{"id":"https://openalex.org/W7207556037","doi":"https://doi.org/10.48550/arxiv.2609.02081","title":"Biquaternion Algebra with Bilinear Multiplication: A General, Elegant, and Computationally Advantageous Framework for Relativistic Electronic Structure Calculations on CPUs and GPUs","display_name":"Biquaternion Algebra with Bilinear Multiplication: A General, Elegant, and Computationally Advantageous Framework for Relativistic Electronic Structure Calculations on CPUs and GPUs","publication_year":2026,"publication_date":"2026-09-02","ids":{"openalex":"https://openalex.org/W7207556037","doi":"https://doi.org/10.48550/arxiv.2609.02081"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2609.02081","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.02081","pdf_url":"https://arxiv.org/pdf/2609.02081","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"volume":null,"type":"preprint","is_traditional":true,"indexed_in":["arxiv","datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/2609.02081","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5149963546","display_name":"Sylvia Kaviraj","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I74788687","display_name":"Comenius University Bratislava","ror":"https://ror.org/0587ef340","country_code":"SK","type":"education","lineage":["https://openalex.org/I74788687"]}],"countries":["SK"],"is_corresponding":false,"raw_author_name":"Kaviraj, Sylvia","raw_affiliation_strings":["Department of Physical and Theoretical Chemistry ,","Faculty of Natural Sciences , Comenius University in Bratislava , Slovakia"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Physical and Theoretical Chemistry ,","institution_ids":[]},{"raw_affiliation_string":"Faculty of Natural Sciences , Comenius University in Bratislava , Slovakia","institution_ids":["https://openalex.org/I74788687"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5091751684","display_name":"Stanislav Komorovsk\u00fd","orcid":"https://orcid.org/0000-0002-5317-7200","observed_orcids":["https://orcid.org/0000-0002-5317-7200"]},"institutions":[{"id":"https://openalex.org/I207624831","display_name":"Slovak Academy of Sciences","ror":"https://ror.org/03h7qq074","country_code":"SK","type":"government","lineage":["https://openalex.org/I207624831"]},{"id":"https://openalex.org/I4210111833","display_name":"Institute of Inorganic Chemistry of the Slovak Academy of Sciences","ror":"https://ror.org/01tpajh44","country_code":"SK","type":"facility","lineage":["https://openalex.org/I207624831","https://openalex.org/I4210111833"]}],"countries":["SK"],"is_corresponding":false,"raw_author_name":"Komorovsky, Stanislav","raw_affiliation_strings":["Institute of Inorganic Chemistry , Slovak Academy of Sciences , Bratislava , Slovakia"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Institute of Inorganic Chemistry , Slovak Academy of Sciences , Bratislava , Slovakia","institution_ids":["https://openalex.org/I207624831","https://openalex.org/I4210111833"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5084672632","display_name":"Trond Saue","orcid":"https://orcid.org/0000-0001-6407-0305","observed_orcids":["https://orcid.org/0000-0001-6407-0305"]},"institutions":[{"id":"https://openalex.org/I1294671590","display_name":"Centre National de la Recherche Scientifique","ror":"https://ror.org/02feahw73","country_code":"FR","type":"government","lineage":["https://openalex.org/I1294671590"]},{"id":"https://openalex.org/I4210144120","display_name":"Laboratoire de Chimie et Physique Quantiques","ror":"https://ror.org/056g7f250","country_code":"FR","type":"facility","lineage":["https://openalex.org/I1294671590","https://openalex.org/I4210128300","https://openalex.org/I4210144120","https://openalex.org/I4405258862"]},{"id":"https://openalex.org/I4405258862","display_name":"Universit\u00e9 de Toulouse","ror":"https://ror.org/01ahyrz84","country_code":"FR","type":"education","lineage":["https://openalex.org/I4405258862"]}],"countries":["FR"],"is_corresponding":false,"raw_author_name":"Saue, Trond","raw_affiliation_strings":["Laboratoire de Chimie et Physique Quantiques ,","UMR 5626 CNRS - Universit\u00e9 de Toulouse , France"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Laboratoire de Chimie et Physique Quantiques ,","institution_ids":["https://openalex.org/I4210144120"]},{"raw_affiliation_string":"UMR 5626 CNRS - Universit\u00e9 de Toulouse , France","institution_ids":["https://openalex.org/I1294671590","https://openalex.org/I4210144120","https://openalex.org/I4405258862"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5039933961","display_name":"Michal Repisk\u00fd","orcid":"https://orcid.org/0000-0003-0776-4137","observed_orcids":["https://orcid.org/0000-0003-0776-4137"]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Repisky, Michal","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":6,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":true,"cited_by_count":0,"citation_normalized_percentile":null,"cited_by_percentile_year":null,"biblio":{"volume":null,"issue":null,"first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T12037","display_name":"Algebraic and Geometric Analysis","score":0.8105000257492065,"subfield":{"id":"https://openalex.org/subfields/2604","display_name":"Applied Mathematics"},"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/T12037","display_name":"Algebraic and Geometric Analysis","score":0.8105000257492065,"subfield":{"id":"https://openalex.org/subfields/2604","display_name":"Applied Mathematics"},"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/T10054","display_name":"Parallel Computing and Optimization Techniques","score":0.09719999879598618,"subfield":{"id":"https://openalex.org/subfields/1708","display_name":"Hardware and Architecture"},"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/T10792","display_name":"Matrix Theory and Algorithms","score":0.05739999935030937,"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/quaternion-algebras","display_name":"quaternion algebras","score":0.7940000295639038},{"id":"https://openalex.org/keywords/relativistic-electronic-structure","display_name":"relativistic electronic structure","score":0.7950000166893005},{"id":"https://openalex.org/keywords/time-reversal-symmetry","display_name":"time-reversal symmetry","score":0.8569999933242798},{"id":"https://openalex.org/keywords/complex-quaternions","display_name":"complex quaternions","score":0.7129999995231628},{"id":"https://openalex.org/keywords/respect","display_name":"respect","score":0.8180000185966492},{"id":"https://openalex.org/keywords/gpu-acceleration","display_name":"GPU acceleration","score":0.7009999752044678},{"id":"https://openalex.org/keywords/hybrid-cpu-gpu-computing","display_name":"CPU-GPU hybrid computing","score":0.7289999723434448},{"id":"https://openalex.org/keywords/matrix-diagonalization","display_name":"diagonalization","score":0.8149999976158142},{"id":"https://openalex.org/keywords/basis-functions","display_name":"basis functions","score":0.46399998664855957},{"id":"https://openalex.org/keywords/expectation-values","display_name":"expectation values","score":0.4519999921321869}],"concepts":[{"id":"https://openalex.org/C200127275","wikidata":"https://www.wikidata.org/wiki/Q173853","display_name":"Quaternion","level":2,"score":0.7921000123023987},{"id":"https://openalex.org/C136119220","wikidata":"https://www.wikidata.org/wiki/Q1000660","display_name":"Algebra over a field","level":2,"score":0.5547999739646912},{"id":"https://openalex.org/C2776359362","wikidata":"https://www.wikidata.org/wiki/Q2145286","display_name":"Representation (politics)","level":3,"score":0.5011000037193298},{"id":"https://openalex.org/C106487976","wikidata":"https://www.wikidata.org/wiki/Q685816","display_name":"Matrix (chemical analysis)","level":2,"score":0.4945000112056732},{"id":"https://openalex.org/C17349429","wikidata":"https://www.wikidata.org/wiki/Q1049914","display_name":"Matrix multiplication","level":3,"score":0.49399998784065247},{"id":"https://openalex.org/C205203396","wikidata":"https://www.wikidata.org/wiki/Q612143","display_name":"Bilinear interpolation","level":2,"score":0.46219998598098755},{"id":"https://openalex.org/C151640129","wikidata":"https://www.wikidata.org/wiki/Q2835967","display_name":"Quaternion algebra","level":5,"score":0.43939998745918274},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.4390999972820282},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.4178999960422516},{"id":"https://openalex.org/C2780595030","wikidata":"https://www.wikidata.org/wiki/Q3860309","display_name":"Multiplication (music)","level":2,"score":0.40290001034736633},{"id":"https://openalex.org/C90805587","wikidata":"https://www.wikidata.org/wiki/Q10944557","display_name":"Word (group theory)","level":2,"score":0.39430001378059387},{"id":"https://openalex.org/C103275481","wikidata":"https://www.wikidata.org/wiki/Q6787889","display_name":"Matrix representation","level":3,"score":0.37959998846054077},{"id":"https://openalex.org/C12426560","wikidata":"https://www.wikidata.org/wiki/Q189569","display_name":"Basis (linear algebra)","level":2,"score":0.37540000677108765},{"id":"https://openalex.org/C33676613","wikidata":"https://www.wikidata.org/wiki/Q13415176","display_name":"Dimension (graph theory)","level":2,"score":0.3531000018119812},{"id":"https://openalex.org/C139352143","wikidata":"https://www.wikidata.org/wiki/Q82571","display_name":"Linear algebra","level":2,"score":0.31779998540878296},{"id":"https://openalex.org/C45374587","wikidata":"https://www.wikidata.org/wiki/Q12525525","display_name":"Computation","level":2,"score":0.31630000472068787},{"id":"https://openalex.org/C26517878","wikidata":"https://www.wikidata.org/wiki/Q228039","display_name":"Key (lock)","level":2,"score":0.31279999017715454},{"id":"https://openalex.org/C75165309","wikidata":"https://www.wikidata.org/wiki/Q2258979","display_name":"Search engine indexing","level":2,"score":0.296099990606308},{"id":"https://openalex.org/C200106649","wikidata":"https://www.wikidata.org/wiki/Q223683","display_name":"Transpose","level":3,"score":0.27459999918937683},{"id":"https://openalex.org/C203249530","wikidata":"https://www.wikidata.org/wiki/Q837414","display_name":"Hypercomplex number","level":3,"score":0.26750001311302185},{"id":"https://openalex.org/C49154492","wikidata":"https://www.wikidata.org/wiki/Q5300","display_name":"Central processing unit","level":2,"score":0.2624000012874603},{"id":"https://openalex.org/C198082693","wikidata":"https://www.wikidata.org/wiki/Q1551631","display_name":"Gr\u00f6bner basis","level":3,"score":0.2531999945640564}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2609.02081","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.02081","pdf_url":"https://arxiv.org/pdf/2609.02081","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},{"id":"doi:10.48550/arxiv.2609.02081","doi":"https://doi.org/10.48550/arxiv.2609.02081","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2609.02081","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:2609.02081","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.02081","pdf_url":"https://arxiv.org/pdf/2609.02081","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"sustainable_development_goals":[],"study_designs":[],"awards":[{"id":"https://openalex.org/G165883175","display_name":null,"funder_award_id":"09I05-03-V02-00034","funder_id":"https://openalex.org/F4320320300","funder_display_name":"European Commission"},{"id":"https://openalex.org/G3281168586","display_name":null,"funder_award_id":"315822","funder_id":"https://openalex.org/F4320323299","funder_display_name":"Norges Forskningsr\u00e5d"},{"id":"https://openalex.org/G481740877","display_name":null,"funder_award_id":"262695","funder_id":"https://openalex.org/F4320323299","funder_display_name":"Norges Forskningsr\u00e5d"},{"id":"https://openalex.org/G5877418170","display_name":null,"funder_award_id":"101019907","funder_id":"https://openalex.org/F4320320300","funder_display_name":"European Commission"},{"id":"https://openalex.org/G5913003182","display_name":"Jadrov\u00e1 magnetick\u00e1 rezonan\u010dn\u00e1 spektroskopia spin-orbit\u00e1lne spriahnut\u00fdch paramagnetick\u00fdch tuh\u00fdch l\u00e1tok","funder_award_id":"1/0670/24","funder_id":"https://openalex.org/F4320323641","funder_display_name":"Vedeck\u00e1 Grantov\u00e1 Agent\u00fara M\u0160VVa\u0160 SR a SAV"},{"id":"https://openalex.org/G6483025576","display_name":"Magnetick\u00e9 materi\u00e1ly pre nanotechnol\u00f3gie","funder_award_id":"DS-FR-24-0045","funder_id":"https://openalex.org/F4320323251","funder_display_name":"Agent\u00fara na Podporu V\u00fdskumu a V\u00fdvoja"},{"id":"https://openalex.org/G6762430579","display_name":null,"funder_award_id":"09I05-03-V02","funder_id":"https://openalex.org/F4320320300","funder_display_name":"European