{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,1]],"date-time":"2025-10-01T16:35:15Z","timestamp":1759336515669,"version":"3.41.0"},"reference-count":21,"publisher":"Association for Computing Machinery (ACM)","issue":"4","license":[{"start":{"date-parts":[[2012,8,1]],"date-time":"2012-08-01T00:00:00Z","timestamp":1343779200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.acm.org\/publications\/policies\/copyright_policy#Background"}],"funder":[{"DOI":"10.13039\/501100000266","name":"Engineering and Physical Sciences Research Council","doi-asserted-by":"publisher","award":["EP\/E045847"],"award-info":[{"award-number":["EP\/E045847"]}],"id":[{"id":"10.13039\/501100000266","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["dl.acm.org"],"crossmark-restriction":true},"short-container-title":["ACM Trans. Math. Softw."],"published-print":{"date-parts":[[2012,8]]},"abstract":"<jats:p>\n            A new solver for nonlinear boundary-value problems (BVPs) in\n            <jats:sc>Matlab<\/jats:sc>\n            is presented, based on the Chebfun software system for representing functions and operators automatically as numerical objects. The solver implements Newton\u2019s method in function space, where instead of the usual Jacobian matrices, the derivatives involved are Fr\u00e9chet derivatives. A major novelty of this approach is the application of automatic differentiation (AD) techniques to compute the operator-valued Fr\u00e9chet derivatives in the continuous context. Other novelties include the use of anonymous functions and numbering of each variable to enable a recursive, delayed evaluation of derivatives with forward mode AD . The AD techniques are applied within a new Chebfun class called which allows users to set up and solve nonlinear BVPs, both scalar and systems of coupled equations, in a few lines of code, using the \u201cnonlinear backslash\u201d operator (\\). This framework enables one to study the behaviour of Newton\u2019s method in function space.\n          <\/jats:p>","DOI":"10.1145\/2331130.2331134","type":"journal-article","created":{"date-parts":[[2012,9,4]],"date-time":"2012-09-04T12:50:47Z","timestamp":1346763047000},"page":"1-29","update-policy":"https:\/\/doi.org\/10.1145\/crossmark-policy","source":"Crossref","is-referenced-by-count":23,"title":["Automatic Fr\u00e9chet Differentiation for the Numerical Solution of Boundary-Value Problems"],"prefix":"10.1145","volume":"38","author":[{"given":"Asgeir","family":"Birkisson","sequence":"first","affiliation":[{"name":"University of Oxford"}]},{"given":"Tobin A.","family":"Driscoll","sequence":"additional","affiliation":[{"name":"University of Delaware"}]}],"member":"320","published-online":{"date-parts":[[2012,8]]},"reference":[{"key":"e_1_2_1_1_1","doi-asserted-by":"crossref","unstructured":"Allgower E. L. and Georg K. 2003. Introduction to Numerical Continuation Methods. SIAM Philadelphia PA. Allgower E. L. and Georg K. 2003. Introduction to Numerical Continuation Methods . SIAM Philadelphia PA.","DOI":"10.1137\/1.9780898719154"},{"key":"e_1_2_1_2_1","doi-asserted-by":"crossref","unstructured":"Ascher U. M. Mattheij R. M. M. and Russell R. D. 1995. Numerical Solution of Boundary Value Problems for Ordinary Differential Equations. SIAM Philadelphia PA. Ascher U. M. Mattheij R. M. M. and Russell R. D. 1995. Numerical Solution of Boundary Value Problems for Ordinary Differential Equations . SIAM Philadelphia PA.","DOI":"10.1137\/1.9781611971231"},{"key":"e_1_2_1_3_1","unstructured":"Bender C. M. and Orszag S. A. 1978. Advanced Mathematical Methods for Scientists and Engineers. McGraw-Hill New York. Bender C. M. and Orszag S. A. 1978. Advanced Mathematical Methods for Scientists and Engineers . McGraw-Hill New York."