{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,20]],"date-time":"2026-04-20T23:40:28Z","timestamp":1776728428831,"version":"3.51.2"},"reference-count":23,"publisher":"American Mathematical Society (AMS)","issue":"240","license":[{"start":{"date-parts":[[2002,11,28]],"date-time":"2002-11-28T00:00:00Z","timestamp":1038441600000},"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                    Let\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"a left-parenthesis t right-parenthesis tilde upper A plus phi left-parenthesis t right-parenthesis sigma-summation Underscript i equals 0 Overscript normal infinity Endscripts beta Subscript i Baseline t Superscript i\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>a<\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>t<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                            <mml:mo>\n                              \u223c\n                              \n                            <\/mml:mo>\n                            <mml:mi>A<\/mml:mi>\n                            <mml:mo>+<\/mml:mo>\n                            <mml:mi>\n                              \u03c6\n                              \n                            <\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>t<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                            <mml:munderover>\n                              <mml:mo>\n                                \u2211\n                                \n                              <\/mml:mo>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi>i<\/mml:mi>\n                                <mml:mo>=<\/mml:mo>\n                                <mml:mn>0<\/mml:mn>\n                              <\/mml:mrow>\n                              <mml:mi mathvariant=\"normal\">\n                                \u221e\n                                \n                              <\/mml:mi>\n                            <\/mml:munderover>\n                            <mml:msub>\n                              <mml:mi>\n                                \u03b2\n                                \n                              <\/mml:mi>\n                              <mml:mi>i<\/mml:mi>\n                            <\/mml:msub>\n                            <mml:msup>\n                              <mml:mi>t<\/mml:mi>\n                              <mml:mi>i<\/mml:mi>\n                            <\/mml:msup>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">a(t)\\sim A+\\varphi (t)\\sum ^\\infty _{i=0}\\beta _it^i<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    as\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"t right-arrow 0 plus\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>t<\/mml:mi>\n                            <mml:mo stretchy=\"false\">\n                              \u2192\n                              \n                            <\/mml:mo>\n                            <mml:mn>0<\/mml:mn>\n                            <mml:mo>+<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">t\\to 0+<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    , where\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"a left-parenthesis t right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>a<\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>t<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">a(t)<\/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=\"phi left-parenthesis t right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>\n                              \u03c6\n                              \n                            <\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>t<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\varphi (t)<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    are known for\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"0 greater-than t less-than-or-equal-to c\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mn>0<\/mml:mn>\n                            <mml:mo>&gt;<\/mml:mo>\n                            <mml:mi>t<\/mml:mi>\n                            <mml:mo>\n                              \u2264\n                              \n                            <\/mml:mo>\n                            <mml:mi>c<\/mml:mi>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">0&gt;t\\leq c<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    for some\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"c greater-than 0\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>c<\/mml:mi>\n                            <mml:mo>&gt;<\/mml:mo>\n                            <mml:mn>0<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">c&gt;0<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    , but\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper A\">\n                        <mml:semantics>\n                          <mml:mi>A<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">A<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    and the\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"beta Subscript i\">\n                        <mml:semantics>\n                          <mml:msub>\n                            <mml:mi>\n                              \u03b2\n                              \n                            <\/mml:mi>\n                            <mml:mi>i<\/mml:mi>\n                          <\/mml:msub>\n                          <mml:annotation encoding=\"application\/x-tex\">\\beta _i<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    are not known. The generalized Richardson extrapolation process GREP\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"Superscript left-parenthesis 1 right-parenthesis\">\n                        <mml:semantics>\n                          <mml:msup>\n                            <mml:mi\/>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mo stretchy=\"false\">(<\/mml:mo>\n                              <mml:mn>1<\/mml:mn>\n                              <mml:mo stretchy=\"false\">)<\/mml:mo>\n                            <\/mml:mrow>\n                          <\/mml:msup>\n                          <mml:annotation