{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,22]],"date-time":"2026-08-22T04:05:46Z","timestamp":1787371546981,"version":"build-2736575974"},"reference-count":27,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"5","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Numer. Anal."],"published-print":{"date-parts":[[2008,1]]},"abstract":"<jats:p>The paper concerns numerical algorithms for solving the Beltrami equation $f_{\\bar{z}}(z)=\\mu(z) f_z(z)$ for a compactly supported $\\mu$. First, we study an efficient algorithm that has been proposed in [P. Daripa, J. Comput. Phys., 106 (1993), pp. 355\u2013365] and [P. Daripa and D. Mashat, Numer. Algorithms, 18 (1998), pp. 133\u2013157] and present its rigorous justification. We then propose a different scheme for solving the Beltrami equation which has a comparable speed and accuracy, but has the virtue of easier implementation by avoiding the use of the Hilbert transform. The present paper can also be viewed as a prologue to one important application of the Beltrami equation: it provides a detailed description of the algorithm that has been used in [D. Gaidashev, Nonlinearity, 20 (1998), pp. 713\u2013741] and [D. Gaidashev and M. Yampolsky, Experiment. Math., 16 (2007), pp. 215\u2013226] to address an important issue in complex dynamics\u2014conjectural universality for Siegel disks.<\/jats:p>","DOI":"10.1137\/050640710","type":"journal-article","created":{"date-parts":[[2008,5,12]],"date-time":"2008-05-12T09:52:42Z","timestamp":1210585962000},"page":"2238-2253","source":"Crossref","is-referenced-by-count":9,"title":["On Numerical Algorithms for the Solution of a Beltrami Equation"],"prefix":"10.1137","volume":"46","author":[{"given":"Denis","family":"Gaidashev","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Dmitry","family":"Khmelev","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2008,5,9]]},"reference":[{"key":"R1","unstructured":"Ada Core Technologies, New York, NY 10003, USA; see also ftp:\/\/cs.nyu.edu\/pub\/gnat."},{"key":"R2","unstructured":"L. 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