{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T07:27:49Z","timestamp":1776842869477,"version":"3.51.2"},"reference-count":25,"publisher":"American Mathematical Society (AMS)","issue":"253","license":[{"start":{"date-parts":[[2006,8,31]],"date-time":"2006-08-31T00:00:00Z","timestamp":1156982400000},"content-version":"am","delay-in-days":365,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>\n                    Class\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"script upper S Subscript m\">\n                        <mml:semantics>\n                          <mml:msub>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">S<\/mml:mi>\n                              <\/mml:mrow>\n                            <\/mml:mrow>\n                            <mml:mi>m<\/mml:mi>\n                          <\/mml:msub>\n                          <mml:annotation encoding=\"application\/x-tex\">{\\mathcal S}_m<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    variable transformations with integer\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"m\">\n                        <mml:semantics>\n                          <mml:mi>m<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">m<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    , for numerical computation of finite-range integrals, were introduced and studied by the author in the paper [A. Sidi, A new variable transformation for numerical integration,\n                    <italic>Numerical Integration IV,<\/italic>\n                    1993 (H. Brass and G. H\u00e4mmerlin, eds.), pp. 359\u2013373.] A representative of this class is the\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"sine Superscript m\">\n                        <mml:semantics>\n                          <mml:msup>\n                            <mml:mi>sin<\/mml:mi>\n                            <mml:mi>m<\/mml:mi>\n                          <\/mml:msup>\n                          <mml:annotation encoding=\"application\/x-tex\">\\sin ^m<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    -transformation that has been used with lattice rules for multidimensional integration. These transformations \u201cperiodize\u201d the integrand functions in a way that enables the trapezoidal rule to achieve very high accuracy, especially with even\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"m\">\n                        <mml:semantics>\n                          <mml:mi>m<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">m<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    . In the present work, we extend these transformations to\n                    <italic>arbitrary<\/italic>\n                    values of\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"m\">\n                        <mml:semantics>\n                          <mml:mi>m<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">m<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    , and give a detailed analysis of the resulting transformed trapezoidal rule approximations. We show that, with suitable\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"m\">\n                        <mml:semantics>\n                          <mml:mi>m<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">m<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    , they can be very useful in different situations. We prove, for example, that if the integrand function is smooth on the interval of integration and vanishes at the endpoints, then results of especially high accuracy are obtained by taking\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"2 m\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mn>2<\/mml:mn>\n                            <mml:mi>m<\/mml:mi>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">2m<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    to be an odd integer. Such a situation can be realized in general by subtracting from the integrand the linear interpolant at the endpoints of the interval of integration. We also illustrate some of the results with numerical examples via the extended\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"sine Superscript m\">\n                        <mml:semantics>\n                          <mml:msup>\n                            <mml:mi>sin<\/mml:mi>\n                            <mml:mi>m<\/mml:mi>\n                          <\/mml:msup>\n                          <mml:annotation encoding=\"application\/x-tex\">\\sin ^m<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    -transformation.\n                  <\/p>","DOI":"10.1090\/s0025-5718-05-01773-4","type":"journal-article","created":{"date-parts":[[2005,11,16]],"date-time":"2005-11-16T10:22:35Z","timestamp":1132136555000},"page":"327-343","source":"Crossref","is-referenced-by-count":30,"title":["Extension of a class of periodizing variable transformations for numerical Integration"],"prefix":"10.1090","volume":"75","author":[{"given":"Avram","family":"Sidi","sequence":"first","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2005,8,31]]},"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","isbn-type":"print","volume-title":"An introduction to numerical analysis","author":"Atkinson, Kendall E.","year":"1989","ISBN":"https:\/\/id.crossref.org\/isbn\/0471624896","edition":"2"},{"key":"3","series-title":"Computer Science and Applied Mathematics","isbn-type":"print","volume-title":"Methods of numerical integration","author":"Davis, Philip J.","year":"1984","ISBN":"https:\/\/id.crossref.org\/isbn\/0122063600","edition":"2"},{"key":"4","first-page":"E77--E137","article-title":"Sigmoidal transformations and the trapezoidal rule","volume":"40","author":"Elliott, David","year":"1998","journal-title":"J. Austral. Math. Soc. Ser. B","ISSN":"https:\/\/id.crossref.org\/issn\/0334-2700","issn-type":"print"},{"key":"5","doi-asserted-by":"crossref","unstructured":"W. Gautschi. Algorithm 726: ORTHPOL \u2013 a package of routines for generating orthogonal polynomials and Gauss-type quadrature rules. ACM Transactions on Mathematical Software, 20:21\u201362, 1994.","DOI":"10.1145\/174603.174605"},{"issue":"1-2","key":"6","doi-asserted-by":"publisher","first-page":"121","DOI":"10.1016\/S0377-0427(99)00217-4","article-title":"d2lri: a nonadaptive algorithm for two-dimensional cubature","volume":"112","author":"Hill, Michael","year":"1999","journal-title":"J. Comput. Appl. