{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,20]],"date-time":"2026-08-20T14:34:01Z","timestamp":1787236441812,"version":"build-2736575974"},"reference-count":23,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"2","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Sci. and Stat. Comput."],"published-print":{"date-parts":[[1991,3]]},"abstract":"<jats:p>This paper reformulates, generalizes, and investigates the stability of the modified Prony algorithm introduced by Osborne [SIAM J. Numer. Anal. 12 (1975), pp. 571\u2013592], with special reference to rational and exponential fitting. The algorithm, originally for exponential functions, is generalized to the least squares fitting of any function which satisfies a linear homogeneous difference equation. Using the difference equation formulation, the problem is expressed as a separable regression, and hence as a nonlinear eigenproblem in terms of the coefficients of the difference equation. The eigenproblem involves finding the null space of a matrix of data differences B, and is solved using a variant of inverse iteration. Stability of the algorithm is shown to depend on the fact that B closely approximates the Hessian of the sum of squares. The expectations of B and the Hessian are evaluated. In the case of rational fitting, the relative difference between B and the Hessian is shown to converge to zero almost surely. Some details of the implementation of the algorithm are given. A simulation study compares the modified Prony algorithm with the Levenberg algorithm on a rational fitting problem, and supports the theoretical results.<\/jats:p>","DOI":"10.1137\/0912020","type":"journal-article","created":{"date-parts":[[2005,3,1]],"date-time":"2005-03-01T03:37:13Z","timestamp":1109648233000},"page":"362-382","source":"Crossref","is-referenced-by-count":68,"title":["A Modified Prony Algorithm for Fitting Functions Defined by Difference Equations"],"prefix":"10.1137","volume":"12","author":[{"given":"M. R.","family":"Osborne","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"G. K.","family":"Smyth","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2006,7,13]]},"reference":[{"key":"R1","doi-asserted-by":"publisher","DOI":"10.1002\/9780470316757"},{"key":"R2","doi-asserted-by":"publisher","DOI":"10.1090\/S0025-5718-1977-0428694-0"},{"key":"R3","volume-title":"Circulant matrices","author":"Davis Philip J.","year":"1979"},{"key":"R4","doi-asserted-by":"publisher","DOI":"10.1137\/1.9781611971811"},{"key":"R5","volume-title":"Practical methods of optimization. Vol. 1","author":"Fletcher R.","year":"1980"},{"key":"R6","unstructured":"A. S. Householder, On Prony's method of fitting exponential decay curves and multiple-hit survival curves, Report, ORNL-455, Oak Ridge National Laboratory, Oak Ridge, TN, 1950"},{"key":"R7","doi-asserted-by":"publisher","DOI":"10.1214\/aoms\/1177697731"},{"key":"R8","unstructured":"R. Kass, G. K. Smytch, Convergence of the scoring algorithm: a geometric interpretation, Tech. Report, Department of Mathematics, University of Queensland, St. Lucia, Australia, 1989"},{"key":"R9","volume-title":"Digital spectral analysis with applications","author":"Marple S. L.","year":"1987"},{"key":"R10","doi-asserted-by":"publisher","DOI":"10.1080\/00207728608926783"},{"key":"R11","unstructured":"NAG FORTRAN Library Manual Mark 10, Numerical Algorithms Group, Oxford, U.K., 1983"},{"key":"R12","volume-title":"Iterative solution of nonlinear equations in several variables","author":"Ortega J. M.","year":"1970"},{"key":"R13","doi-asserted-by":"publisher","DOI":"10.1137\/0712044"},{"key":"R14","doi-asserted-by":"publisher","DOI":"10.1017\/S033427000000120X"},{"key":"R15","unstructured":"M. R. Osborne, Inverse iteration, Newton's method, and nonlinear eigenvalue problems, Proc. Conference in Honour of J. H. Wilkinson, Vol. 19, Institute of Mathematics and Its Applications, Royal Society, London. Pub., 1978, 21\u2013530433.65021"},{"key":"R16","doi-asserted-by":"publisher","DOI":"10.2307\/3214370"},{"key":"R17","first-page":"24","volume":"1","author":"Prony R.","year":"1795","journal-title":"J. de L'\u00c9cole Polytechnique"},{"key":"R18","volume-title":"Nonlinear Regression Modelling: A Unified Practical Approach","author":"Ratkowsky D. A.","year":"1983"},{"key":"R19","unstructured":"G. K. Smyth, Ph.D. Thesis, Coupled and separable iterations in nonlinear estimation, Australian National University, Canberra, Australia, 1985"},{"key":"R20","unstructured":"G. K. Smyth, Curvature and convergence, Proc. Statistical Computing Section, American Statistical Society, Alexandria, VA, 1987, 278\u2013283"},{"key":"R21","doi-asserted-by":"crossref","first-page":"1549","DOI":"10.1214\/aoms\/1177698136","volume":"39","author":"Stout William F.","year":"1968","journal-title":"Ann. Math. 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H.","year":"1965"}],"container-title":["SIAM Journal on Scientific and Statistical Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/0912020","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,20]],"date-time":"2026-08-20T13:29:29Z","timestamp":1787232569000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/0912020"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1991,3]]},"references-count":23,"journal-issue":{"issue":"2","published-print":{"date-parts":[[1991,3]]}},"alternative-id":["10.1137\/0912020"],"URL":"https:\/\/doi.org\/10.1137\/0912020","relation":{},"ISSN":["0196-5204","2168-3417"],"issn-type":[{"value":"0196-5204","type":"print"},{"value":"2168-3417","type":"electronic"}],"subject":[],"published":{"date-parts":[[1991,3]]}}}