{"id":"https://openalex.org/W4387406806","doi":"https://doi.org/10.1137/22m1515173","title":"Convergence Analysis of Leapfrog for Geodesics","display_name":"Convergence Analysis of Leapfrog for Geodesics","publication_year":2023,"publication_date":"2023-10-06","ids":{"openalex":"https://openalex.org/W4387406806","doi":"https://doi.org/10.1137/22m1515173"},"language":"en","primary_location":{"id":"doi:10.1137/22m1515173","is_oa":true,"landing_page_url":"https://doi.org/10.1137/22m1515173","pdf_url":null,"source":{"id":"https://openalex.org/S203348814","display_name":"SIAM Journal on Numerical Analysis","issn_l":"0036-1429","issn":["0036-1429","1095-7170"],"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 Numerical Analysis","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":true,"oa_status":"hybrid","oa_url":"https://doi.org/10.1137/22m1515173","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5014124196","display_name":"Erchuan Zhang","orcid":"https://orcid.org/0000-0002-4005-5431"},"institutions":[{"id":"https://openalex.org/I12079687","display_name":"Edith Cowan University","ror":"https://ror.org/05jhnwe22","country_code":"AU","type":"education","lineage":["https://openalex.org/I12079687"]},{"id":"https://openalex.org/I177877127","display_name":"The University of Western Australia","ror":"https://ror.org/047272k79","country_code":"AU","type":"education","lineage":["https://openalex.org/I177877127"]}],"countries":["AU"],"is_corresponding":true,"raw_author_name":"Erchuan Zhang","raw_affiliation_strings":["Department of Mathematics and Statistics, The University of Western Australia, 35 Stirling Highway, Crawley, WA, 6009, Australia","School of Science, Edith Cowan University, 270 Joondalup Drive, Joondalup, WA, 6027, Australia"],"raw_orcid":"https://orcid.org/0000-0002-4005-5431","affiliations":[{"raw_affiliation_string":"Department of Mathematics and Statistics, The University of Western Australia, 35 Stirling Highway, Crawley, WA, 6009, Australia","institution_ids":["https://openalex.org/I177877127"]},{"raw_affiliation_string":"School of Science, Edith Cowan University, 270 Joondalup Drive, Joondalup, WA, 6027, Australia","institution_ids":["https://openalex.org/I12079687"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5001541606","display_name":"Lyle Noakes","orcid":"https://orcid.org/0000-0001-9595-0735"},"institutions":[{"id":"https://openalex.org/I177877127","display_name":"The University of Western Australia","ror":"https://ror.org/047272k79","country_code":"AU","type":"education","lineage":["https://openalex.org/I177877127"]}],"countries":["AU"],"is_corresponding":false,"raw_author_name":"Lyle Noakes","raw_affiliation_strings":["Department of Mathematics and Statistics, The University of Western Australia, 35 Stirling Highway, Crawley, WA, 6009, Australia"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics and Statistics, The University of Western Australia, 35 Stirling Highway, Crawley, WA, 6009, Australia","institution_ids":["https://openalex.org/I177877127"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5014124196"],"corresponding_institution_ids":["https://openalex.org/I12079687","https://openalex.org/I177877127"],"apc_list":null,"apc_paid":null,"fwci":0.0,"has_fulltext":false,"cited_by_count":0,"citation_normalized_percentile":{"value":0.21867268,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":null,"biblio":{"volume":"61","issue":"5","first_page":"2261","last_page":"2284"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T12417","display_name":"Morphological variations and asymmetry","score":0.9980000257492065,"subfield":{"id":"https://openalex.org/subfields/2608","display_name":"Geometry and Topology"},"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/T12417","display_name":"Morphological variations and asymmetry","score":0.9980000257492065,"subfield":{"id":"https://openalex.org/subfields/2608","display_name":"Geometry and