{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,25]],"date-time":"2026-09-25T17:54:59Z","timestamp":1790358899499,"version":"4.1.0"},"reference-count":19,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"4","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Numer. Anal."],"published-print":{"date-parts":[[2008,1]]},"abstract":"<jats:p>A Newton\u2013Krylov method is an implementation of Newton's method in which a Krylov subspace method is used to solve approximately the linear systems that characterize steps of Newton's method. Newton\u2013Krylov methods are often implemented in \u201cmatrix-free\u201d form, in which the Jacobian-vector products required by the Krylov solver are approximated by finite differences. Here we consider using approximate function values in these finite differences. We first formulate a finite-difference Arnoldi process that uses approximate function values. We then outline a Newton\u2013Krylov method that uses an implementation of the GMRES or Arnoldi method based on this process, and we develop a local convergence analysis for it, giving sufficient conditions on the approximate function values for desirable local convergence properties to hold. We conclude with numerical experiments involving particular function-value approximations suitable for nonlinear diffusion problems. For this case, conditions are given for meeting the convergence assumptions for both lagging and linearizing the nonlinearity in the function evaluation.<\/jats:p>","DOI":"10.1137\/060652749","type":"journal-article","created":{"date-parts":[[2008,4,18]],"date-time":"2008-04-18T18:06:43Z","timestamp":1208542003000},"page":"1892-1911","source":"Crossref","is-referenced-by-count":13,"title":["On Using Approximate Finite Differences in Matrix-Free Newton\u2013Krylov Methods"],"prefix":"10.1137","volume":"46","author":[{"given":"Peter N.","family":"Brown","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Homer F.","family":"Walker","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Rebecca","family":"Wasyk","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Carol S.","family":"Woodward","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2008,4,18]]},"reference":[{"key":"R1","doi-asserted-by":"publisher","DOI":"10.1090\/qam\/42792"},{"key":"R2","doi-asserted-by":"publisher","DOI":"10.1137\/0911026"},{"key":"R3","doi-asserted-by":"publisher","DOI":"10.1137\/0724031"},{"key":"R4","doi-asserted-by":"publisher","DOI":"10.1023\/A:1017583307974"},{"key":"R5","doi-asserted-by":"publisher","DOI":"10.1137\/0719025"},{"key":"R6","unstructured":"J. E. Dennis, Jr., and R. B. Schnabel,\n                      Numerical Methods for Unconstrained Optimization and Nonlinear Equations\n                      , Series in Automatic Computation, Prentice-Hall, Englewood Cliffs, NJ, 1983."},{"key":"R7","doi-asserted-by":"publisher","DOI":"10.1137\/0917003"},{"key":"R8","doi-asserted-by":"publisher","DOI":"10.1017\/S0962492900002245"},{"key":"R9","unstructured":"G. H. Golub and C. Van Loan,\n                      Matrix Computations\n                      , 3rd ed., Johns Hopkins University Press, Baltimore, 1996."},{"key":"R10","doi-asserted-by":"publisher","DOI":"10.1145\/1089014.1089020"},{"key":"R11","doi-asserted-by":"crossref","unstructured":"C. T. Kelley,\n                      Solving Nonlinear Equations with Newton's Method\n                      , Fundam. Algorithms 1, SIAM, Philadelphia, PA, 2003.","DOI":"10.1137\/1.9780898718898"},{"key":"R12","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2003.08.010"},{"key":"R13","doi-asserted-by":"publisher","DOI":"10.1029\/98WR01673"},{"key":"R14","unstructured":"J. M. Ortega and W. C. Rheinboldt,\n                      Iterative Solution of Nonlinear Equations in Several Variables\n                      , Comput. Sci. Appl. Math., Academic Press, New York, 1970."},{"key":"R15","doi-asserted-by":"publisher","DOI":"10.1137\/S0036144504443511"},{"key":"R16","doi-asserted-by":"publisher","DOI":"10.1137\/0907058"},{"key":"R17","doi-asserted-by":"publisher","DOI":"10.1090\/S0025-5718-1981-0616364-6"},{"key":"R18","unstructured":"Y. Saad,\n                      Iterative Methods for Sparse Linear Systems\n                      , PWS Publishing, Boston, 1996."},{"key":"R19","doi-asserted-by":"crossref","unstructured":"H. A. van der Vorst,\n                      Iterative Krylov Methods for Large Linear Systems\n                      , Cambridge University Press, Cambridge, 2003.","DOI":"10.1017\/CBO9780511615115"}],"container-title":["SIAM Journal on Numerical Analysis"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/060652749","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T15:14:09Z","timestamp":1787325249000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/060652749"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2008,1]]},"references-count":19,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2008,1]]}},"alternative-id":["10.1137\/060652749"],"URL":"https:\/\/doi.org\/10.1137\/060652749","relation":{},"ISSN":["0036-1429","1095-7170"],"issn-type":[{"value":"0036-1429","type":"print"},{"value":"1095-7170","type":"electronic"}],"subject":[],"published":{"date-parts":[[2008,1]]}}}