{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,20]],"date-time":"2026-04-20T21:52:26Z","timestamp":1776721946026,"version":"3.51.2"},"reference-count":13,"publisher":"American Mathematical Society (AMS)","issue":"216","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>\n                    Many verification algorithms use an expansion\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"f left-parenthesis x right-parenthesis element-of f left-parenthesis x overTilde right-parenthesis plus upper S dot left-parenthesis x minus x overTilde right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>f<\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>x<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                            <mml:mo>\n                              \u2208\n                              \n                            <\/mml:mo>\n                            <mml:mi>f<\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mover>\n                                <mml:mi>x<\/mml:mi>\n                                <mml:mo stretchy=\"false\">\n                                  ~\n                                  \n                                <\/mml:mo>\n                              <\/mml:mover>\n                            <\/mml:mrow>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                            <mml:mo>+<\/mml:mo>\n                            <mml:mi>S<\/mml:mi>\n                            <mml:mo>\n                              \u22c5\n                              \n                            <\/mml:mo>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>x<\/mml:mi>\n                            <mml:mo>\n                              \u2212\n                              \n                            <\/mml:mo>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mover>\n                                <mml:mi>x<\/mml:mi>\n                                <mml:mo stretchy=\"false\">\n                                  ~\n                                  \n                                <\/mml:mo>\n                              <\/mml:mover>\n                            <\/mml:mrow>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">f(x) \\in f(\\tilde {x}) + S \\cdot (x - \\tilde {x})<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    ,\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"f colon double-struck upper R Superscript n Baseline right-arrow double-struck upper R Superscript n\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>f<\/mml:mi>\n                            <mml:mo>:<\/mml:mo>\n                            <mml:msup>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi mathvariant=\"double-struck\">R<\/mml:mi>\n                              <\/mml:mrow>\n                              <mml:mi>n<\/mml:mi>\n                            <\/mml:msup>\n                            <mml:mo stretchy=\"false\">\n                              \u2192\n                              \n                            <\/mml:mo>\n                            <mml:msup>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi mathvariant=\"double-struck\">R<\/mml:mi>\n                              <\/mml:mrow>\n                              <mml:mi>n<\/mml:mi>\n                            <\/mml:msup>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">f : \\mathbb {R}^n \\rightarrow \\mathbb {R}^n<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    for\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"x element-of upper X\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>x<\/mml:mi>\n                            <mml:mo>\n                              \u2208\n                              \n                            <\/mml:mo>\n                            <mml:mi>X<\/mml:mi>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">x \\in X<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    , where the set of matrices\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper S\">\n                        <mml:semantics>\n                          <mml:mi>S<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">S<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    is usually computed as a gradient or by means of slopes. In the following, an expansion scheme is described which frequently yields sharper inclusions for\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper S\">\n                        <mml:semantics>\n                          <mml:mi>S<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">S<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    . This allows also to compute sharper inclusions for the range of\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"f\">\n                        <mml:semantics>\n                          <mml:mi>f<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">f<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    over a domain. Roughly speaking,\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"f\">\n                        <mml:semantics>\n                          <mml:mi>f<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">f<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    has to be given by means of a computer