},{"key":"e_1_2_1_4_1","article-title":"Computing derivatives of computer programs. In Modern Methods and Algorithms of Quantum Chemistry","volume":"3","author":"Bischof C. H.","year":"2000","journal-title":"Proceedings 2nd Ed. J. Grotendorst Ed., NIC Series"},{"volume-title":"Proceedings of the 2nd IEEE International Workshop on Source Code Analysis and Manipulation (SCAM\u201902)","author":"Bischof C. H.","key":"e_1_2_1_5_1"},{"volume-title":"Proceedings of the 19th IFIP TC7 Conference on System Modeling and Optimization. M. J. Powell and S. Scholtes Eds., Kluwer, B.V.","author":"Bock H. G.","key":"e_1_2_1_6_1"},{"key":"e_1_2_1_7_1","unstructured":"Deuflhard P. 2006. Newton Methods for Nonlinear Problems. Springer. Deuflhard P. 2006. Newton Methods for Nonlinear Problems . Springer."},{"key":"e_1_2_1_8_1","doi-asserted-by":"publisher","DOI":"10.1137\/S1064827594274607"},{"key":"e_1_2_1_9_1","doi-asserted-by":"publisher","DOI":"10.1007\/s10543-008-0198-4"},{"key":"e_1_2_1_10_1","doi-asserted-by":"publisher","DOI":"10.1145\/1141885.1141888"},{"key":"e_1_2_1_11_1","unstructured":"Forth S. A. and Shampine L. F. 2005. bvp4cAD: An automatic differentiation enabled boundary value solver. http:\/\/www.amorg.co.uk\/AD\/ADODE\/bvp4cAD\/index.html. Forth S. A. and Shampine L. F. 2005. bvp4cAD: An automatic differentiation enabled boundary value solver. http:\/\/www.amorg.co.uk\/AD\/ADODE\/bvp4cAD\/index.html."},{"key":"e_1_2_1_12_1","doi-asserted-by":"publisher","DOI":"10.1137\/1.9780898717761"},{"key":"e_1_2_1_13_1","unstructured":"Hutson V. Pym J. S. and Cloud M. J. 2005. Applications of Functional Analysis and Operator Theory 2nd Ed. Elsevier. Hutson V. Pym J. S. and Cloud M. J. 2005. Applications of Functional Analysis and Operator Theory 2nd Ed. Elsevier."},{"key":"e_1_2_1_14_1","doi-asserted-by":"publisher","DOI":"10.1007\/11758549_77"},{"key":"e_1_2_1_15_1","doi-asserted-by":"publisher","DOI":"10.1145\/502800.502801"},{"key":"e_1_2_1_16_1","unstructured":"Kierzenka J. and Shampine L. F. 2007. A BVP solver that controls residual and error. http:\/\/faculty.smu.edu\/shampine\/finalbvp5c.pdf. Kierzenka J. and Shampine L. F. 2007. A BVP solver that controls residual and error. http:\/\/faculty.smu.edu\/shampine\/finalbvp5c.pdf."},{"key":"e_1_2_1_17_1","doi-asserted-by":"publisher","DOI":"10.1145\/258726.258799"},{"volume-title":"Developments in Reliable Computing","author":"Rump S.","key":"e_1_2_1_18_1"},{"key":"e_1_2_1_19_1","doi-asserted-by":"publisher","DOI":"10.1137\/0907058"},{"key":"e_1_2_1_20_1","doi-asserted-by":"publisher","DOI":"10.1145\/1055531.1055535"},{"key":"e_1_2_1_21_1","unstructured":"Trefethen L. N. etal 2011. Chebfun Version 4.0. The Chebfun Development Team. http:\/\/www.maths.ox.ac.uk\/chebfun\/. Trefethen L. N. et al. 2011. Chebfun Version 4.0. The Chebfun Development Team. http:\/\/www.maths.ox.ac.uk\/chebfun\/."}],"container-title":["ACM Transactions on Mathematical Software"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/dl.acm.org\/doi\/10.1145\/2331130.2331134","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/dl.acm.org\/doi\/pdf\/10.1145\/2331130.2331134","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,6,18]],"date-time":"2025-06-18T08:49:12Z","timestamp":1750236552000},"score":1,"resource":{"primary":{"URL":"https:\/\/dl.acm.org\/doi\/10.1145\/2331130.2331134"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2012,8]]},"references-count":21,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2012,8]]}},"alternative-id":["10.1145\/2331130.2331134"],"URL":"https:\/\/doi.org\/10.1145\/2331130.2331134","relation":{},"ISSN":["0098-3500","1557-7295"],"issn-type":[{"type":"print","value":"0098-3500"},{"type":"electronic","value":"1557-7295"}],"subject":[],"published":{"date-parts":[[2012,8]]},"assertion":[{"value":"2010-08-01","order":0,"name":"received","label":"Received","group":{"name":"publication_history","label":"Publication History"}},{"value":"2011-12-01","order":1,"name":"accepted","label":"Accepted","group":{"name":"publication_history","label":"Publication History"}},{"value":"2012-08-01","order":2,"name":"published","label":"Published","group":{"name":"publication_history","label":"Publication History"}}]}}