{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T04:13:42Z","timestamp":1760242422178,"version":"build-2065373602"},"reference-count":18,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2017,7,10]],"date-time":"2017-07-10T00:00:00Z","timestamp":1499644800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Algorithms"],"abstract":"<jats:p>The nonlinear least squares problem     m i  n  y , z    \u2225 A  ( y )  z + b  ( y )  \u2225     , where     A ( y )     is a full-rank     ( N + \u2113 ) \u00d7 N     matrix,     y \u2208  R n     ,     z \u2208  R N      and     b  ( y )  \u2208  R  N + \u2113       with     \u2113 \u2265 n    , can be solved by first solving a reduced problem     m i  n y   \u2225 f  ( y )  \u2225      to find the optimal value     y *     of y, and then solving the resulting linear least squares problem     m i  n z   \u2225 A  (  y *  )  z + b  (  y *  )  \u2225      to find the optimal value     z *     of z. We have previously justified the use of the reduced function     f  ( y )  =  C T   ( y )  b  ( y )     , where     C ( y )     is a matrix whose columns form an orthonormal basis for the nullspace of      A T   ( y )     , and presented a quadratically convergent Gauss\u2013Newton type method for solving     m i  n y   \u2225  C T   ( y )  b  ( y )  \u2225      based on the use of QR factorization. In this note, we show how LU factorization can replace the QR factorization in those computations, halving the associated computational cost while also providing opportunities to exploit sparsity and thus further enhance computational efficiency.<\/jats:p>","DOI":"10.3390\/a10030078","type":"journal-article","created":{"date-parts":[[2017,7,10]],"date-time":"2017-07-10T11:05:31Z","timestamp":1499684731000},"page":"78","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["An Efficient Algorithm for the Separable Nonlinear Least Squares Problem"],"prefix":"10.3390","volume":"10","author":[{"given":"Yunqiu","family":"Shen","sequence":"first","affiliation":[{"name":"Department of Mathematics, Western Washington University, Bellingham, WA 98225-9063, USA"}]},{"given":"Tjalling","family":"Ypma","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Western Washington University, Bellingham, WA 98225-9063, USA"}]}],"member":"1968","published-online":{"date-parts":[[2017,7,10]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"R1","DOI":"10.1088\/0266-5611\/19\/2\/201","article-title":"Separable nonlinear least squares: the variable projection method and its applications. Topic Review","volume":"19","author":"Golub","year":"2003","journal-title":"Inverse Probl."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"319","DOI":"10.1007\/s11075-008-9235-2","article-title":"The variable projection algorithm in time-resolved spectroscopy, microscopy and mass spectrometry applications","volume":"51","author":"Mullen","year":"2009","journal-title":"Numer. Algorithms"},{"key":"ref_3","first-page":"258072","article-title":"Solving separable nonlinear equations using LU factorization","volume":"2013","author":"Shen","year":"2013","journal-title":"ISRN Math. Anal."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"413","DOI":"10.1137\/0710036","article-title":"The differentiation of pseudo-inverses and nonlinear least squares problems whose variables separate","volume":"10","author":"Golub","year":"1973","journal-title":"SIAM J. Numer. Anal."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"235","DOI":"10.1016\/0377-0427(90)90031-T","article-title":"Solving nonlinear systems of equations with only one nonlinear variable","volume":"30","author":"Shen","year":"1990","journal-title":"J. Comput. Appl. Math."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"259","DOI":"10.1007\/BF02262221","article-title":"Solving N+m nonlinear equations with only m nonlinear variables","volume":"44","author":"Ypma","year":"1990","journal-title":"Computing"},{"key":"ref_7","unstructured":"Lukeman, G.G. (2009). Separable Overdetermined Nonlinear Systems: An Application of the Shen-Ypma Algorithm, VDM Verlag."},{"key":"ref_8","unstructured":"Shen, Y., and Ypma, T.J. Solving Separable Least Squares Problems using QR factorization. J. Comp. Appl. Math., submitted."},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Dennis, J.E., and Schnabel, R.B. (1996). Numerical Methods for Unconstrained Optimization and Nonlinear Equations, Corrected Reprint of the 1983 Original, SIAM.","DOI":"10.1137\/1.9781611971200"},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Deuflhard, P., and Hohmann, A. (2003). Numerical Analysis in Modern Scientific Computing, Springer. [2nd ed.].","DOI":"10.1007\/978-0-387-21584-6"},{"key":"ref_11","unstructured":"Ortega, J.M., and Rheinboldt, W.C. (1970). Iterative Solution of Nonlinear Equations in Several Variables, Academic Press."},{"key":"ref_12","unstructured":"Golub, G.H., and Van Loan, C.F. (1996). Matrix Computations, Johns Hopkins. [3rd ed.]."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"256","DOI":"10.1016\/j.apnum.2004.09.029","article-title":"Newton\u2019s method for singular nonlinear equations using approximate left and right nullspaces of the Jacobian","volume":"54","author":"Shen","year":"2005","journal-title":"Appl. Numer. Math."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"609","DOI":"10.1016\/j.apnum.2006.07.025","article-title":"Solving rank-deficient separable nonlinear equations","volume":"57","author":"Shen","year":"2007","journal-title":"Appl. Numer. Math."},{"key":"ref_15","first-page":"31","article-title":"Numerical bifurcation of separable parameterized equations","volume":"34","author":"Shen","year":"2009","journal-title":"Elect. Trans. Numer. Anal."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"271","DOI":"10.1090\/mcom\/2968","article-title":"Rank deficiency and bifurcation into affine subspaces for separable parametrized equations","volume":"85","author":"Shen","year":"2016","journal-title":"Math. Comp."},{"key":"ref_17","unstructured":"Smith, G.D. (1985). Numerical Solution of Partial Differential Equations: Finite Difference Methods, Oxford University Press. [3rd ed.]."},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Li, Z., and Ito, K. (2006). The Immersed Interface Method, SIAM.","DOI":"10.1137\/1.9780898717464"}],"container-title":["Algorithms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1999-4893\/10\/3\/78\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T18:42:09Z","timestamp":1760208129000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1999-4893\/10\/3\/78"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2017,7,10]]},"references-count":18,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2017,9]]}},"alternative-id":["a10030078"],"URL":"https:\/\/doi.org\/10.3390\/a10030078","relation":{},"ISSN":["1999-4893"],"issn-type":[{"type":"electronic","value":"1999-4893"}],"subject":[],"published":{"date-parts":[[2017,7,10]]}}}