{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,12]],"date-time":"2026-06-12T10:22:02Z","timestamp":1781259722243,"version":"3.54.1"},"reference-count":12,"publisher":"American Mathematical Society (AMS)","issue":"227","license":[{"start":{"date-parts":[[2000,2,10]],"date-time":"2000-02-10T00:00:00Z","timestamp":950140800000},"content-version":"am","delay-in-days":365,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>\n                    Finite difference methods, such as the mid-point rule, have been applied successfully to the numerical solution of ordinary and partial differential equations. If such formulas are applied to observational data, in order to determine derivatives, the results can be disastrous. The reason for this is that measurement errors, and even rounding errors in computer approximations, are strongly amplified in the differentiation process, especially if small step-sizes are chosen and higher derivatives are required. A number of authors have examined the use of various forms of averaging which allows the stable computation of low order derivatives from observational data. The size of the averaging set acts like a regularization parameter and has to be chosen as a function of the grid size\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"h\">\n                        <mml:semantics>\n                          <mml:mi>h<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">h<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    . In this paper, it is initially shown how first (and higher) order single-variate numerical differentiation of higher dimensional observational data can be stabilized with a reduced loss of accuracy than occurs for the corresponding differentiation of one-dimensional data. The result is then extended to the multivariate differentiation of higher dimensional data. The nature of the trade-off between convergence and stability is explicitly characterized, and the complexity of various implementations is examined.\n                  <\/p>","DOI":"10.1090\/s0025-5718-99-01033-9","type":"journal-article","created":{"date-parts":[[2002,7,26]],"date-time":"2002-07-26T18:14:44Z","timestamp":1027707284000},"page":"1121-1141","source":"Crossref","is-referenced-by-count":70,"title":["For numerical differentiation, dimensionality can be a blessing!"],"prefix":"10.1090","volume":"68","author":[{"given":"Robert","family":"Anderssen","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Markus","family":"Hegland","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"14","published-online":{"date-parts":[[1999,2,10]]},"reference":[{"key":"1","unstructured":"R.S. Anderssen, F. de Hoog, and M. Hegland, A stable finite difference ansatz for higher order differentiation of non-exact data, Mathematics Research Report MRR96-023, ANU, School of Mathematical Sciences, 1996, \\url{ftp:\/\/nimbus.anu.edu.au:\/pub\/Hegland\/andhh96.ps.gz}."},{"issue":"3-4","key":"2","doi-asserted-by":"publisher","first-page":"259","DOI":"10.1007\/BF02242272","article-title":"Finite difference methods for the numerical differentiation of nonexact data","volume":"33","author":"Anderssen, R. S.","year":"1984","journal-title":"Computing","ISSN":"https:\/\/id.crossref.org\/issn\/0010-485X","issn-type":"print"},{"key":"3","doi-asserted-by":"crossref","unstructured":"S. S. Choi and R. S. Anderssen, Determination of the transmissitivity zonata using a linear function strategy, Inverse Problems 7 (1991), 831\u2013851.","DOI":"10.1088\/0266-5611\/7\/6\/007"},{"key":"4","doi-asserted-by":"crossref","unstructured":"R. R. Clements, An inviscid model of two-dimensional vortex shedding, J. Fluid Mech. 75 (1976), 209\u2013231.","DOI":"10.1017\/S0022112076000189"},{"key":"5","series-title":"Wiley Series in Probability and Mathematical Statistics: Applied Probability and Statistics","isbn-type":"print","volume-title":"Statistics for spatial data","author":"Cressie, Noel A. C.","year":"1991","ISBN":"https:\/\/id.crossref.org\/isbn\/0471843369"},{"key":"6","doi-asserted-by":"crossref","unstructured":"M. E. Davies, A comparison of the wake structure of a stationary and oscillating bluff body, using a conditional averaging technique, J. Fluid Mech. 75 (1976), 209\u2013231.","DOI":"10.1017\/S0022112076000189"},{"key":"7","unstructured":"D.J. 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P.","year":"1995","ISBN":"https:\/\/id.crossref.org\/isbn\/0412552701"}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.ams.org\/mcom\/1999-68-227\/S0025-5718-99-01033-9\/S0025-5718-99-01033-9.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/1999-68-227\/S0025-5718-99-01033-9\/S0025-5718-99-01033-9.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,20]],"date-time":"2026-04-20T22:05:54Z","timestamp":1776722754000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/1999-68-227\/S0025-5718-99-01033-9\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1999,2,10]]},"references-count":12,"journal-issue":{"issue":"227","published-print":{"date-parts":[[1999,7]]}},"alternative-id":["S0025-5718-99-01033-9"],"URL":"https:\/\/doi.org\/10.1090\/s0025-5718-99-01033-9","archive":["CLOCKSS","Portico"],"relation":{},"ISSN":["1088-6842","0025-5718"],"issn-type":[{"value":"1088-6842","type":"electronic"},{"value":"0025-5718","type":"print"}],"subject":[],"published":{"date-parts":[[1999,2,10]]}}}