{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,5]],"date-time":"2026-10-05T15:15:29Z","timestamp":1791213329458,"version":"4.1.0"},"reference-count":22,"publisher":"American Mathematical Society (AMS)","issue":"275","license":[{"start":{"date-parts":[[2012,2,1]],"date-time":"2012-02-01T00:00:00Z","timestamp":1328054400000},"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                    We developed an efficient procedure to evaluate two auxiliary complete elliptic integrals of the second kind\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper B left-parenthesis m right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>B<\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>m<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">B(m)<\/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=\"upper D left-parenthesis m right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>D<\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>m<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">D(m)<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    by using their Taylor series expansions, the definition of Jacobi\u2019s nome, and Legendre\u2019s relation. The developed procedure is more precise than the existing ones in the sense that the maximum relative errors are 1-3 machine epsilons, and it runs drastically faster; around 5 times faster than Bulirsch\u2019s\n                    <monospace>cel2<\/monospace>\n                    and 16 times faster than Carlson\u2019s\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper R Subscript upper F\">\n                        <mml:semantics>\n                          <mml:msub>\n                            <mml:mi>R<\/mml:mi>\n                            <mml:mi>F<\/mml:mi>\n                          <\/mml:msub>\n                          <mml:annotation encoding=\"application\/x-tex\">R_F<\/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=\"upper R Subscript upper D\">\n                        <mml:semantics>\n                          <mml:msub>\n                            <mml:mi>R<\/mml:mi>\n                            <mml:mi>D<\/mml:mi>\n                          <\/mml:msub>\n                          <mml:annotation encoding=\"application\/x-tex\">R_D<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    .\n                  <\/p>","DOI":"10.1090\/s0025-5718-2011-02455-5","type":"journal-article","created":{"date-parts":[[2011,2,1]],"date-time":"2011-02-01T07:33:16Z","timestamp":1296545596000},"page":"1725-1743","source":"Crossref","is-referenced-by-count":27,"title":["Precise and fast computation of the general complete elliptic integral of the second kind"],"prefix":"10.1090","volume":"80","author":[{"given":"Toshio","family":"Fukushima","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"14","published-online":{"date-parts":[[2011,2,1]]},"reference":[{"key":"1","unstructured":"Abramowitz, M., & Stegun, I.A. 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