"https://openalex.org/keywords/generalized-least-squares","display_name":"Generalized least squares","score":0.47137224674224854},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.4544714689254761},{"id":"https://openalex.org/keywords/linear-system","display_name":"Linear system","score":0.44612541794776917},{"id":"https://openalex.org/keywords/iteratively-reweighted-least-squares","display_name":"Iteratively reweighted least squares","score":0.4149879813194275},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.3349931836128235},{"id":"https://openalex.org/keywords/explained-sum-of-squares","display_name":"Explained sum of squares","score":0.23100990056991577},{"id":"https://openalex.org/keywords/statistics","display_name":"Statistics","score":0.21146148443222046},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.1904067099094391},{"id":"https://openalex.org/keywords/linear-model","display_name":"Linear model","score":0.1803724467754364},{"id":"https://openalex.org/keywords/eigenvalues-and-eigenvectors","display_name":"Eigenvalues and eigenvectors","score":0.07363036274909973}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8457095623016357},{"id":"https://openalex.org/C49712288","wikidata":"https://www.wikidata.org/wiki/Q77601250","display_name":"Positive-definite matrix","level":3,"score":0.6576215028762817},{"id":"https://openalex.org/C9936470","wikidata":"https://www.wikidata.org/wiki/Q6510405","display_name":"Least-squares function approximation","level":3,"score":0.5998495817184448},{"id":"https://openalex.org/C45923927","wikidata":"https://www.wikidata.org/wiki/Q3319230","display_name":"Non-linear least squares","level":3,"score":0.5469480156898499},{"id":"https://openalex.org/C58663186","wikidata":"https://www.wikidata.org/wiki/Q6553462","display_name":"Linear least squares","level":3,"score":0.5447171926498413},{"id":"https://openalex.org/C94375191","wikidata":"https://www.wikidata.org/wiki/Q11205","display_name":"Arithmetic","level":1,"score":0.5183971524238586},{"id":"https://openalex.org/C188649462","wikidata":"https://www.wikidata.org/wiki/Q2246261","display_name":"Generalized least squares","level":3,"score":0.47137224674224854},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.4544714689254761},{"id":"https://openalex.org/C6802819","wikidata":"https://www.wikidata.org/wiki/Q1072174","display_name":"Linear system","level":2,"score":0.44612541794776917},{"id":"https://openalex.org/C126090379","wikidata":"https://www.wikidata.org/wiki/Q6094424","display_name":"Iteratively reweighted least squares","level":4,"score":0.4149879813194275},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.3349931836128235},{"id":"https://openalex.org/C49847556","wikidata":"https://www.wikidata.org/wiki/Q3964631","display_name":"Explained sum of squares","level":2,"score":0.23100990056991577},{"id":"https://openalex.org/C105795698","wikidata":"https://www.wikidata.org/wiki/Q12483","display_name":"Statistics","level":1,"score":0.21146148443222046},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.1904067099094391},{"id":"https://openalex.org/C163175372","wikidata":"https://www.wikidata.org/wiki/Q3339222","display_name":"Linear model","level":2,"score":0.1803724467754364},{"id":"https://openalex.org/C158693339","wikidata":"https://www.wikidata.org/wiki/Q190524","display_name":"Eigenvalues and eigenvectors","level":2,"score":0.07363036274909973},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.0},{"id":"https://openalex.org/C185429906","wikidata":"https://www.wikidata.org/wiki/Q1130160","display_name":"Estimator","level":2,"score":0.0},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.0}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.1137/19m1298263","is_oa":true,"landing_page_url":"https://doi.org/10.1137/19m1298263","pdf_url":"https://epubs.siam.org/doi/pdf/10.1137/19M1298263","source":{"id":"https://openalex.org/S165512578","display_name":"SIAM Journal on Scientific