Commission"},{"id":"https://openalex.org/G7374094195","display_name":null,"funder_award_id":"09I05-03-V02","funder_id":"https://openalex.org/F4320323251","funder_display_name":"Agent\u00fara na Podporu V\u00fdskumu a V\u00fdvoja"}],"funders":[{"id":"https://openalex.org/F4320320300","display_name":"European Commission","ror":"https://ror.org/00k4n6c32"},{"id":"https://openalex.org/F4320323251","display_name":"Agent\u00fara na Podporu V\u00fdskumu a V\u00fdvoja","ror":"https://ror.org/037nx0e70"},{"id":"https://openalex.org/F4320323299","display_name":"Norges Forskningsr\u00e5d","ror":"https://ror.org/00epmv149"},{"id":"https://openalex.org/F4320323641","display_name":"Vedeck\u00e1 Grantov\u00e1 Agent\u00fara M\u0160VVa\u0160 SR a SAV","ror":"https://ror.org/044gwpv05"},{"id":"https://openalex.org/F4320331617","display_name":"National Supercomputing Centre Singapore","ror":"https://ror.org/05qbsk739"}],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7207556037.pdf","grobid_xml":"https://content.openalex.org/works/W7207556037.grobid-xml"},"referenced_works_count":0,"referenced_works":[],"related_works":["https://openalex.org/W4396639216","https://openalex.org/W4406277792","https://openalex.org/W7163036503","https://openalex.org/W752783541","https://openalex.org/W7143694747","https://openalex.org/W2199871499","https://openalex.org/W2038232699","https://openalex.org/W4390064738","https://openalex.org/W2920802704","https://openalex.org/W2316223668"],"abstract_inverted_index":{"Quaternion":[0],"algebra":[1,31,193],"provides":[2],"a":[3,48,70,90,195],"natural":[4],"representation":[5,20,118],"of":[6,101],"time-reversal-symmetric":[7],"matrix":[8,72,124,161,172],"structures":[9,17],"in":[10],"relativistic":[11,203],"electronic-structure":[12,204],"theory,":[13],"whereas":[14],"complementary":[15],"time-reversal-antisymmetric":[16],"extend":[18],"this":[19,63],"to":[21,107,181],"complex":[22],"quaternions,":[23],"or":[24],"biquaternions.":[25],"Here,":[26],"we":[27,66],"explicitly":[28],"exploit":[29],"biquaternion":[30,71,95,117,143,192],"for":[32,79,93,201],"fundamental":[33],"objects,":[34],"including":[35],"operators,":[36],"kinetically":[37],"and":[38,43,54,82,137,171,177,185,197],"magnetically":[39],"balanced":[40],"basis":[41],"functions,":[42],"their":[44],"expectation":[45],"values,":[46],"within":[47,75],"unified":[49],"framework":[50,64,200],"encompassing":[51],"real,":[52],"complex,":[53],"real":[55,102],"quaternion":[56],"subalgebras":[57],"as":[58,194],"special":[59],"cases.":[60],"To":[61],"realize":[62],"computationally,":[65],"have":[67],"developed":[68],"HMATLIB,":[69],"library":[73],"implemented":[74],"the":[76,99,111,116,121,127,142,152],"ReSpect":[77],"package":[78],"pure":[80,168],"CPU":[81,136,169,182],"hybrid":[83,156],"CPU/GPU":[84,157],"execution.":[85],"A":[86],"key":[87],"development":[88],"is":[89],"bilinear":[91,160],"algorithm":[92],"matrix-valued":[94],"multiplication":[96,162],"that":[97,141],"reduces":[98],"number":[100],"matrix--matrix":[103],"multiplications":[104],"from":[105],"64":[106],"24":[108],"compared":[109],"with":[110],"conventional":[112],"component-wise":[113],"approach,":[114],"while":[115],"effectively":[119],"doubles":[120],"maximum":[122],"accessible":[123],"dimension":[125],"under":[126],"same":[128],"32-bit":[129],"indexing":[130],"constraint.":[131],"Numerical":[132],"benchmarks":[133],"on":[134],"modern":[135,202],"GPU":[138],"architectures":[139],"demonstrate":[140],"formulation":[144],"consistently":[145],"outperforms":[146],"its":[147],"isomorphic":[148],"complex-algebra":[149],"counterpart.":[150],"For":[151],"largest":[153],"matrices":[154],"reported,":[155],"execution":[158,170],"accelerates":[159],"by":[163,174],"approximately":[164,175],"5-8":[165],"times":[166,179],"over":[167],"diagonalization":[173],"12":[176],"61":[178],"relative":[180],"oneAPI":[183],"MKL":[184],"NVHPC":[186],"OpenBLAS,":[187],"respectively.":[188],"These":[189],"results":[190],"establish":[191],"general":[196],"computationally":[198],"advantageous":[199],"calculations.":[205]},"counts_by_year":[],"updated_date":"2026-11-06T12:05:46.655746","created_date":"2026-09-04T00:00:00"}