encoding=\"application\/x-tex\">^{(1)}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    is used in obtaining good approximations to\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper A\">\n                        <mml:semantics>\n                          <mml:mi>A<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">A<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    , the limit or antilimit of\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"a left-parenthesis t right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>a<\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>t<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">a(t)<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    as\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"t right-arrow 0 plus\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>t<\/mml:mi>\n                            <mml:mo stretchy=\"false\">\n                              \u2192\n                              \n                            <\/mml:mo>\n                            <mml:mn>0<\/mml:mn>\n                            <mml:mo>+<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">t\\to 0+<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    . The approximations\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper A Subscript n Superscript left-parenthesis j right-parenthesis\">\n                        <mml:semantics>\n                          <mml:msubsup>\n                            <mml:mi>A<\/mml:mi>\n                            <mml:mi>n<\/mml:mi>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mo stretchy=\"false\">(<\/mml:mo>\n                              <mml:mi>j<\/mml:mi>\n                              <mml:mo stretchy=\"false\">)<\/mml:mo>\n                            <\/mml:mrow>\n                          <\/mml:msubsup>\n                          <mml:annotation encoding=\"application\/x-tex\">A^{(j)}_n<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    to\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper A\">\n                        <mml:semantics>\n                          <mml:mi>A<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">A<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    obtained via GREP\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"Superscript left-parenthesis 1 right-parenthesis\">\n                        <mml:semantics>\n                          <mml:msup>\n                            <mml:mi\/>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mo stretchy=\"false\">(<\/mml:mo>\n                              <mml:mn>1<\/mml:mn>\n                              <mml:mo stretchy=\"false\">)<\/mml:mo>\n                            <\/mml:mrow>\n                          <\/mml:msup>\n                          <mml:annotation encoding=\"application\/x-tex\">^{(1)}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    are defined by the linear systems\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"a left-parenthesis t Subscript l Baseline right-parenthesis equals upper A Subscript n Superscript left-parenthesis j right-parenthesis Baseline plus phi left-parenthesis t Subscript l Baseline right-parenthesis sigma-summation Underscript i equals 0 Overscript n minus 1 Endscripts beta overbar Subscript i Baseline t Subscript l Superscript i\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>a<\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:msub>\n                              <mml:mi>t<\/mml:mi>\n                              <mml:mi>l<\/mml:mi>\n                            <\/mml:msub>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:msubsup>\n                              <mml:mi>A<\/mml:mi>\n                              <mml:mi>n<\/mml:mi>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mo stretchy=\"false\">(<\/mml:mo>\n                                <mml:mi>j<\/mml:mi>\n                                <mml:mo stretchy=\"false\">)<\/mml:mo>\n                              <\/mml:mrow>\n                            <\/mml:msubsup>\n                            <mml:mo>+<\/mml:mo>\n                            <mml:mi>\n                              \u03c6\n                              \n                            <\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:msub>\n                              <mml:mi>t<\/mml:mi>\n                              <mml:mi>l<\/mml:mi>\n                            <\/mml:msub>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                            <mml:munderover>\n                              <mml:mo>\n                                \u2211\n                                \n                              <\/mml:mo>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi>i<\/mml:mi>\n                                <mml:mo>=<\/mml:mo>\n                                <mml:mn>0<\/mml:mn>\n                              <\/mml:mrow>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi>n<\/mml:mi>\n                                <mml:mo>\n                                  \u2212\n                                  \n                                <\/mml:mo>\n                                <mml:mn>1<\/mml:mn>\n                              <\/mml:mrow>\n                            <\/mml:munderover>\n                            <mml:msub>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mover>\n                                  <mml:mi>\n                                    \u03b2\n                                    \n                                  <\/mml:mi>\n                                  <mml:mo stretchy=\"false\">\n                                    \u00af\n                                    \n                                  <\/mml:mo>\n                                <\/mml:mover>\n                              <\/mml:mrow>\n                              <mml:mi>i<\/mml:mi>\n                            <\/mml:msub>\n                            <mml:msubsup>\n                              <mml:mi>t<\/mml:mi>\n                              <mml:mi>l<\/mml:mi>\n                              <mml:mi>i<\/mml:mi>\n                            <\/mml:msubsup>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">a(t_l)=A^{(j)}_n+\\varphi (t_l) \\sum ^{n-1}_{i=0}\\bar {\\beta }_it_l^i<\/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=\"l equals j comma j plus 1 comma ellipsis comma j plus n\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>l<\/mml:mi>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:mi>j<\/mml:mi>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mi>j<\/mml:mi>\n                            <mml:mo>+<\/mml:mo>\n                            <mml:mn>1<\/mml:mn>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mo>\n                              \u2026\n                              \n                            <\/mml:mo>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mi>j<\/mml:mi>\n                            <mml:mo>+<\/mml:mo>\n                            <mml:mi>n<\/mml:mi>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">l=j,j+1,\\ldots ,j+n<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    , where\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"left-brace t Subscript l Baseline right-brace Subscript l equals 0 Superscript normal infinity\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mo fence=\"false\" stretchy=\"false\">{<\/mml:mo>\n                            <mml:msub>\n                              <mml:mi>t<\/mml:mi>\n                              <mml:mi>l<\/mml:mi>\n                            <\/mml:msub>\n                            <mml:msubsup>\n                              <mml:mo fence=\"false\" stretchy=\"false\">}<\/mml:mo>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi>l<\/mml:mi>\n                                <mml:mo>=<\/mml:mo>\n                                <mml:mn>0<\/mml:mn>\n                              <\/mml:mrow>\n                              <mml:mi mathvariant=\"normal\">\n                                \u221e\n                                \n                              <\/mml:mi>\n                            <\/mml:msubsup>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\{t_l\\}^\\infty _{l=0}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    is a decreasing positive sequence with limit zero. The study of GREP\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"Superscript left-parenthesis 1 right-parenthesis\">\n                        <mml:semantics>\n                          <mml:msup>\n                            <mml:mi\/>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mo stretchy=\"false\">(<\/mml:mo>\n                              <mml:mn>1<\/mml:mn>\n                              <mml:mo stretchy=\"false\">)<\/mml:mo>\n                            <\/mml:mrow>\n                          <\/mml:msup>\n                          <mml:annotation encoding=\"application\/x-tex\">^{(1)}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    for slowly varying functions\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"a left-parenthesis t right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>a<\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>t<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">a(t)<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    was begun in two recent papers by the author. For such\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"a left-parenthesis t right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>a<\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>t<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">a(t)<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    we have\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"phi left-parenthesis t right-parenthesis tilde alpha t Superscript delta\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>\n                              \u03c6\n                              \n                            <\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>t<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                            <mml:mo>\n                              \u223c\n                              \n                            <\/mml:mo>\n                            <mml:mi>\n                              \u03b1\n                              \n                            <\/mml:mi>\n                            <mml:msup>\n                              <mml:mi>t<\/mml:mi>\n                              <mml:mi>\n                                \u03b4\n                                \n                              <\/mml:mi>\n                            <\/mml:msup>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\varphi (t)\\sim \\alpha t^\\delta<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    as\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"t right-arrow 0 plus\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>t<\/mml:mi>\n                            <mml:mo stretchy=\"false\">\n                              \u2192\n                              \n                            <\/mml:mo>\n                            <mml:mn>0<\/mml:mn>\n                            <mml:mo>+<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">t\\to 0+<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    with\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"delta\">\n                        <mml:semantics>\n                          <mml:mi>\n                            \u03b4\n                            \n                          <\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">\\delta<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    possibly complex and\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"delta not-equals 0\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>\n                              \u03b4\n                              \n                            <\/mml:mi>\n                            <mml:mo>\n                              \u2260\n                              \n                            <\/mml:mo>\n                            <mml:mn>0<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\delta \\neq 0<\/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=\"negative 1\">\n                        <mml:semantics>\n                          <mml:mrow>\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\">-1<\/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=\"negative 2\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mo>\n                              \u2212\n                              \n                            <\/mml:mo>\n                            <mml:mn>2<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">-2<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    , \u2026. In the present work we continue to study the convergence and stability of GREP\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"Superscript left-parenthesis 1 right-parenthesis\">\n                        <mml:semantics>\n                          <mml:msup>\n                            <mml:mi\/>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mo stretchy=\"false\">(<\/mml:mo>\n                              <mml:mn>1<\/mml:mn>\n                              <mml:mo stretchy=\"false\">)<\/mml:mo>\n                            <\/mml:mrow>\n                          <\/mml:msup>\n                          <mml:annotation encoding=\"application\/x-tex\">^{(1)}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    as it is applied to such\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"a left-parenthesis t right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>a<\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>t<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">a(t)<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    with different sets of collocation points\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"t Subscript l\">\n                        <mml:semantics>\n                          <mml:msub>\n                            <mml:mi>t<\/mml:mi>\n                            <mml:mi>l<\/mml:mi>\n                          <\/mml:msub>\n                          <mml:annotation encoding=\"application\/x-tex\">t_l<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    that have been used in practical situations. In particular, we consider the cases in which (i)\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"t Subscript l\">\n                        <mml:semantics>\n                          <mml:msub>\n                            <mml:mi>t<\/mml:mi>\n                            <mml:mi>l<\/mml:mi>\n                          <\/mml:msub>\n                          <mml:annotation encoding=\"application\/x-tex\">t_l<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    are arbitrary, (ii)\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"limit Underscript l right-arrow normal infinity Endscripts t Subscript l plus 1 Baseline slash t Subscript l Baseline equals 1\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:munder>\n                              <mml:mo movablelimits=\"true\" form=\"prefix\">lim<\/mml:mo>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi>l<\/mml:mi>\n                                <mml:mo stretchy=\"false\">\n                                  \u2192\n                                  \n                                <\/mml:mo>\n                                <mml:mi mathvariant=\"normal\">\n                                  \u221e\n                                  \n                                <\/mml:mi>\n                              <\/mml:mrow>\n                            <\/mml:munder>\n                            <mml:msub>\n                              <mml:mi>t<\/mml:mi>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi>l<\/mml:mi>\n                                <mml:mo>+<\/mml:mo>\n                                <mml:mn>1<\/mml:mn>\n                              <\/mml:mrow>\n                            <\/mml:msub>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mo>\/<\/mml:mo>\n                            <\/mml:mrow>\n                            <mml:msub>\n                              <mml:mi>t<\/mml:mi>\n                              <mml:mi>l<\/mml:mi>\n                            <\/mml:msub>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:mn>1<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\lim _{l\\to \\infty }t_{l+1}\/t_l=1<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    , (iii)\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"t Subscript l Baseline tilde c l Superscript negative q\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:msub>\n                              <mml:mi>t<\/mml:mi>\n                              <mml:mi>l<\/mml:mi>\n                            <\/mml:msub>\n                            <mml:mo>\n                              \u223c\n                              \n                            <\/mml:mo>\n                            <mml:mi>c<\/mml:mi>\n                            <mml:msup>\n                              <mml:mi>l<\/mml:mi>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mo>\n                                  \u2212\n                                  \n                                <\/mml:mo>\n                                <mml:mi>q<\/mml:mi>\n                              <\/mml:mrow>\n                            <\/mml:msup>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">t_l\\sim cl^{-q}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    as\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"l right-arrow normal infinity\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>l<\/mml:mi>\n                            <mml:mo stretchy=\"false\">\n                              \u2192\n                              \n                            <\/mml:mo>\n                            <mml:mi mathvariant=\"normal\">\n                              \u221e\n                              \n                            <\/mml:mi>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">l\\to \\infty<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    for some\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"c comma q greater-than 0\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>c<\/mml:mi>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mi>q<\/mml:mi>\n                            <mml:mo>&gt;<\/mml:mo>\n                            <mml:mn>0<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">c, q&gt;0<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    , (iv)\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"t Subscript l plus 1 Baseline slash t Subscript l Baseline less-than-or-equal-to omega element-of left-parenthesis 0 comma 1 right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:msub>\n                              <mml:mi>t<\/mml:mi>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi>l<\/mml:mi>\n                                <mml:mo>+<\/mml:mo>\n                                <mml:mn>1<\/mml:mn>\n                              <\/mml:mrow>\n                            <\/mml:msub>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mo>\/<\/mml:mo>\n                            <\/mml:mrow>\n                            <mml:msub>\n                              <mml:mi>t<\/mml:mi>\n                              <mml:mi>l<\/mml:mi>\n                            <\/mml:msub>\n                            <mml:mo>\n                              \u2264\n                              \n                            <\/mml:mo>\n                            <mml:mi>\n                              \u03c9\n                              \n                            <\/mml:mi>\n                            <mml:mo>\n                              \u2208\n                              \n                            <\/mml:mo>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mn>0<\/mml:mn>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mn>1<\/mml:mn>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">t_{l+1}\/t_l\\leq \\omega \\in (0,1)<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    for all\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"l\">\n                        <mml:semantics>\n                          <mml:mi>l<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">l<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    , (v)\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"limit Underscript l right-arrow normal infinity Endscripts t Subscript l plus 1 Baseline slash t Subscript l Baseline equals omega element-of left-parenthesis 0 comma 1 right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:munder>\n                              <mml:mo movablelimits=\"true\" form=\"prefix\">lim<\/mml:mo>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi>l<\/mml:mi>\n                                <mml:mo stretchy=\"false\">\n                                  \u2192\n                                  \n                                <\/mml:mo>\n                                <mml:mi mathvariant=\"normal\">\n                                  \u221e\n                                  \n                                <\/mml:mi>\n                              <\/mml:mrow>\n                            <\/mml:munder>\n                            <mml:msub>\n                              <mml:mi>t<\/mml:mi>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi>l<\/mml:mi>\n                                <mml:mo>+<\/mml:mo>\n                                <mml:mn>1<\/mml:mn>\n                              <\/mml:mrow>\n                            <\/mml:msub>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mo>\/<\/mml:mo>\n                            <\/mml:mrow>\n                            <mml:msub>\n                              <mml:mi>t<\/mml:mi>\n                              <mml:mi>l<\/mml:mi>\n                            <\/mml:msub>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:mi>\n                              \u03c9\n                              \n                            <\/mml:mi>\n                            <mml:mo>\n                              \u2208\n                              \n                            <\/mml:mo>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mn>0<\/mml:mn>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mn>1<\/mml:mn>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\lim _{l\\to \\infty }t_{l+1}\/t_l= \\omega \\in (0,1)<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    , and (vi)\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"t Subscript l plus 1 Baseline slash t Subscript l Baseline equals omega element-of left-parenthesis 0 comma 1 right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:msub>\n                              <mml:mi>t<\/mml:mi>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi>l<\/mml:mi>\n                                <mml:mo>+<\/mml:mo>\n                                <mml:mn>1<\/mml:mn>\n                              <\/mml:mrow>\n                            <\/mml:msub>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mo>\/<\/mml:mo>\n                            <\/mml:mrow>\n                            <mml:msub>\n                              <mml:mi>t<\/mml:mi>\n                              <mml:mi>l<\/mml:mi>\n                            <\/mml:msub>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:mi>\n                              \u03c9\n                              \n                            <\/mml:mi>\n                            <mml:mo>\n                              \u2208\n                              \n                            <\/mml:mo>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mn>0<\/mml:mn>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mn>1<\/mml:mn>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">t_{l+1}\/t_l=\\omega \\in (0,1)<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    for all\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"l\">\n                        <mml:semantics>\n                          <mml:mi>l<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">l<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    .