program. The process of expanding\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"f\">\n                        <mml:semantics>\n                          <mml:mi>f<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">f<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    can then be fully automatized. The function\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"f\">\n                        <mml:semantics>\n                          <mml:mi>f<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">f<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    need not be differentiable. For locally convex or concave functions special improvements are described. Moreover, in contrast to other methods,\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"x overTilde intersection upper X\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mover>\n                                <mml:mi>x<\/mml:mi>\n                                <mml:mo stretchy=\"false\">\n                                  ~\n                                  \n                                <\/mml:mo>\n                              <\/mml:mover>\n                            <\/mml:mrow>\n                            <mml:mo>\n                              \u2229\n                              \n                            <\/mml:mo>\n                            <mml:mi>X<\/mml:mi>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\tilde {x} \\cap X<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    may be empty without implying large overestimations for\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper S\">\n                        <mml:semantics>\n                          <mml:mi>S<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">S<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    . This may be advantageous in practical applications.\n                  <\/p>","DOI":"10.1090\/s0025-5718-96-00773-9","type":"journal-article","created":{"date-parts":[[2002,7,26]],"date-time":"2002-07-26T18:14:44Z","timestamp":1027707284000},"page":"1503-1512","source":"Crossref","is-referenced-by-count":25,"title":["Expansion and estimation of the range of nonlinear functions"],"prefix":"10.1090","volume":"65","author":[{"given":"S.","family":"Rump","sequence":"first","affiliation":[]}],"member":"14","published-online":{"date-parts":[[1996]]},"reference":[{"key":"1","isbn-type":"print","first-page":"63","article-title":"Intervallanalytische Methoden bei nichtlinearen Gleichungen","author":"Alefeld, G\u00f6tz","year":"1979","ISBN":"https:\/\/id.crossref.org\/isbn\/3411015594"},{"key":"2","series-title":"Computer Science and Applied Mathematics","isbn-type":"print","volume-title":"Introduction to interval computations","author":"Alefeld, G\u00f6tz","year":"1983","ISBN":"https:\/\/id.crossref.org\/isbn\/0120498200"},{"key":"3","doi-asserted-by":"publisher","first-page":"94","DOI":"10.1093\/comjnl\/12.1.94","article-title":"A new method of solving nonlinear simultaneous equations","volume":"12","author":"Broyden, C. G.","year":"1969","journal-title":"Comput. J.","ISSN":"https:\/\/id.crossref.org\/issn\/0010-4620","issn-type":"print"},{"key":"4","isbn-type":"print","first-page":"83","article-title":"On automatic differentiation","author":"Griewank, Andreas","year":"1989","ISBN":"https:\/\/id.crossref.org\/isbn\/079230490X"},{"key":"5","doi-asserted-by":"crossref","unstructured":"E. Hansen, A generalized interval arithmetic, Interval Mathematics (K. Nickel, editor), vol. 29, Springer, Berlin, 1975, pp. 7\u201318.","DOI":"10.1007\/3-540-07170-9_2"},{"key":"6","doi-asserted-by":"publisher","first-page":"374","DOI":"10.2307\/2004667","article-title":"On solving systems of equations using interval arithmetic","volume":"22","author":"Hansen, Eldon R.","year":"1968","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"key":"7","unstructured":"G. Heindl, Zur numerischen Einschlie\u00dfung der Werte von Peanofunktionalen, Z. Angew. Math. Mech. 75 (1995), S637\u2013S638."},{"key":"8","unstructured":"D. Husung, Precompiler for scientific computation (TPX), Technical Report 91.1, Inst. for Computer Science III, TU Hamburg-Harburg, 1989."},{"issue":"3","key":"9","doi-asserted-by":"publisher","first-page":"604","DOI":"10.1137\/0722037","article-title":"Interval slopes for rational functions and associated centered forms","volume":"22","author":"Krawczyk, R.","year":"1985","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"key":"10","volume-title":"Interval analysis","author":"Moore, Ramon E.","year":"1966"},{"key":"11","series-title":"Encyclopedia of Mathematics and its Applications","isbn-type":"print","doi-asserted-by":"crossref","DOI":"10.1007\/978-94-011-3330-2","volume-title":"Interval methods for systems of equations","volume":"37","author":"Neumaier, Arnold","year":"1990","ISBN":"https:\/\/id.crossref.org\/isbn\/052133196X"},{"key":"12","unstructured":"L.B. Rall, Automatic differentiation: techniques and applications, Lecture Notes in Comput. Sci., vol. 120, Springer Verlag, Berlin\u2013Heidelberg\u2013New York, 1981."},{"key":"13","unstructured":"J.W. Schmidt, Die Regula Falsi f\u00fcr Operatioren in Banachr\u00e4umen, Z. Angew. Math. 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