{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T18:34:02Z","timestamp":1787337242437,"version":"build-2736575974"},"reference-count":41,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"3","funder":[{"DOI":"10.13039\/100000070","name":"National Institute of Biomedical Imaging and Bioengineering","doi-asserted-by":"publisher","award":["NIBIB-U24EB029012"],"award-info":[{"award-number":["NIBIB-U24EB029012"]}],"id":[{"id":"10.13039\/100000070","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["DMS-1848508"],"award-info":[{"award-number":["DMS-1848508"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["DMS-1720416"],"award-info":[{"award-number":["DMS-1720416"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Sci. Comput."],"published-print":{"date-parts":[[2023,6,30]]},"abstract":"<jats:p>Abstract.<\/jats:p>\n                  <jats:p>Orthogonal polynomials of several variables have a vector-valued three-term recurrence relation, much like the corresponding one-dimensional relation. This relation requires only knowledge of certain recurrence matrices, and allows simple and stable evaluation of multivariate orthogonal polynomials. In the univariate case, various algorithms can stably and accurately evaluate the recurrence coefficients given the ability to compute polynomial moments, but it is difficult to identify analogous procedures in multiple dimensions. We present a new Multivariate Stieltjes (MS) algorithm that fills this gap in the multivariate case, allowing computation of recurrence matrices assuming moments are available. The algorithm is essentially explicit in two and three dimensions, but requires the numerical solution to a nonconvex problem in more than three dimensions. Compared to direct Gram\u2013Schmidt-type orthogonalization, we demonstrate on several examples in up to three dimensions that the MS algorithm is far more stable, and allows accurate computation of orthogonal bases in the multivariate setting, in contrast to direct orthogonalization approaches.<\/jats:p>","DOI":"10.1137\/22m1477131","type":"journal-article","created":{"date-parts":[[2023,6,1]],"date-time":"2023-06-01T16:03:32Z","timestamp":1685635412000},"page":"A1125-A1147","source":"Crossref","is-referenced-by-count":4,"title":["A Stieltjes Algorithm for Generating Multivariate Orthogonal Polynomials"],"prefix":"10.1137","volume":"45","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-3409-5709","authenticated-orcid":true,"given":"Zexin","family":"Liu","sequence":"first","affiliation":[{"name":"Department of Mathematics, and Scientific Computing and Imaging (SCI) Institute, the University of Utah, Salt Lake City, UT 84112 USA."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5914-4207","authenticated-orcid":true,"given":"Akil","family":"Narayan","sequence":"additional","affiliation":[{"name":"Department of Mathematics, and Scientific Computing and Imaging (SCI) Institute, the University of Utah, Salt Lake City, UT 84112 USA."}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2023,6,1]]},"reference":[{"key":"ref1","doi-asserted-by":"publisher","DOI":"10.1142\/9789812770752_0058"},{"key":"ref2","doi-asserted-by":"publisher","DOI":"10.1093\/oso\/9780198506720.001.0001"},{"key":"ref3","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-540-36716-1_1"},{"key":"ref4","doi-asserted-by":"publisher","DOI":"10.1017\/S0962492900002701"},{"key":"ref5","doi-asserted-by":"publisher","DOI":"10.1137\/S1064827501387826"},{"key":"ref6","doi-asserted-by":"publisher","DOI":"10.1137\/19M1245888"},{"key":"ref7","doi-asserted-by":"publisher","DOI":"10.1016\/j.cma.2019.03.049"},{"key":"ref8","doi-asserted-by":"publisher","DOI":"10.1007\/s00365-018-9441-7"},{"key":"ref9","doi-asserted-by":"publisher","DOI":"10.1017\/S0962492920000045"},{"key":"ref10","doi-asserted-by":"publisher","DOI":"10.1137\/19M1279459"},{"key":"ref11","doi-asserted-by":"publisher","DOI":"10.1093\/imanum\/draa023"},{"key":"ref12","doi-asserted-by":"publisher","DOI":"10.1215\/S0012-7094-36-00233-8"},{"key":"ref13","doi-asserted-by":"publisher","DOI":"10.1007\/BF02412238"},{"key":"ref14","doi-asserted-by":"publisher","DOI":"10.1137\/0513022"},{"key":"ref15","doi-asserted-by":"publisher","DOI":"10.1137\/0513023"},{"key":"ref16","doi-asserted-by":"publisher","DOI":"10.1137\/0524048"},{"key":"ref17","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-1994-1169912-X"},{"key":"ref18","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9781107786134"},{"key":"ref19","doi-asserted-by":"publisher","DOI":"10.1007\/s10915-021-01586-w"},{"key":"ref20","volume-title":"Inzell Lectures on Orthogonal Polynomials","author":"zu Castell W.","year":"2005"},{"key":"ref21","doi-asserted-by":"publisher","DOI":"10.1007\/s00365-010-9114-7"},{"key":"ref22","first-page":"409","volume-title":"Annales scientifiques de l\u2019\u00c9cole Normale Sup\u00e9rieure, 1","author":"Stieltjes T. 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