{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,20]],"date-time":"2026-04-20T21:49:47Z","timestamp":1776721787863,"version":"3.51.2"},"reference-count":19,"publisher":"American Mathematical Society (AMS)","issue":"214","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>\n                    We present some relations that allow the efficient approximate inversion of linear differential operators with rational function coefficients. We employ expansions in terms of a large class of orthogonal polynomial families, including all the classical orthogonal polynomials. These families obey a simple 3-term recurrence relation for differentiation, which implies that on an appropriately restricted domain the differentiation operator has a unique banded inverse. The inverse is an integration operator for the family, and it is simply the tridiagonal coefficient matrix for the recurrence. Since in these families convolution operators (i.e., matrix representations of multiplication by a function) are banded for polynomials, we are able to obtain a banded representation for linear differential operators with rational coefficients. This leads to a method of solution of initial or boundary value problems that, besides having an operation count that scales linearly with the order of truncation\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper N\">\n                        <mml:semantics>\n                          <mml:mi>N<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">N<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    , is computationally well conditioned. Among the applications considered is the use of rational maps for the resolution of sharp interior layers.\n                  <\/p>","DOI":"10.1090\/s0025-5718-96-00704-1","type":"journal-article","created":{"date-parts":[[2002,7,26]],"date-time":"2002-07-26T18:14:44Z","timestamp":1027707284000},"page":"611-635","source":"Crossref","is-referenced-by-count":68,"title":["An efficient spectral method for ordinary differential equations with rational function coefficients"],"prefix":"10.1090","volume":"65","author":[{"given":"Evangelos","family":"Coutsias","sequence":"first","affiliation":[]},{"given":"Thomas","family":"Hagstrom","sequence":"additional","affiliation":[]},{"given":"David","family":"Torres","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[1996]]},"reference":[{"key":"1","series-title":"National Bureau of Standards Applied Mathematics Series, No. 55","volume-title":"Handbook of mathematical functions with formulas, graphs, and mathematical tables","author":"Abramowitz, Milton","year":"1964"},{"key":"2","doi-asserted-by":"crossref","unstructured":"A. Bayliss, D. Gottlieb, B. Matkowsky and M. Minkoff, An adaptive pseudo-spectral method for reaction diffusion problems, J. Comp. Phys., 81, (1989), 421-443.","DOI":"10.1016\/0021-9991(89)90215-5"},{"key":"3","doi-asserted-by":"crossref","unstructured":"A. Bayliss and E. Turkel, Mappings and accuracy for Chebyshev pseudo-spectral computations, J. Comp. Phys., 101, (1992), 349\u2013359.","DOI":"10.1016\/0021-9991(92)90012-N"},{"issue":"1-2","key":"4","doi-asserted-by":"publisher","first-page":"53","DOI":"10.1016\/0377-0427(92)90259-Z","article-title":"Polynomial interpolation results in Sobolev spaces","volume":"43","author":"Bernardi, Christine","year":"1992","journal-title":"J. Comput. Appl. 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