Total positivity and least squares problems in the Lagrange basis
Summary The problem of polynomial least squares fitting in the standard Lagrange basis is addressed in this work. Although the matrices involved in the corresponding overdetermined linear systems are not totally positive, rectangular totally positive Lagrange‐Vandermonde matrices are used to take ad...
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| Veröffentlicht in: | Numerical linear algebra with applications 2024-08, Vol.31 (4), p.n/a |
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| creator | Marco, Ana Martínez, José‐Javier Viaña, Raquel |
| description | Summary
The problem of polynomial least squares fitting in the standard Lagrange basis is addressed in this work. Although the matrices involved in the corresponding overdetermined linear systems are not totally positive, rectangular totally positive Lagrange‐Vandermonde matrices are used to take advantage of total positivity in the construction of accurate algorithms to solve the considered problem. In particular, a fast and accurate algorithm to compute the bidiagonal decomposition of such rectangular totally positive matrices is crucial to solve the problem. This algorithm also allows the accurate computation of the Moore‐Penrose inverse and the projection matrix of the collocation matrices involved in these problems. Numerical experiments showing the good behaviour of the proposed algorithms are included. |
| doi_str_mv | 10.1002/nla.2554 |
| format | Article |
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The problem of polynomial least squares fitting in the standard Lagrange basis is addressed in this work. Although the matrices involved in the corresponding overdetermined linear systems are not totally positive, rectangular totally positive Lagrange‐Vandermonde matrices are used to take advantage of total positivity in the construction of accurate algorithms to solve the considered problem. In particular, a fast and accurate algorithm to compute the bidiagonal decomposition of such rectangular totally positive matrices is crucial to solve the problem. This algorithm also allows the accurate computation of the Moore‐Penrose inverse and the projection matrix of the collocation matrices involved in these problems. Numerical experiments showing the good behaviour of the proposed algorithms are included.</description><identifier>ISSN: 1070-5325</identifier><identifier>EISSN: 1099-1506</identifier><identifier>DOI: 10.1002/nla.2554</identifier><language>eng</language><publisher>Oxford: Wiley Subscription Services, Inc</publisher><subject>Algorithms ; bidiagonal decomposition ; high relative accuracy ; Lagrange basis ; Least squares ; Linear systems ; Moore‐Penrose inverse ; Polynomials ; projection matrix ; total positivity</subject><ispartof>Numerical linear algebra with applications, 2024-08, Vol.31 (4), p.n/a</ispartof><rights>2024 The Authors. published by John Wiley & Sons Ltd.</rights><rights>2024. This article is published under http://creativecommons.org/licenses/by-nc-nd/4.0/ (the “License”). Notwithstanding the ProQuest Terms and Conditions, you may use this content in accordance with the terms of the License.</rights><lds50>peer_reviewed</lds50><oa>free_for_read</oa><woscitedreferencessubscribed>false</woscitedreferencessubscribed><cites>FETCH-LOGICAL-c2884-8e44a06842675d56f07c3c1ecbded08b5141d243b3b7cf81c92a4c1f93d2fd8c3</cites><orcidid>0009-0003-6517-0174 ; 0000-0002-8322-0361 ; 0000-0003-4662-6327</orcidid></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktopdf>$$Uhttps://onlinelibrary.wiley.com/doi/pdf/10.1002%2Fnla.2554$$EPDF$$P50$$Gwiley$$Hfree_for_read</linktopdf><linktohtml>$$Uhttps://onlinelibrary.wiley.com/doi/full/10.1002%2Fnla.2554$$EHTML$$P50$$Gwiley$$Hfree_for_read</linktohtml><link.rule.ids>314,776,780,1411,27901,27902,45550,45551</link.rule.ids></links><search><creatorcontrib>Marco, Ana</creatorcontrib><creatorcontrib>Martínez, José‐Javier</creatorcontrib><creatorcontrib>Viaña, Raquel</creatorcontrib><title>Total positivity and least squares problems in the Lagrange basis</title><title>Numerical linear algebra with applications</title><description>Summary
The problem of polynomial least squares fitting in the standard Lagrange basis is addressed in this work. Although the matrices involved in the corresponding overdetermined linear systems are not totally positive, rectangular totally positive Lagrange‐Vandermonde matrices are used to take advantage of total positivity in the construction of accurate algorithms to solve the considered problem. In particular, a fast and accurate algorithm to compute the bidiagonal decomposition of such rectangular totally positive matrices is crucial to solve the problem. This algorithm also allows the accurate computation of the Moore‐Penrose inverse and the projection matrix of the collocation matrices involved in these problems. Numerical experiments showing the good behaviour of the proposed algorithms are included.