Computing","issn_l":"1064-8275","issn":["1064-8275","1095-7197"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320508","host_organization_name":"Society for Industrial and Applied Mathematics","host_organization_lineage":["https://openalex.org/P4310320508"],"host_organization_lineage_names":["Society for Industrial and Applied Mathematics"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"SIAM Journal on Scientific Computing","raw_type":"journal-article"},{"id":"pmh:oai:pure.atira.dk:openaire_cris_publications/e5ba680e-85ec-499b-982a-8dbd2c309c15","is_oa":false,"landing_page_url":"https://research.manchester.ac.uk/en/publications/e5ba680e-85ec-499b-982a-8dbd2c309c15","pdf_url":null,"source":{"id":"https://openalex.org/S4306400662","display_name":"Research Explorer (The University of Manchester)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I28407311","host_organization_name":"University of Manchester","host_organization_lineage":["https://openalex.org/I28407311"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Higham, N & Pranesh, S 2020, 'EXPLOITING LOWER PRECISION ARITHMETIC IN SOLVING SYMMETRIC POSITIVE DEFINITE LINEAR SYSTEMS AND LEAST SQUARES PROBLEMS', SIAM Journal on Scientific Computing.","raw_type":"info:eu-repo/semantics/publishedVersion"},{"id":"pmh:oai:eprints.maths.manchester.ac.uk.MIMS.EPrints:2736","is_oa":false,"landing_page_url":null,"pdf_url":null,"source":{"id":"https://openalex.org/S4306400452","display_name":"MIMS EPrints (University of Southampton)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I43439940","host_organization_name":"University of Southampton","host_organization_lineage":["https://openalex.org/I43439940"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"acceptedVersion","is_accepted":true,"is_published":false,"raw_source_name":"","raw_type":"MIMS Preprint"}],"best_oa_location":{"id":"doi:10.1137/19m1298263","is_oa":true,"landing_page_url":"https://doi.org/10.1137/19m1298263","pdf_url":"https://epubs.siam.org/doi/pdf/10.1137/19M1298263","source":{"id":"https://openalex.org/S165512578","display_name":"SIAM Journal on Scientific Computing","issn_l":"1064-8275","issn":["1064-8275","1095-7197"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320508","host_organization_name":"Society for Industrial and Applied Mathematics","host_organization_lineage":["https://openalex.org/P4310320508"],"host_organization_lineage_names":["Society for Industrial and Applied Mathematics"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"SIAM Journal on Scientific Computing","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G1322834978","display_name":"Inference, COmputation and Numerics for Insights into Cities (ICONIC)","funder_award_id":"EP/P020720/1","funder_id":"https://openalex.org/F4320334627","funder_display_name":"Engineering and Physical Sciences Research Council"},{"id":"https://openalex.org/G1935751110","display_name":null,"funder_award_id":"EP/P020720/1","funder_id":"https://openalex.org/F4320334627","funder_display_name":"Engineering and Physical Sciences Research Council"}],"funders":[{"id":"https://openalex.org/F4320315144","display_name":"MathWorks","ror":"https://ror.org/01n8qtk87"},{"id":"https://openalex.org/F4320320006","display_name":"Royal Society","ror":"https://ror.org/03wnrjx87"},{"id":"https://openalex.org/F4320334627","display_name":"Engineering and Physical Sciences Research Council","ror":"https://ror.org/0439y7842"}],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":30,"referenced_works":["https://openalex.org/W36826159","https://openalex.org/W1483820693","https://openalex.org/W1506342804","https://openalex.org/W1558296484","https://openalex.org/W1985536181","https://openalex.org/W2035080386","https://openalex.org/W2041033288","https://openalex.org/W2053627399","https://openalex.org/W2078643711","https://openalex.org/W2078675418","https://openalex.org/W2109616222","https://openalex.org/W2140918795","https://openalex.org/W2162322364","https://openalex.org/W2169150754","https