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0377-0427","issn-type":"print"},{"issue":"1-2","key":"7","doi-asserted-by":"publisher","first-page":"3","DOI":"10.1016\/0377-0427(87)90034-3","article-title":"On a certain quadrature formula","volume":"17","author":"Iri, Masao","year":"1987","journal-title":"J. Comput. Appl. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0377-0427","issn-type":"print"},{"issue":"10","key":"8","doi-asserted-by":"publisher","first-page":"1709","DOI":"10.1002\/(SICI)1097-0207(20000410)47:10<1709::AID-NME852>3.0.CO;2-V","article-title":"Semi-sigmoidal transformations for evaluating weakly singular boundary element integrals","volume":"47","author":"Johnston, Peter R.","year":"2000","journal-title":"Internat. J. Numer. Methods Engrg.","ISSN":"https:\/\/id.crossref.org\/issn\/0029-5981","issn-type":"print"},{"key":"9","volume-title":"{\\cyr Teoretiko}-{\\cyr chislovye metody v priblizhennom analize}","author":"Korobov, N. M.","year":"1963"},{"issue":"1-2","key":"10","doi-asserted-by":"publisher","first-page":"337","DOI":"10.1016\/0377-0427(95)00196-4","article-title":"Periodizing transformations for numerical integration","volume":"66","author":"Laurie, Dirk P.","year":"1996","journal-title":"J. Comput. Appl. 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Sci.","ISSN":"https:\/\/id.crossref.org\/issn\/0034-5318","issn-type":"print"},{"key":"13","series-title":"International Series in Pure and Applied Mathematics","isbn-type":"print","volume-title":"A first course in numerical analysis","author":"Ralston, Anthony","year":"1978","ISBN":"https:\/\/id.crossref.org\/isbn\/0070511586","edition":"2"},{"key":"14","doi-asserted-by":"crossref","unstructured":"I. Robinson and M. Hill. Algorithm 816: r2d2lri: An algorithm for automatic two-dimensional cubature. ACM Trans. Math. Software, 28:75\u2013100, 2002.","DOI":"10.1145\/513001.513006"},{"key":"15","doi-asserted-by":"publisher","first-page":"245","DOI":"10.2307\/2003298","article-title":"Numerical evaluation of high-dimensional integrals","volume":"18","author":"Sag, T. W.","year":"1964","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"key":"16","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1002\/sapm19553411","article-title":"Non-linear transformations of divergent and slowly convergent sequences","volume":"34","author":"Shanks, Daniel","year":"1955","journal-title":"J. Math. and Phys.","ISSN":"https:\/\/id.crossref.org\/issn\/0097-1421","issn-type":"print"},{"key":"17","isbn-type":"print","first-page":"359","article-title":"A new variable transformation for numerical integration","author":"Sidi, Avram","year":"1993","ISBN":"https:\/\/id.crossref.org\/isbn\/376432922X"},{"key":"18","unstructured":"A. Sidi. Class \ud835\udcae\u2098 variable transformations and application to numerical integration over smooth surfaces in \u211d\u00b3. Preprint. Computer Science Dept., Technion\u2013Israel Institute of Technology, 2003."},{"key":"19","series-title":"Cambridge Monographs on Applied and Computational Mathematics","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511546815","volume-title":"Practical extrapolation methods","volume":"10","author":"Sidi, Avram","year":"2003","ISBN":"https:\/\/id.crossref.org\/isbn\/0521661595"},{"issue":"2","key":"20","doi-asserted-by":"publisher","first-page":"371","DOI":"10.1007\/s00211-004-0539-4","article-title":"Euler-Maclaurin expansions for integrals with endpoint singularities: a new perspective","volume":"98","author":"Sidi, Avram","year":"2004","journal-title":"Numer. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0029-599X","issn-type":"print"},{"key":"21","doi-asserted-by":"crossref","unstructured":"A. Sidi, Numerical integration over smooth surfaces in \u211d\u00b3 via class \ud835\udcae\u2098 variable transformations. Part I: Smooth integrands. Appl. Math. Comp., 2005. In press. First published electronically on April 11, 2005.","DOI":"10.1016\/j.amc.2005.01.077"},{"key":"22","unstructured":"A. Sidi, Application of class \ud835\udcae\u2098 variable transformations to numerical integration over surfaces of spheres. J. Comp. Appl. Math., 2005. In press. First published electronically on April 11, 2005."},{"key":"23","series-title":"Oxford Science Publications","isbn-type":"print","doi-asserted-by":"crossref","DOI":"10.1093\/oso\/9780198534723.001.0001","volume-title":"Lattice methods for multiple integration","author":"Sloan, I. H.","year":"1994","ISBN":"https:\/\/id.crossref.org\/isbn\/0198534728"},{"issue":"3-4","key":"24","doi-asserted-by":"publisher","first-page":"321","DOI":"10.1023\/A:1019155601289","article-title":"Error expansions for multidimensional trapezoidal rules with Sidi transformations","volume":"16","author":"Verlinden, P.","year":"1997","journal-title":"Numer. 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Tables Aids Comput.","ISSN":"https:\/\/id.crossref.org\/issn\/0891-6837","issn-type":"print"}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.ams.org\/mcom\/2006-75-253\/S0025-5718-05-01773-4\/S0025-5718-05-01773-4.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/2006-75-253\/S0025-5718-05-01773-4\/S0025-5718-05-01773-4.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T14:33:43Z","timestamp":1776782023000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2006-75-253\/S0025-5718-05-01773-4\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2005,8,31]]},"references-count":25,"journal-issue":{"issue":"253","published-print":{"date-parts":[[2006,1]]}},"alternative-id":["S0025-5718-05-01773-4"],"URL":"https:\/\/doi.org\/10.1090\/s0025-5718-05-01773-4","archive":["CLOCKSS","Portico"],"relation":{},"ISSN":["1088-6842","0025-5718"],"issn-type":[{"value":"1088-6842","type":"electronic"},{"value":"0025-5718","type":"print"}],"subject":[],"published":{"date-parts":[[2005,8,31]]}}}