When Newton\u2019s method is applied to find the maximal symmetric solution of an algebraic Riccati equation, convergence can be guaranteed under moderate conditions. In particular, the initial guess need not be close to the solution. The convergence is quadratic if the Fr\u00e9chet derivative is invertible at the solution. In this paper we examine the behaviour of the Newton iteration when the derivative is not invertible at the solution. We find that a simple modification can improve the performance of the Newton iteration dramatically.<\/p>","DOI":"10.1090\/s0025-5718-98-00947-8","type":"journal-article","created":{"date-parts":[[2002,7,26]],"date-time":"2002-07-26T22:14:44Z","timestamp":1027721684000},"page":"1089-1105","source":"Crossref","is-referenced-by-count":52,"title":["Analysis and modificaton of Newton\u2019s method for algebraic Riccati equations"],"prefix":"10.1090","volume":"67","author":[{"given":"Chun-Hua","family":"Guo","sequence":"first","affiliation":[]},{"given":"Peter","family":"Lancaster","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[1998]]},"reference":[{"key":"1","doi-asserted-by":"crossref","unstructured":"R. H. Bartels and G. W. Stewart, Solution of the matrix equation \ud835\udc34\ud835\udc4b+\ud835\udc4b\ud835\udc35=\ud835\udc36, Comm. ACM 15 (1972), 820\u2013826.","DOI":"10.1145\/361573.361582"},{"key":"2","unstructured":"P. Benner and R. Byers, An exact line search method for solving generalized continuous-time algebraic Riccati equations, IEEE Trans. Autom. Control (to appear)."},{"key":"3","unstructured":"P. Benner, A. J. Laub and V. Mehrmann, A collection of benchmark examples for the numerical solution of algebraic Riccati equations I: continuous-time case, Technical Report SPC 95-22, Fakult\u00e4t f\u00fcr Mathematik, Technische Universit\u00e4t Chemnitz-Zwickau, FRG, 1995."},{"key":"4","doi-asserted-by":"publisher","first-page":"377","DOI":"10.1017\/S0004972700041071","article-title":"Matrix quadratic equations","volume":"10","author":"Coppel, W. A.","year":"1974","journal-title":"Bull. Austral. Math. Soc.","ISSN":"http:\/\/id.crossref.org\/issn\/0004-9727","issn-type":"print"},{"issue":"2","key":"5","doi-asserted-by":"publisher","first-page":"296","DOI":"10.1137\/0720020","article-title":"Convergence rates for Newton\u2019s method at singular points","volume":"20","author":"Decker, D. 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Control","ISSN":"http:\/\/id.crossref.org\/issn\/0018-9286","issn-type":"print"},{"key":"10","series-title":"International Series of Monographs in Pure and Applied Mathematics, Vol. 46","volume-title":"Functional analysis in normed spaces","author":"Kantorovich, L. V.","year":"1964"},{"issue":"176","key":"11","doi-asserted-by":"publisher","first-page":"609","DOI":"10.2307\/2008176","article-title":"A Shamanski\u012d-like acceleration scheme for nonlinear equations at singular roots","volume":"47","author":"Kelley, C. T.","year":"1986","journal-title":"Math. Comp.","ISSN":"http:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"5","key":"12","doi-asserted-by":"publisher","first-page":"1001","DOI":"10.1137\/0720070","article-title":"A new acceleration method for Newton\u2019s method at singular points","volume":"20","author":"Kelley, C. T.","year":"1983","journal-title":"SIAM J. Numer. 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