Topology"},"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/T10719","display_name":"3D Shape Modeling and Analysis","score":0.9933000206947327,"subfield":{"id":"https://openalex.org/subfields/2206","display_name":"Computational Mechanics"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11830","display_name":"Point processes and geometric inequalities","score":0.9908999800682068,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/geodesic","display_name":"Geodesic","score":0.9162603616714478},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.8199734687805176},{"id":"https://openalex.org/keywords/manifold","display_name":"Manifold (fluid mechanics)","score":0.6700179576873779},{"id":"https://openalex.org/keywords/riemannian-manifold","display_name":"Riemannian manifold","score":0.6233519911766052},{"id":"https://openalex.org/keywords/piecewise","display_name":"Piecewise","score":0.6129698157310486},{"id":"https://openalex.org/keywords/convergence","display_name":"Convergence (economics)","score":0.5459009408950806},{"id":"https://openalex.org/keywords/bounded-function","display_name":"Bounded function","score":0.48112955689430237},{"id":"https://openalex.org/keywords/geodesic-map","display_name":"Geodesic map","score":0.464147686958313},{"id":"https://openalex.org/keywords/polynomial","display_name":"Polynomial","score":0.45681488513946533},{"id":"https://openalex.org/keywords/curvature","display_name":"Curvature","score":0.4520851671695709},{"id":"https://openalex.org/keywords/constant-curvature","display_name":"Constant curvature","score":0.44893795251846313},{"id":"https://openalex.org/keywords/topology","display_name":"Topology (electrical circuits)","score":0.4455031454563141},{"id":"https://openalex.org/keywords/sequence","display_name":"Sequence (biology)","score":0.43671682476997375},{"id":"https://openalex.org/keywords/path","display_name":"Path (computing)","score":0.4158664345741272},{"id":"https://openalex.org/keywords/constant","display_name":"Constant (computer programming)","score":0.4151243567466736},{"id":"https://openalex.org/keywords/surface","display_name":"Surface (topology)","score":0.41234296560287476},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.33547714352607727},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.33314889669418335},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.3253245949745178},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.2269723117351532},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.1356227695941925}],"concepts":[{"id":"https://openalex.org/C165818556","wikidata":"https://www.wikidata.org/wiki/Q213488","display_name":"Geodesic","level":2,"score":0.9162603616714478},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8199734687805176},{"id":"https://openalex.org/C529865628","wikidata":"https://www.wikidata.org/wiki/Q1790740","display_name":"Manifold (fluid mechanics)","level":2,"score":0.6700179576873779},{"id":"https://openalex.org/C2779593128","wikidata":"https://www.wikidata.org/wiki/Q632814","display_name":"Riemannian manifold","level":2,"score":0.6233519911766052},{"id":"https://openalex.org/C164660894","wikidata":"https://www.wikidata.org/wiki/Q2037833","display_name":"Piecewise","level":2,"score":0.6129698157310486},{"id":"https://openalex.org/C2777303404","wikidata":"https://www.wikidata.org/wiki/Q759757","display_name":"Convergence (economics)","level":2,"score":0.5459009408950806},{"id":"https://openalex.org/C34388435","wikidata":"https://www.wikidata.org/wiki/Q2267362","display_name":"Bounded function","level":2,"score":0.48112955689430237},{"id":"https://openalex.org/C162269530","wikidata":"https://www.wikidata.org/wiki/Q17100560","display_name":"Geodesic