{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T15:35:25Z","timestamp":1776785725864,"version":"3.51.2"},"reference-count":10,"publisher":"American Mathematical Society (AMS)","issue":"254","license":[{"start":{"date-parts":[[2007,1,3]],"date-time":"2007-01-03T00:00:00Z","timestamp":1167782400000},"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                    Here we propose a new method based on projections for the approximate solution of eigenvalue problems. For an integral operator with a smooth kernel, using an interpolatory projection at Gauss points onto the space of (discontinuous) piecewise polynomials of degree\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"less-than-or-equal-to r minus 1\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mo>\n                              \u2264\n                              \n                            <\/mml:mo>\n                            <mml:mi>r<\/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\">\\leq r-1<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    , we show that the proposed method exhibits an error of the order of\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"4 r\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mn>4<\/mml:mn>\n                            <mml:mi>r<\/mml:mi>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">4r<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    for eigenvalue approximation and of the order of\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"3 r\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mn>3<\/mml:mn>\n                            <mml:mi>r<\/mml:mi>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">3r<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    for spectral subspace approximation. In the case of a simple eigenvalue, we show that by using an iteration technique, an eigenvector approximation of the order\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"4 r\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mn>4<\/mml:mn>\n                            <mml:mi>r<\/mml:mi>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">4r<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    can be obtained. This improves upon the order\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"2 r\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mn>2<\/mml:mn>\n                            <mml:mi>r<\/mml:mi>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">2r<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    for eigenvalue approximation in the collocation\/iterated collocation method and the orders\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"r\">\n                        <mml:semantics>\n                          <mml:mi>r<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">r<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    and\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"2 r\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mn>2<\/mml:mn>\n                            <mml:mi>r<\/mml:mi>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">2r<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    for spectral subspace approximation in the collocation method and the iterated collocation method, respectively. We illustrate this improvement in the order of convergence by numerical examples.\n                  <\/p>","DOI":"10.1090\/s0025-5718-06-01871-0","type":"journal-article","created":{"date-parts":[[2006,2,15]],"date-time":"2006-02-15T11:05:20Z","timestamp":1140001520000},"page":"847-857","source":"Crossref","is-referenced-by-count":7,"title":["A new superconvergent collocation method for eigenvalue problems"],"prefix":"10.1090","volume":"75","author":[{"given":"Rekha","family":"Kulkarni","sequence":"first","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2006,1,3]]},"reference":[{"key":"1","series-title":"Cambridge Monographs on Applied and Computational Mathematics","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511626340","volume-title":"The numerical solution of integral equations of the second kind","volume":"4","author":"Atkinson, Kendall E.","year":"1997","ISBN":"https:\/\/id.crossref.org\/isbn\/0521583918"},{"issue":"1","key":"2","doi-asserted-by":"publisher","first-page":"172","DOI":"10.1137\/0720012","article-title":"Piecewise continuous collocation for integral equations","volume":"20","author":"Atkinson, K.","year":"1983","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"key":"3","series-title":"Monographs on Numerical Analysis","isbn-type":"print","volume-title":"The numerical treatment of integral equations","author":"Baker, Christopher T. H.","year":"1977","ISBN":"https:\/\/id.crossref.org\/isbn\/019853406X"},{"key":"4","series-title":"Computer Science and Applied Mathematics","isbn-type":"print","volume-title":"Spectral approximation of linear operators","author":"Chatelin, Fran\u00e7oise","year":"1983","ISBN":"https:\/\/id.crossref.org\/isbn\/0121706206"},{"issue":"1","key":"5","first-page":"71","article-title":"Superconvergence results for the iterated projection method applied to a Fredholm integral equation of the second kind and the corresponding eigenvalue problem","volume":"6","author":"Chatelin, Fran\u00e7oise","year":"1984","journal-title":"J. Integral Equations","ISSN":"https:\/\/id.crossref.org\/issn\/0163-5549","issn-type":"print"},{"key":"6","doi-asserted-by":"publisher","first-page":"582","DOI":"10.1137\/0710052","article-title":"Collocation at Gaussian points","volume":"10","author":"de Boor, Carl","year":"1973","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"issue":"151","key":"7","doi-asserted-by":"publisher","first-page":"679","DOI":"10.2307\/2006187","article-title":"Collocation approximation to eigenvalues of an ordinary differential equation: the principle of the thing","volume":"35","author":"de Boor, Carl","year":"1980","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"1-2","key":"8","doi-asserted-by":"publisher","first-page":"75","DOI":"10.1081\/NFA-120020246","article-title":"A new superconvergent projection method for approximate solutions of eigenvalue problems","volume":"24","author":"Kulkarni, Rekha P.","year":"2003","journal-title":"Numer. Funct. Anal. Optim.","ISSN":"https:\/\/id.crossref.org\/issn\/0163-0563","issn-type":"print"},{"key":"9","doi-asserted-by":"crossref","first-page":"712","DOI":"10.1090\/S0025-5718-1975-0383117-3","article-title":"Spectral approximation for compact operators","volume":"29","author":"Osborn, John E.","year":"1975","journal-title":"Math. Comput."},{"key":"10","isbn-type":"print","first-page":"35","article-title":"Superconvergence","author":"Sloan, I. H.","year":"1990","ISBN":"https:\/\/id.crossref.org\/isbn\/0306432625"}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.ams.org\/mcom\/2006-75-254\/S0025-5718-06-01871-0\/S0025-5718-06-01871-0.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/2006-75-254\/S0025-5718-06-01871-0\/S0025-5718-06-01871-0.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T14:35:16Z","timestamp":1776782116000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2006-75-254\/S0025-5718-06-01871-0\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2006,1,3]]},"references-count":10,"journal-issue":{"issue":"254","published-print":{"date-parts":[[2006,4]]}},"alternative-id":["S0025-5718-06-01871-0"],"URL":"https:\/\/doi.org\/10.1090\/s0025-5718-06-01871-0","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":[[2006,1,3]]}}}