://openalex.org/W2481847753","https://openalex.org/W2599973081","https://openalex.org/W2752879223","https://openalex.org/W2808102735","https://openalex.org/W2895305554","https://openalex.org/W2903589674","https://openalex.org/W2910374893","https://openalex.org/W2937715180","https://openalex.org/W2999942548","https://openalex.org/W3006375150","https://openalex.org/W3007862902","https://openalex.org/W3009762362","https://openalex.org/W3022909171","https://openalex.org/W3035977869","https://openalex.org/W3036263969","https://openalex.org/W3108141724"],"related_works":["https://openalex.org/W4380487384","https://openalex.org/W1546276525","https://openalex.org/W4378552425","https://openalex.org/W1959514624","https://openalex.org/W2034819163","https://openalex.org/W2368345218","https://openalex.org/W79007405","https://openalex.org/W2388513917","https://openalex.org/W2026673180","https://openalex.org/W4319880936"],"abstract_inverted_index":{"What":[0],"is":[1,20,30,131,199],"the":[2,55,97,102,114,118,134,143,157,170,189,197,209,221,247,250],"fastest":[3],"way":[4],"to":[5,133,152,172,228,253],"solve":[6,173],"a":[7,15,93,121,140,174,180],"linear":[8,175],"system":[9],"$Ax=":[10],"b$":[11],"in":[12,106,150,163,241,257],"arithmetic":[13,50],"of":[14,51,63,72,113,124,137,142,159,196,249,263],"given":[16,48],"precision":[17,52,57,99,166,203],"when":[18,161],"$A$":[19,36],"symmetric":[21],"positive":[22],"definite":[23],"and":[24,76,79,86,100,155,187,208],"otherwise":[25],"unstructured?":[26],"The":[27,66],"usual":[28],"answer":[29],"by":[31,120,139,185,220],"Cholesky":[32,94],"factorization,":[33],"assuming":[34],"that":[35,44,205,233,259],"can":[37,45],"be":[38,46,71,229],"factorized.":[39],"We":[40,91,127,146,168,215],"develop":[41],"an":[42,49],"algorithm":[43,171],"faster,":[47],"lower":[53,98],"than":[54],"working":[56,162],"as":[58,60,104],"well":[59,181],"(optionally)":[61],"one":[62],"higher":[64],"precision.":[65],"arithmetics":[67],"might,":[68],"for":[69],"example,":[70],"precisions":[73,262],"half,":[74],"single,":[75],"double;":[77],"half":[78,165],"double,":[80,87],"possibly":[81,88],"with":[82,89,179],"quadruple;":[83],"or":[84],"single":[85],"quadruple.":[90],"compute":[92],"factorization":[95,115],"at":[96,201],"use":[101],"factors":[103],"preconditioners":[105],"GMRES-based":[107],"iterative":[108,206,211],"refinement.":[109],"To":[110],"avoid":[111,153],"breakdown":[112],"we":[116],"shift":[117],"matrix":[119,184],"small":[122],"multiple":[123,141,261],"its":[125],"diagonal.":[126],"explain":[128,216],"why":[129,217],"this":[130,234],"preferable":[132],"common":[135],"approach":[136],"shifting":[138],"identity":[144],"matrix,":[145],"also":[147],"incorporate":[148],"scaling":[149],"order":[151],"overflow":[154],"reduce":[156],"chance":[158],"underflow":[160],"IEEE":[164],"arithmetic.":[167,264],"extend":[169],"least":[176],"squares":[177],"problem":[178],"conditioned":[182],"coefficient":[183],"forming":[186],"solving":[188],"normal":[190],"equations.":[191],"In":[192],"both":[193],"algorithms":[194,252],"most":[195],"work":[198],"done":[200],"low":[202],"provided":[204],"refinement":[207],"inner":[210],"solver":[212],"converge":[213],"quickly.":[214],"replacing":[218],"GMRES":[219],"conjugate":[222],"gradient":[223],"method":[224],"causes":[225],"convergence":[226,240],"guarantees":[227],"lost,":[230],"but":[231],"show":[232],"change":[235],"has":[236],"little":[237],"effect":[238],"on":[239],"practice.":[242],"Our":[243],"numerical":[244],"experiments":[245],"confirm":[246],"potential":[248],"new":[251],"provide":[254],"faster":[255],"solutions":[256],"environments":[258],"support":[260]},"counts_by_year":[{"year":2026,"cited_by_count":1},{"year":2025,"cited_by_count":9},{"year":2024,"cited_by_count":6},{"year":2023,"cited_by_count":5},{"year":2022,"cited_by_count":4},{"year":2021,"cited_by_count":5},{"year":2020,"cited_by_count":3}],"updated_date":"2026-06-11T09:08:48.828518","created_date":"2025-10-10T00:00:00"}