map","level":3,"score":0.464147686958313},{"id":"https://openalex.org/C90119067","wikidata":"https://www.wikidata.org/wiki/Q43260","display_name":"Polynomial","level":2,"score":0.45681488513946533},{"id":"https://openalex.org/C195065555","wikidata":"https://www.wikidata.org/wiki/Q214881","display_name":"Curvature","level":2,"score":0.4520851671695709},{"id":"https://openalex.org/C2779668893","wikidata":"https://www.wikidata.org/wiki/Q2250422","display_name":"Constant curvature","level":3,"score":0.44893795251846313},{"id":"https://openalex.org/C184720557","wikidata":"https://www.wikidata.org/wiki/Q7825049","display_name":"Topology (electrical circuits)","level":2,"score":0.4455031454563141},{"id":"https://openalex.org/C2778112365","wikidata":"https://www.wikidata.org/wiki/Q3511065","display_name":"Sequence (biology)","level":2,"score":0.43671682476997375},{"id":"https://openalex.org/C2777735758","wikidata":"https://www.wikidata.org/wiki/Q817765","display_name":"Path (computing)","level":2,"score":0.4158664345741272},{"id":"https://openalex.org/C2777027219","wikidata":"https://www.wikidata.org/wiki/Q1284190","display_name":"Constant (computer programming)","level":2,"score":0.4151243567466736},{"id":"https://openalex.org/C2776799497","wikidata":"https://www.wikidata.org/wiki/Q484298","display_name":"Surface (topology)","level":2,"score":0.41234296560287476},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.33547714352607727},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.33314889669418335},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.3253245949745178},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.2269723117351532},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.1356227695941925},{"id":"https://openalex.org/C127413603","wikidata":"https://www.wikidata.org/wiki/Q11023","display_name":"Engineering","level":0,"score":0.0},{"id":"https://openalex.org/C78519656","wikidata":"https://www.wikidata.org/wiki/Q101333","display_name":"Mechanical engineering","level":1,"score":0.0},{"id":"https://openalex.org/C54355233","wikidata":"https://www.wikidata.org/wiki/Q7162","display_name":"Genetics","level":1,"score":0.0},{"id":"https://openalex.org/C50522688","wikidata":"https://www.wikidata.org/wiki/Q189833","display_name":"Economic growth","level":1,"score":0.0},{"id":"https://openalex.org/C162324750","wikidata":"https://www.wikidata.org/wiki/Q8134","display_name":"Economics","level":0,"score":0.0},{"id":"https://openalex.org/C86803240","wikidata":"https://www.wikidata.org/wiki/Q420","display_name":"Biology","level":0,"score":0.0},{"id":"https://openalex.org/C199360897","wikidata":"https://www.wikidata.org/wiki/Q9143","display_name":"Programming language","level":1,"score":0.0}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.1137/22m1515173","is_oa":true,"landing_page_url":"https://doi.org/10.1137/22m1515173","pdf_url":null,"source":{"id":"https://openalex.org/S203348814","display_name":"SIAM Journal on Numerical Analysis","issn_l":"0036-1429","issn":["0036-1429","1095-7170"],"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 Numerical Analysis","raw_type":"journal-article"},{"id":"pmh:oai:pure.atira.dk:publications/c7402f7a-4f2b-4076-9d38-05f619251da2","is_oa":false,"landing_page_url":"https://research-repository.uwa.edu.au/en/publications/c7402f7a-4f2b-4076-9d38-05f619251da2","pdf_url":null,"source":{"id":"https://openalex.org/S4306402492","display_name":"UWA Profiles and Research Repository (UWA)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I177877127","host_organization_name":"The University of Western Australia","host_organization_lineage":["https://openalex.org/I177877127"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Zhang , E & Noakes , L 2023 , ' CONVERGENCE ANALYSIS OF LEAPFROG FOR GEODESICS ' , SIAM Journal on Numerical Analysis , vol. 61 , no. 5 , pp. 2261-2284 . https://doi.org/10.1137/22M1515173","raw_type":"article"},{"id":"pmh:oai:ro.ecu.edu.au:ecuworks2022-2026-4291","is_oa":true,"landing_page_url":"https://ro.ecu.edu.au/ecuworks2022-2026/3290","pdf_url":null,"source":{"id":"https://openalex.org/S2765015692","display_name":"Australasian