{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,2]],"date-time":"2026-10-02T19:42:20Z","timestamp":1790970140475,"version":"4.1.0"},"reference-count":26,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"2","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Numer. Anal."],"published-print":{"date-parts":[[2008,1]]},"abstract":"<jats:p>This article treats numerical methods for tracking an implicitly defined path. The numerical precision required to successfully track such a path is difficult to predict a priori, and indeed it may change dramatically through the course of the path. In current practice, one must either choose a conservatively large numerical precision at the outset or rerun paths multiple times in successively higher precision until success is achieved. To avoid unnecessary computational cost, it would be preferable to adaptively adjust the precision as the tracking proceeds in response to the local conditioning of the path. We present an algorithm that can be set to either reactively adjust precision in response to step failure or proactively set the precision using error estimates. We then test the relative merits of reactive and proactive adaptation on several examples arising as homotopies for solving systems of polynomial equations.<\/jats:p>","DOI":"10.1137\/060658862","type":"journal-article","created":{"date-parts":[[2008,2,8]],"date-time":"2008-02-08T18:10:00Z","timestamp":1202494200000},"page":"722-746","source":"Crossref","is-referenced-by-count":75,"title":["Adaptive Multiprecision Path Tracking"],"prefix":"10.1137","volume":"46","author":[{"given":"Daniel J.","family":"Bates","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Jonathan D.","family":"Hauenstein","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Andrew J.","family":"Sommese","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"suffix":"II","given":"Charles W.","family":"Wampler","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2008,2,8]]},"reference":[{"key":"R1","doi-asserted-by":"crossref","unstructured":"E. L. Allgower and K. Georg,\n                      Numerical Continuation Methods. An Introduction\n                      , Springer Ser. Comput. Math. 13, Springer-Verlag, Berlin, Heidelberg, New York, 1990; reprinted as\n                      Introduction to Numerical Continuation Methods\n                      , Classics Appl. Math. 45, SIAM, Philadelphia, 2003.","DOI":"10.1007\/978-3-642-61257-2_1"},{"key":"R2","unstructured":"D. J. Bates, J. D. Hauenstein, A. J. Sommese, and C. W. Wampler.\n                      Bertini: Software for Numerical Algebraic Geometry\n                      , available online from www.nd.edu\/~sommese\/bertini."},{"key":"R3","doi-asserted-by":"crossref","unstructured":"J. W. Demmel,\n                      Applied Numerical Linear Algebra\n                      , SIAM, Philadelphia, 1997.","DOI":"10.1137\/1.9781611971446"},{"key":"R4","doi-asserted-by":"crossref","unstructured":"I. Gargantini and P. Henrici,\n                      Circular arithmetic and the determination of polynomial zeros\n                      , Numer. Math., 18 (1971\/1972), pp. 305\u2013320.","DOI":"10.1007\/BF01404681"},{"key":"R5","doi-asserted-by":"publisher","DOI":"10.1145\/78928.78931"},{"key":"R6","doi-asserted-by":"publisher","DOI":"10.1137\/0731048"},{"key":"R7","unstructured":"D. E. Knuth,\n                      The Art of Computer Programming, Vol.\n                      2:\n                      Seminumerical Algorithms\n                      , Addison-Wesley, Reading, MA, 1969."},{"key":"R8","unstructured":"A. Leykin, J. Verschelde, and A. Zhao,\n                      Evaluation of Jacobian matrices for Newton's method with deflation for isolated singularities of polynomial systems\n                      , in Proceedings of the International Workshop on Symbolic-Numeric Computation (SNC 2005), D. Wang and L. Zhi, eds., Xi'an, China, 2005, pp. 19\u201328."},{"key":"R9","doi-asserted-by":"publisher","DOI":"10.1017\/S0962492900002749"},{"key":"R10","doi-asserted-by":"crossref","unstructured":"T. Y. Li,\n                      Numerical solution of polynomial systems by homotopy continuation methods\n                      , in Handbook of Numerical Analysis. Volume XI. Special Volume: Foundations of Computational Mathematics, F. Cucker, ed., North-Holland, Amsterdam, 2003, pp. 209\u2013304.","DOI":"10.1016\/S1570-8659(02)11004-0"},{"key":"R11","doi-asserted-by":"publisher","DOI":"10.1137\/S0895479803435282"},{"key":"R12","doi-asserted-by":"publisher","DOI":"10.1006\/jsco.2000.0429"},{"key":"R13","unstructured":"A. P. Morgan,\n                      Solving Polynomial Systems Using Continuation for Engineering and Scientific Problems\n                      , Prentice-Hall, Englewood Cliffs, NJ, 1987."},{"key":"R14","first-page":"209","volume":"51","author":"Morgan A. 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Wilkinson,\n                      Rounding Errors in Algebraic Processes\n                      , Dover, New York, 1994; reprint of the 1963 Prentice-Hall original."}],"container-title":["SIAM Journal on Numerical Analysis"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/060658862","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T15:11:49Z","timestamp":1787325109000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/060658862"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2008,1]]},"references-count":26,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2008,1]]}},"alternative-id":["10.1137\/060658862"],"URL":"https:\/\/doi.org\/10.1137\/060658862","relation":{},"ISSN":["0036-1429","1095-7170"],"issn-type":[{"value":"0036-1429","type":"print"},{"value":"1095-7170","type":"electronic"}],"subject":[],"published":{"date-parts":[[2008,1]]}}}