</description><subject>Algorithms</subject><subject>bidiagonal decomposition</subject><subject>high relative accuracy</subject><subject>Lagrange basis</subject><subject>Least squares</subject><subject>Linear systems</subject><subject>Moore‐Penrose inverse</subject><subject>Polynomials</subject><subject>projection matrix</subject><subject>total positivity</subject><issn>1070-5325</issn><issn>1099-1506</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2024</creationdate><recordtype>article</recordtype><sourceid>24P</sourceid><recordid>eNp10D1PwzAQBmALgUQpSPwESywsKeevxBmrii8pgqXMlmM7xVWatL4U1H9PSlmZ7oZH751eQm4ZzBgAf-haO-NKyTMyYVCWGVOQnx_3AjIluLokV4hrAMhVKSZkvuwH29Jtj3GIX3E4UNt52gaLA8Xd3qaAdJv6ug0bpLGjw2eglV0l260CrS1GvCYXjW0x3PzNKfl4elwuXrLq_fl1Ma8yx7WWmQ5SWsi15HmhvMobKJxwLLjaBw-6Vkwyz6WoRV24RjNXcisda0rheeO1E1Nyd8od39ntAw5m3e9TN540AjRoVRYKRnV_Ui71iCk0ZpvixqaDYWCOBZmxIHMsaKTZiX7HNhz-deatmv_6H6ITZkc</recordid><startdate>202408</startdate><enddate>202408</enddate><creator>Marco, Ana</creator><creator>Martínez, José‐Javier</creator><creator>Viaña, Raquel</creator><general>Wiley Subscription Services, Inc</general><scope>24P</scope><scope>AAYXX</scope><scope>CITATION</scope><scope>7SC</scope><scope>7TB</scope><scope>8FD</scope><scope>FR3</scope><scope>JQ2</scope><scope>KR7</scope><scope>L7M</scope><scope>L~C</scope><scope>L~D</scope><orcidid>https://orcid.org/0009-0003-6517-0174</orcidid><orcidid>https://orcid.org/0000-0002-8322-0361</orcidid><orcidid>https://orcid.org/0000-0003-4662-6327</orcidid></search><sort><creationdate>202408</creationdate><title>Total positivity and least squares problems in the Lagrange basis</title><author>Marco, Ana ; Martínez, José‐Javier ; Viaña, Raquel</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c2884-8e44a06842675d56f07c3c1ecbded08b5141d243b3b7cf81c92a4c1f93d2fd8c3</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2024</creationdate><topic>Algorithms</topic><topic>bidiagonal decomposition</topic><topic>high relative accuracy</topic><topic>Lagrange basis</topic><topic>Least squares</topic><topic>Linear systems</topic><topic>Moore‐Penrose inverse</topic><topic>Polynomials</topic><topic>projection matrix</topic><topic>total positivity</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Marco, Ana</creatorcontrib><creatorcontrib>Martínez, José‐Javier</creatorcontrib><creatorcontrib>Viaña, Raquel</creatorcontrib><collection>Wiley-Blackwell Open Access Titles</collection><collection>CrossRef</collection><collection>Computer and Information Systems Abstracts</collection><collection>Mechanical & Transportation Engineering Abstracts</collection><collection>Technology Research Database</collection><collection>Engineering Research Database</collection><collection>ProQuest Computer Science Collection</collection><collection>Civil Engineering Abstracts</collection><collection>Advanced Technologies Database with Aerospace</collection><collection>Computer and Information Systems Abstracts Academic</collection><collection>Computer and Information Systems Abstracts Professional</collection><jtitle>Numerical linear algebra with applications</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Marco, Ana</au><au>Martínez, José‐Javier</au><au>Viaña, Raquel</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Total positivity and least squares problems in the Lagrange basis</atitle><jtitle>Numerical linear algebra with applications</jtitle><date>2024-08</date><risdate>2024</risdate><volume>31</volume><issue>4</issue><epage>n/a</epage><issn>1070-5325</issn><eissn>1099-1506</eissn><abstract>Summary
The problem of polynomial least squares fitting in the standard Lagrange basis is addressed in this work. Although the matrices involved in the corresponding overdetermined linear systems are not totally positive, rectangular totally positive Lagrange‐Vandermonde matrices are used to take advantage of total positivity in the construction of accurate algorithms to solve the considered problem. In particular, a fast and accurate algorithm to compute the bidiagonal decomposition of such rectangular totally positive matrices is crucial to solve the problem. This algorithm also allows the accurate computation of the Moore‐Penrose inverse and the projection matrix of the collocation matrices involved in these problems. Numerical experiments showing the good behaviour of the proposed algorithms are included.</abstract><cop>Oxford</cop><pub>Wiley Subscription Services, Inc</pub><doi>10.1002/nla.2554</doi><tpages>14</tpages><orcidid>https://orcid.org/0009-0003-6517-0174</orcidid><orcidid>https://orcid.org/0000-0002-8322-0361</orcidid><orcidid>https://orcid.org/0000-0003-4662-6327</orcidid><oa>free_for_read</oa></addata></record> |
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| language | eng |
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| source | Wiley Online Library Journals Frontfile Complete |
| subjects | Algorithms bidiagonal decomposition high relative accuracy Lagrange basis Least squares Linear systems Moore‐Penrose inverse Polynomials projection matrix total positivity |
| title | Total positivity and least squares problems in the Lagrange basis |
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