Journal of Paramedicine","issn_l":"2202-7270","issn":["2202-7270"],"is_oa":true,"is_in_doaj":false,"is_core":true,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Research outputs 2022 to 2026","raw_type":"text"}],"best_oa_location":{"id":"doi:10.1137/22m1515173","is_oa":true,"landing_page_url":"https://doi.org/10.1137/22m1515173","pdf_url":null,"source":{"id":"https://openalex.org/S203348814","display_name":"SIAM Journal on Numerical Analysis","issn_l":"0036-1429","issn":["0036-1429","1095-7170"],"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 Numerical Analysis","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":16,"referenced_works":["https://openalex.org/W62340590","https://openalex.org/W1499095749","https://openalex.org/W1972923431","https://openalex.org/W1983496390","https://openalex.org/W2008080706","https://openalex.org/W2059140354","https://openalex.org/W2066146666","https://openalex.org/W2105866209","https://openalex.org/W2125949583","https://openalex.org/W2583998085","https://openalex.org/W2765699677","https://openalex.org/W3137801329","https://openalex.org/W3172165933","https://openalex.org/W4285243727","https://openalex.org/W4288751817","https://openalex.org/W4293385897"],"related_works":["https://openalex.org/W2149451321","https://openalex.org/W4377245811","https://openalex.org/W4281256013","https://openalex.org/W2941285539","https://openalex.org/W286947431","https://openalex.org/W2154521153","https://openalex.org/W4298140757","https://openalex.org/W3127827089","https://openalex.org/W2946352054","https://openalex.org/W2950657073"],"abstract_inverted_index":{"Geodesics":[0],"are":[1,183],"of":[2,83,110,127,130,141,168,198],"fundamental":[3],"interest":[4],"in":[5,20],"mathematics,":[6],"physics,":[7],"computer":[8],"science,":[9],"and":[10,45,63,72,100,195],"many":[11],"other":[12],"subjects.":[13],"The":[14,53,102],"so-called":[15],"leapfrog":[16,187],"algorithm":[17,93],"was":[18,78],"proposed":[19],"[L.":[21],"Noakes,":[22],"J.":[23],"Aust.":[24],"Math.":[25],"Soc.,":[26],"65":[27],"(1998),":[28],"pp.":[29],"37-50]":[30],"(but":[31],"not":[32],"named":[33],"there":[34],"as":[35,178],"such)":[36],"to":[37,57,95],"find":[38],"geodesics":[39,70,85],"joining":[40,98],"two":[41,189],"given":[42],"points":[43],"x0":[44,62,99],"x1":[46,64],"on":[47,114,165,188],"a":[48,96,128,159],"path-connected":[49],"complete":[50],"Riemannian":[51,190],"manifold.":[52],"basic":[54],"idea":[55],"is":[56,120,138,149,158],"choose":[58],"some":[59],"junctions":[60],"between":[61],"that":[65,80,175],"can":[66],"be":[67],"joined":[68],"by":[69,91,152,185],"locally":[71],"then":[73],"adjust":[74],"these":[75],"junctions.":[76],"It":[77],"proved":[79],"the":[81,115,123,139,146,166,169,192,196],"sequence":[82],"piecewise":[84],"{":[86],"k}k":[87],"\u2265":[88],"1":[89],"generated":[90],"this":[92],"converges":[94],"geodesic":[97,142],"x1.":[101],"present":[103],"paper":[104],"investigates":[105],"leapfrog's":[106],"convergence":[107],"rate":[108],"i,n":[109,148],"ith":[111],"junction":[112],"depending":[113,164],"manifold":[116,170,197],"M.":[117,171],"A":[118],"relationship":[119],"found":[121],"with":[122],"maximal":[124],"root":[125],"n":[126,134,176,179],"polynomial":[129],"degree":[131],"n-3,":[132],"where":[133,156],"(n":[135],">":[136],"3)":[137],"number":[140],"segments.":[143],"That":[144],"is,":[145],"minimal":[147],"upper":[150],"bounded":[151],"n(1":[153],"+":[154],"c+),":[155],"c+":[157],"sufficiently":[160],"small":[161],"positive":[162,204],"constant":[163],"curvature":[167],"Moreover,":[172],"we":[173],"show":[174],"increases":[177],"increases.":[180],"These":[181],"results":[182],"illustrated":[184],"implementing":[186],"manifolds:":[191],"unit":[193],"2-sphere":[194],"all":[199],"2":[200,202],"\u00d7":[201],"symmetric":[203],"definite":[205],"matrices.":[206]},"counts_by_year":[],"updated_date":"2025-12-25T23:11:45.687758","created_date":"2025-10-10T00:00:00"}