{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T04:38:56Z","timestamp":1760243936905,"version":"build-2065373602"},"reference-count":18,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2010,8,20]],"date-time":"2010-08-20T00:00:00Z","timestamp":1282262400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Algorithms"],"abstract":"<jats:p>We compare univariate L1 interpolating splines calculated on 5-point windows, on 7-point windows and on global data sets using four different spline functionals, namely, ones based on the second derivative, the first derivative, the function value and the antiderivative. Computational results indicate that second-derivative-based 5-point-window L1 splines preserve shape as well as or better than the other types of L1 splines. To calculate second-derivative-based 5-point-window L1 splines, we introduce an analysis-based, parallelizable algorithm. This algorithm is orders of magnitude faster than the previously widely used primal affine algorithm.<\/jats:p>","DOI":"10.3390\/a3030311","type":"journal-article","created":{"date-parts":[[2011,1,14]],"date-time":"2011-01-14T15:36:17Z","timestamp":1295019377000},"page":"311-328","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":12,"title":["Univariate Cubic L1 Interpolating Splines: Spline Functional, Window Size and Analysis-based Algorithm"],"prefix":"10.3390","volume":"3","author":[{"given":"Lu","family":"Yu","sequence":"first","affiliation":[{"name":"Industrial and Systems Engineering Department, North Carolina State University, Raleigh, NC 27695-7906, USA"}]},{"given":"Qingwei","family":"Jin","sequence":"additional","affiliation":[{"name":"Industrial and Systems Engineering Department, North Carolina State University, Raleigh, NC 27695-7906, USA"}]},{"given":"John E.","family":"Lavery","sequence":"additional","affiliation":[{"name":"Industrial and Systems Engineering Department, North Carolina State University, Raleigh, NC 27695-7906, USA"},{"name":"Mathematical Sciences Division, Army Research Office, Army Research Laboratory, P.O. Box 12211, Research Triangle Park, NC 27709-2211, USA"}]},{"given":"Shu-Cherng","family":"Fang","sequence":"additional","affiliation":[{"name":"Industrial and Systems Engineering Department, North Carolina State University, Raleigh, NC 27695-7906, USA"}]}],"member":"1968","published-online":{"date-parts":[[2010,8,20]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"373","DOI":"10.1016\/j.cagd.2007.04.007","article-title":"C1 and C2-continuous polynomial parametric Lp splines (p \u2265 1)","volume":"24","author":"Auquiert","year":"2007","journal-title":"Comput. Aided Geom. Design"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"321","DOI":"10.1007\/s11075-007-9140-0","article-title":"On the cubic L1 spline interpolant to the Heaviside function","volume":"46","author":"Auquiert","year":"2007","journal-title":"Numer. Algorithms"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"219","DOI":"10.1023\/B:COAP.0000042031.03626.1b","article-title":"An efficient algorithm for generating univariate cubic L1 splines","volume":"29","author":"Cheng","year":"2004","journal-title":"Comput. Optim. Appl."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"229","DOI":"10.1007\/s10479-004-5035-9","article-title":"A geometric programming framework for univariate cubic L1 smoothing splines","volume":"133","author":"Cheng","year":"2005","journal-title":"Ann. Oper. Res."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"319","DOI":"10.1016\/S0167-8396(00)00003-0","article-title":"Univariate cubic Lp splines and shape-preserving, multiscale interpolation by univariate cubic L1 splines","volume":"17","author":"Lavery","year":"2000","journal-title":"Comput. Aided Geom. Design"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"715","DOI":"10.1016\/S0167-8396(00)00025-X","article-title":"Shape-preserving, multiscale fitting of univariate data by cubic L1 smoothing splines","volume":"17","author":"Lavery","year":"2000","journal-title":"Comput. Aided Geom. Design"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"321","DOI":"10.1016\/S0167-8396(01)00034-6","article-title":"Shape-preserving, multiscale interpolation by bi- and multivariate cubic L1 splines","volume":"18","author":"Lavery","year":"2001","journal-title":"Comput. Aided Geom. Design"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"43","DOI":"10.1016\/j.cagd.2003.07.002","article-title":"Shape-preserving approximation of multiscale univariate data by cubic L1 spline fits","volume":"21","author":"Lavery","year":"2004","journal-title":"Comput. Aided Geom. Design"},{"key":"ref_9","unstructured":"Lucian, M.L., and Neamtu, M. (, January November). The state of the art in shape preserving, multiscale modeling by L1 splines. Proceedings of SIAM Conference on Geometric Design and Computing, Seattle, WA, USA."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"276","DOI":"10.1016\/j.cagd.2005.11.002","article-title":"Shape-preserving, first-derivative-based parametric and nonparametric cubic L1 spline curves","volume":"23","author":"Lavery","year":"2006","journal-title":"Comput. Aided Geom. Design"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.cagd.2008.01.004","article-title":"Shape-preserving univariate cubic and higher-degree L1 splines with function-value-based and multistep minimization principles","volume":"26","author":"Lavery","year":"2009","journal-title":"Comput. Aided Geom. Design"},{"key":"ref_12","unstructured":"Lin, Y.-M., Zhang, W., Wang, Y., Fang, S.-C., and Lavery, J.E. (, January November). 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Appl."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"589","DOI":"10.1007\/s10898-006-9124-y","article-title":"Geometric dual formulation for first-derivative-based univariate cubic L1 splines","volume":"40","author":"Zhao","year":"2008","journal-title":"J. Global Optim."},{"key":"ref_16","unstructured":"Auquiert, P., Gibaru, O., and Nyiri, E. (2010). Fast L1\u2013Ck polynomial spline interpolation algorithm with shape-preserving properties. Comput. Aided Geom. Design, in press."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"276","DOI":"10.3390\/a3030276","article-title":"Univariate cubic L1 interpolating splines: Analytical results for linearity, convexity and oscillation on 5-point windows","volume":"3","author":"Jin","year":"2010","journal-title":"Algorithms"},{"key":"ref_18","unstructured":"Bertsekas, D.P., Nedi\u0107, A., and Ozdaglar, A.E. (2003). 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