{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T18:39:18Z","timestamp":1776796758633,"version":"3.51.2"},"reference-count":30,"publisher":"American Mathematical Society (AMS)","issue":"285","license":[{"start":{"date-parts":[[2014,5,22]],"date-time":"2014-05-22T00:00:00Z","timestamp":1400716800000},"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                    Cross-derivatives are mixed partial derivatives involving at most one differentiation in each one of\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                    coordinate directions. They are a computational tool in combinatorics and of potential use in high-dimensional integration. Here we present two methods that evaluate all\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"2 Superscript n\">\n                        <mml:semantics>\n                          <mml:msup>\n                            <mml:mn>2<\/mml:mn>\n                            <mml:mi>n<\/mml:mi>\n                          <\/mml:msup>\n                          <mml:annotation encoding=\"application\/x-tex\">2^n<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    cross-derivatives at a given point. The computational complexity is, respectively,\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"3 Superscript n\">\n                        <mml:semantics>\n                          <mml:msup>\n                            <mml:mn>3<\/mml:mn>\n                            <mml:mi>n<\/mml:mi>\n                          <\/mml:msup>\n                          <mml:annotation encoding=\"application\/x-tex\">3^n<\/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=\"n squared 2 Superscript n\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:msup>\n                              <mml:mi>n<\/mml:mi>\n                              <mml:mn>2<\/mml:mn>\n                            <\/mml:msup>\n                            <mml:msup>\n                              <mml:mn>2<\/mml:mn>\n                              <mml:mi>n<\/mml:mi>\n                            <\/mml:msup>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">n^2 2^n<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    times that of the underlying function. The asymptotically faster method involves a final interpolation step, which can easily be carried out using extra-accurate subtractions to reduce the effect of numerical round-off. Further complexity reductions for large\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                    can be obtained through faster polynomial multiplications, e.g., Karatsuba\u2019s method or FFT.\n                  <\/p>","DOI":"10.1090\/s0025-5718-2013-02717-2","type":"journal-article","created":{"date-parts":[[2013,5,22]],"date-time":"2013-05-22T13:29:14Z","timestamp":1369229354000},"page":"251-274","source":"Crossref","is-referenced-by-count":6,"title":["Automatic evaluations of cross-derivatives"],"prefix":"10.1090","volume":"83","author":[{"given":"Andreas","family":"Griewank","sequence":"first","affiliation":[]},{"given":"Lutz","family":"Lehmann","sequence":"additional","affiliation":[]},{"given":"Hernan","family":"Leovey","sequence":"additional","affiliation":[]},{"given":"Marat","family":"Zilberman","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2013,5,22]]},"reference":[{"key":"1","doi-asserted-by":"crossref","unstructured":"[BGLS10] Torsten Bosse, Andreas Griewank, Lutz Lehmann, and Volker Schlo\u00dfhauer, On Hessian- and Jacobian-free SQP methods \u2013 a total quasi-Newton scheme with compact storage, Recent Advances in Optimization and its Applications in Engineering (Moritz Diehl, Francois Glineur, Elias Jarlebring, and Wim Michiels, eds.), Springer Berlin Heidelberg, 2010, pp. 63\u201372.","DOI":"10.1007\/978-3-642-12598-0_6"},{"key":"2","isbn-type":"print","doi-asserted-by":"publisher","first-page":"67","DOI":"10.1145\/1250790.1250801","article-title":"Fourier meets M\u00f6bius: fast subset convolution","author":"Bj\u00f6rklund, Andreas","year":"2007","ISBN":"https:\/\/id.crossref.org\/isbn\/9781595936318"},{"key":"3","doi-asserted-by":"crossref","unstructured":"[CU09] Isabelle Charpentier and Jean Utke, Fast higher-order derivative tensors with Rapsodia, Optimization Methods Software 24 (2009), no. 1, 1\u201314.","DOI":"10.1080\/10556780802413769"},{"key":"4","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511761188","volume-title":"Digital nets and sequences","author":"Dick, Josef","year":"2010","ISBN":"https:\/\/id.crossref.org\/isbn\/9780521191593"},{"key":"5","series-title":"A Series of Books in the Mathematical Sciences","isbn-type":"print","volume-title":"Computers and intractability","author":"Garey, Michael R.","year":"1979","ISBN":"https:\/\/id.crossref.org\/isbn\/0716710455"},{"issue":"5","key":"6","doi-asserted-by":"publisher","first-page":"523","DOI":"10.1016\/j.jco.2010.04.003","article-title":"The smoothing effect of the ANOVA decomposition","volume":"26","author":"Griebel, Michael","year":"2010","journal-title":"J. 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