{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T07:31:20Z","timestamp":1776843080282,"version":"3.51.2"},"reference-count":20,"publisher":"American Mathematical Society (AMS)","issue":"214","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>\n                    An anti-Gaussian quadrature formula is an\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"left-parenthesis n plus 1 right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>n<\/mml:mi>\n                            <mml:mo>+<\/mml:mo>\n                            <mml:mn>1<\/mml:mn>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">(n+1)<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    -point formula of degree\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"2 n minus 1\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mn>2<\/mml:mn>\n                            <mml:mi>n<\/mml:mi>\n                            <mml:mo>\n                              \u2212\n                              \n                            <\/mml:mo>\n                            <mml:mn>1<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">2n-1<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    which integrates polynomials of degree up to\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"2 n plus 1\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mn>2<\/mml:mn>\n                            <mml:mi>n<\/mml:mi>\n                            <mml:mo>+<\/mml:mo>\n                            <mml:mn>1<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">2n+1<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    with an error equal in magnitude but of opposite sign to that of the\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"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                    -point Gaussian formula. Its intended application is to estimate the error incurred in Gaussian integration by halving the difference between the results obtained from the two formulas. We show that an anti-Gaussian formula has positive weights, and that its nodes are in the integration interval and are interlaced by those of the corresponding Gaussian formula. Similar results for Gaussian formulas with respect to a positive weight are given, except that for some weight functions, at most two of the nodes may be outside the integration interval. The anti-Gaussian formula has only interior nodes in many cases when the Kronrod extension does not, and is as easy to compute as the\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"left-parenthesis n plus 1 right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>n<\/mml:mi>\n                            <mml:mo>+<\/mml:mo>\n                            <mml:mn>1<\/mml:mn>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">(n+1)<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    -point Gaussian formula.\n                  <\/p>","DOI":"10.1090\/s0025-5718-96-00713-2","type":"journal-article","created":{"date-parts":[[2002,7,26]],"date-time":"2002-07-26T18:14:44Z","timestamp":1027707284000},"page":"739-747","source":"Crossref","is-referenced-by-count":92,"title":["Anti-Gaussian quadrature formulas"],"prefix":"10.1090","volume":"65","author":[{"given":"Dirk","family":"Laurie","sequence":"first","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"},{"issue":"8","key":"2","doi-asserted-by":"publisher","first-page":"807","DOI":"10.1007\/BF01385655","article-title":"Suboptimal Kronrod extension formulae for numerical quadrature","volume":"58","author":"Begumisa, Annuntiato","year":"1991","journal-title":"Numer. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0029-599X","issn-type":"print"},{"key":"3","doi-asserted-by":"publisher","first-page":"197","DOI":"10.1007\/BF01386223","article-title":"A method for numerical integration on an automatic computer","volume":"2","author":"Clenshaw, C. W.","year":"1960","journal-title":"Numer. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0029-599X","issn-type":"print"},{"key":"4","first-page":"254","article-title":"On two-sided approximation in the numerical integration of ordinary differential equations","volume":"3","author":"Devjatko, V. I.","year":"1963","journal-title":"\\v{Z}. Vy\\v{c}isl. Mat i Mat. 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Sixteen-place tables","author":"Kronrod, Aleksandr Semenovich","year":"1965"},{"issue":"2","key":"12","doi-asserted-by":"publisher","first-page":"258","DOI":"10.1007\/BF02218446","article-title":"Sharper error estimates in adaptive quadrature","volume":"23","author":"Laurie, D. P.","year":"1983","journal-title":"BIT","ISSN":"https:\/\/id.crossref.org\/issn\/0006-3835","issn-type":"print"},{"key":"13","doi-asserted-by":"crossref","unstructured":"D. P. Laurie, Practical error estimation in numerical integration, J. Computat. Appl. Math, 12&13:258\u2013261, 1985.","DOI":"10.1016\/0377-0427(85)90036-6"},{"issue":"3","key":"14","doi-asserted-by":"crossref","first-page":"365","DOI":"10.1080\/16073606.1992.9631697","article-title":"Stratified sequences of nested quadrature formulas","volume":"15","author":"Laurie, Dirk P.","year":"1992","journal-title":"Quaestiones Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0379-9468","issn-type":"print"},{"key":"15","series-title":"Prentice-Hall Series in Computational Mathematics","isbn-type":"print","volume-title":"The symmetric eigenvalue problem","author":"Parlett, Beresford N.","year":"1980","ISBN":"https:\/\/id.crossref.org\/isbn\/0138800472"},{"issue":"2","key":"16","doi-asserted-by":"publisher","first-page":"123","DOI":"10.1145\/63522.63523","article-title":"An algorithm for generating interpolatory quadrature rules of the highest degree of precision with preassigned nodes for general weight functions","volume":"15","author":"Patterson, T. N. L.","year":"1989","journal-title":"ACM Trans. Math. Software","ISSN":"https:\/\/id.crossref.org\/issn\/0098-3500","issn-type":"print"},{"key":"17","doi-asserted-by":"crossref","unstructured":"T. N. L. Patterson, Modified optimal quadrature extensions, Numer. Math., 60:511\u2013520, 1993.","DOI":"10.1007\/BF01388702"},{"key":"18","series-title":"Springer Series in Computational Mathematics","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-61786-7","volume-title":"QUADPACK","volume":"1","author":"Piessens, Robert","year":"1983","ISBN":"https:\/\/id.crossref.org\/isbn\/3540125531"},{"key":"19","first-page":"947","article-title":"Some properties of solutions of systems of ordinary differential equations by one-step methods of numerical integration","volume":"1","author":"Rakitski\u012d, Ju. V.","year":"1961","journal-title":"\\v{Z}. Vy\\v{c}isl. Mat i Mat. Fiz.","ISSN":"https:\/\/id.crossref.org\/issn\/0044-4669","issn-type":"print"},{"key":"20","first-page":"515","article-title":"Polar difference methods for solving the Cauchy problem for a system of ordinary differential equations","volume":"2","author":"Salihov, N. P.","year":"1962","journal-title":"\\v{Z}. Vy\\v{c}isl. Mat i Mat. 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