\n                  <\/p>","DOI":"10.1090\/s0025-5718-01-01384-9","type":"journal-article","created":{"date-parts":[[2002,9,20]],"date-time":"2002-09-20T14:12:48Z","timestamp":1032531168000},"page":"1569-1596","source":"Crossref","is-referenced-by-count":3,"title":["New convergence results on the generalized Richardson extrapolation process GREP\u207d\u00b9\u207e for logarithmic sequences"],"prefix":"10.1090","volume":"71","author":[{"given":"Avram","family":"Sidi","sequence":"first","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2001,11,28]]},"reference":[{"key":"1","isbn-type":"print","volume-title":"An introduction to numerical analysis","author":"Atkinson, Kendall E.","year":"1978","ISBN":"https:\/\/id.crossref.org\/isbn\/0471029858"},{"key":"2","first-page":"A727--A730","article-title":"Acc\u00e9l\u00e9ration de suites \u00e0 convergence logarithmique","volume":"273","author":"Brezinski, Claude","year":"1971","journal-title":"C. R. Acad. Sci. Paris S\\'{e}r. A-B","ISSN":"https:\/\/id.crossref.org\/issn\/0151-0509","issn-type":"print"},{"key":"3","doi-asserted-by":"publisher","first-page":"413","DOI":"10.1007\/BF01386092","article-title":"Fehlerabsch\u00e4tzungen und Extrapolation mit rationalen Funktionen bei Verfahren vom Richardson-Typus","volume":"6","author":"Bulirsch, Roland","year":"1964","journal-title":"Numer. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0029-599X","issn-type":"print"},{"issue":"5","key":"4","doi-asserted-by":"publisher","first-page":"1212","DOI":"10.1137\/0724080","article-title":"An algorithm for a generalization of the Richardson extrapolation process","volume":"24","author":"Ford, William F.","year":"1987","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"key":"5","doi-asserted-by":"publisher","first-page":"600","DOI":"10.2307\/1968944","article-title":"The lattice theory of ova","volume":"40","author":"Ward, Morgan","year":"1939","journal-title":"Ann. of Math. (2)","ISSN":"https:\/\/id.crossref.org\/issn\/0003-486X","issn-type":"print"},{"key":"6","first-page":"1435","article-title":"Un th\u00e9or\u00e8me de convergence pour le proc\u00e9d\u00e9 d\u2019extrapolation de Richardson","volume":"256","author":"Laurent, Pierre-Jean","year":"1963","journal-title":"C. R. Acad. Sci. Paris","ISSN":"https:\/\/id.crossref.org\/issn\/0001-4036","issn-type":"print"},{"issue":"3","key":"7","doi-asserted-by":"publisher","first-page":"277","DOI":"10.1007\/bf01933660","article-title":"Propagation of initial rounding error in Romberg-like quadrature","volume":"15","author":"Laurie, D. P.","year":"1975","journal-title":"Nordisk Tidskr. Informationsbehandling (BIT)","ISSN":"https:\/\/id.crossref.org\/issn\/0901-246X","issn-type":"print"},{"key":"8","doi-asserted-by":"publisher","first-page":"371","DOI":"10.1080\/00207167308803075","article-title":"Development of non-linear transformations of improving convergence of sequences","volume":"3","author":"Levin, David","year":"1973","journal-title":"Internat. J. Comput. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0020-7160","issn-type":"print"},{"issue":"3","key":"9","doi-asserted-by":"publisher","first-page":"175","DOI":"10.1016\/0096-3003(81)90028-X","article-title":"Two new classes of nonlinear transformations for accelerating the convergence of infinite integrals and series","volume":"9","author":"Levin, David","year":"1981","journal-title":"Appl. Math. Comput.","ISSN":"https:\/\/id.crossref.org\/issn\/0096-3003","issn-type":"print"},{"key":"10","unstructured":"[PS] R.E. Powell and S.M. Shah, Summability Theory and Its Applications, Van Nostrand Rheinhold, London, 1972."},{"issue":"3","key":"11","doi-asserted-by":"crossref","first-page":"327","DOI":"10.1093\/imamat\/24.3.327","article-title":"Some properties of a generalization of the Richardson extrapolation process","volume":"24","author":"Sidi, A.","year":"1979","journal-title":"J. Inst. Math. Appl.","ISSN":"https:\/\/id.crossref.org\/issn\/0020-2932","issn-type":"print"},{"issue":"145","key":"12","doi-asserted-by":"publisher","first-page":"315","DOI":"10.2307\/2006044","article-title":"Convergence properties of some nonlinear sequence transformations","volume":"33","author":"Sidi, Avram","year":"1979","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"151","key":"13","doi-asserted-by":"publisher","first-page":"833","DOI":"10.2307\/2006198","article-title":"Analysis of convergence of the \ud835\udc47-transformation for power series","volume":"35","author":"Sidi, Avram","year":"1980","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"3","key":"14","doi-asserted-by":"publisher","first-page":"299","DOI":"10.1007\/BF01396434","article-title":"An algorithm for a special case of generalization of the Richardson extrapolation process","volume":"38","author":"Sidi, Avram","year":"1981","journal-title":"Numer. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0029-599X","issn-type":"print"},{"issue":"4","key":"15","doi-asserted-by":"publisher","first-page":"365","DOI":"10.1007\/BF01386416","article-title":"On a generalization of the Richardson extrapolation process","volume":"57","author":"Sidi, Avram","year":"1990","journal-title":"Numer. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0029-599X","issn-type":"print"},{"issue":"212","key":"16","doi-asserted-by":"publisher","first-page":"1627","DOI":"10.2307\/2153375","article-title":"Convergence analysis for a generalized Richardson extrapolation process with an application to the \ud835\udc51\u207d\u00b9\u207e-transformation on convergent and divergent logarithmic sequences","volume":"64","author":"Sidi, Avram","year":"1995","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"1-2","key":"17","doi-asserted-by":"publisher","first-page":"269","DOI":"10.1016\/S0377-0427(99)90226-1","article-title":"Further convergence and stability results for the generalized Richardson extrapolation process \ud835\udc3a\ud835\udc45\ud835\udc38\ud835\udc43\u207d\u00b9\u207e with an application to the \ud835\udc37\u207d\u00b9\u207e-transformation for infinite integrals","volume":"112","author":"Sidi, Avram","year":"1999","journal-title":"J. Comput. Appl. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0377-0427","issn-type":"print"},{"issue":"5","key":"18","doi-asserted-by":"publisher","first-page":"1729","DOI":"10.1137\/S0036142998340137","article-title":"The Richardson extrapolation process with a harmonic sequence of collocation points","volume":"37","author":"Sidi, Avram","year":"2000","journal-title":"SIAM J. Numer. 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