{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,29]],"date-time":"2026-04-29T18:20:55Z","timestamp":1777486855636,"version":"3.51.4"},"reference-count":9,"publisher":"American Mathematical Society (AMS)","issue":"246","license":[{"start":{"date-parts":[[2004,10,17]],"date-time":"2004-10-17T00:00:00Z","timestamp":1097971200000},"content-version":"am","delay-in-days":366,"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=\"upper B element-of upper M Subscript n Baseline left-parenthesis bold upper C right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>B<\/mml:mi>\n                            <mml:mo>\n                              \u2208\n                              \n                            <\/mml:mo>\n                            <mml:msub>\n                              <mml:mi>M<\/mml:mi>\n                              <mml:mi>n<\/mml:mi>\n                            <\/mml:msub>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi mathvariant=\"bold\">C<\/mml:mi>\n                              <\/mml:mrow>\n                            <\/mml:mrow>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">B\\in M_n({\\mathbf {C}})<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    be a row diagonally dominant matrix, i.e.,\n                    <disp-formula content-type=\"math\/mathml\">\n                      \\[\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"sigma Subscript i Baseline StartAbsoluteValue b Subscript i i Baseline EndAbsoluteValue equals sigma-summation Underscript StartLayout 1st Row  j equals 1 j not-equals i EndLayout Overscript n Endscripts StartAbsoluteValue b Subscript i j Baseline EndAbsoluteValue comma i equals 1 comma ellipsis comma n comma\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:msub>\n                              <mml:mi>\n                                \u03c3\n                                \n                              <\/mml:mi>\n                              <mml:mi>i<\/mml:mi>\n                            <\/mml:msub>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mo stretchy=\"false\">|<\/mml:mo>\n                            <\/mml:mrow>\n                            <mml:msub>\n                              <mml:mi>b<\/mml:mi>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi>i<\/mml:mi>\n                                <mml:mi>i<\/mml:mi>\n                              <\/mml:mrow>\n                            <\/mml:msub>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mo stretchy=\"false\">|<\/mml:mo>\n                            <\/mml:mrow>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:munderover>\n                              <mml:mo>\n                                \u2211\n                                \n                              <\/mml:mo>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mstyle scriptlevel=\"1\">\n                                  <mml:mtable rowspacing=\"0.1em\" columnspacing=\"0em\" displaystyle=\"false\">\n                                    <mml:mtr>\n                                      <mml:mtd>\n                                        <mml:mi>j<\/mml:mi>\n                                        <mml:mo>=<\/mml:mo>\n                                        <mml:mn>1<\/mml:mn>\n                                        <mml:mi>j<\/mml:mi>\n                                        <mml:mo>\n                                          \u2260\n                                          \n                                        <\/mml:mo>\n                                        <mml:mi>i<\/mml:mi>\n                                      <\/mml:mtd>\n                                    <\/mml:mtr>\n                                  <\/mml:mtable>\n                                <\/mml:mstyle>\n                              <\/mml:mrow>\n                              <mml:mi>n<\/mml:mi>\n                            <\/mml:munderover>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mo stretchy=\"false\">|<\/mml:mo>\n                            <\/mml:mrow>\n                            <mml:msub>\n                              <mml:mi>b<\/mml:mi>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi>i<\/mml:mi>\n                                <mml:mi>j<\/mml:mi>\n                              <\/mml:mrow>\n                            <\/mml:msub>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mo stretchy=\"false\">|<\/mml:mo>\n                            <\/mml:mrow>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mspace width=\"1em\"\/>\n                            <mml:mi>i<\/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>n<\/mml:mi>\n                            <mml:mo>,<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\sigma _i |b_{ii}| = \\sum _{\\substack {j=1 j\\ne i }}^n |b_{ij}|, \\quad i = 1,\\ldots ,n,<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                      \\]\n                    <\/disp-formula>\n                    where\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"0 less-than-or-equal-to sigma Subscript i Baseline greater-than 1 comma i equals 1 comma ellipsis comma n comma\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mn>0<\/mml:mn>\n                            <mml:mo>\n                              \u2264\n                              \n                            <\/mml:mo>\n                            <mml:msub>\n                              <mml:mi>\n                                \u03c3\n                                \n                              <\/mml:mi>\n                              <mml:mi>i<\/mml:mi>\n                            <\/mml:msub>\n                            <mml:mo>&gt;<\/mml:mo>\n                            <mml:mn>1<\/mml:mn>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mtext>\u00a0<\/mml:mtext>\n                            <mml:mi>i<\/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>n<\/mml:mi>\n                            <mml:mo>,<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">0 \\le \\sigma _i &gt; 1,\\ i= 1,\\ldots ,n,<\/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=\"sigma equals max Underscript 1 less-than-or-equal-to i less-than-or-equal-to n Endscripts sigma Subscript i Baseline period\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>\n                              \u03c3\n                              \n                            <\/mml:mi>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:munder>\n                              <mml:mo movablelimits=\"true\" form=\"prefix\">max<\/mml:mo>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mn>1<\/mml:mn>\n                                <mml:mo>\n                                  \u2264\n                                  \n                                <\/mml:mo>\n                                <mml:mi>i<\/mml:mi>\n                                <mml:mo>\n                                  \u2264\n                                  \n                                <\/mml:mo>\n                                <mml:mi>n<\/mml:mi>\n                              <\/mml:mrow>\n                            <\/mml:munder>\n                            <mml:msub>\n                              <mml:mi>\n                                \u03c3\n                                \n                              <\/mml:mi>\n                              <mml:mi>i<\/mml:mi>\n                            <\/mml:msub>\n                            <mml:mo>.<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\sigma = \\max _{1\\le i \\le n} \\sigma _i.<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    We show that no pivoting is necessary when Gaussian elimination is applied to\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper A equals upper B Superscript negative 1 Baseline period\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>A<\/mml:mi>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:msup>\n                              <mml:mi>B<\/mml:mi>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mo>\n                                  \u2212\n                                  \n                                <\/mml:mo>\n                                <mml:mn>1<\/mml:mn>\n                              <\/mml:mrow>\n                            <\/mml:msup>\n                            <mml:mo>.<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">A = B^{-1}.<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    Moreover, the growth factor for\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                    does not exceed\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"1 plus sigma period\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mn>1<\/mml:mn>\n                            <mml:mo>+<\/mml:mo>\n                            <mml:mi>\n                              \u03c3\n                              \n                            <\/mml:mi>\n                            <mml:mo>.<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">1 + \\sigma .<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    The same results are true with row diagonal dominance being replaced by column diagonal dominance.\n                  <\/p>","DOI":"10.1090\/s0025-5718-03-01591-6","type":"journal-article","created":{"date-parts":[[2004,4,21]],"date-time":"2004-04-21T12:29:08Z","timestamp":1082550548000},"page":"653-657","source":"Crossref","is-referenced-by-count":9,"title":["Gaussian elimination is stable for the inverse of a diagonally dominant matrix"],"prefix":"10.1090","volume":"73","author":[{"given":"Alan","family":"George","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Khakim","family":"Ikramov","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"14","published-online":{"date-parts":[[2003,10,17]]},"reference":[{"issue":"224","key":"1","doi-asserted-by":"publisher","first-page":"1591","DOI":"10.1090\/S0025-5718-98-00978-8","article-title":"Factorizing complex symmetric matrices with positive definite real and imaginary parts","volume":"67","author":"Higham, Nicholas 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Paris","ISSN":"https:\/\/id.crossref.org\/issn\/0001-4036","issn-type":"print"},{"key":"4","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511810817","volume-title":"Matrix analysis","author":"Horn, Roger A.","year":"1985","ISBN":"https:\/\/id.crossref.org\/isbn\/0521305861"},{"issue":"1","key":"5","doi-asserted-by":"publisher","first-page":"102","DOI":"10.1007\/BF01934520","article-title":"Backward error analysis for linear systems associated with inverses of \ud835\udc3b-matrices","volume":"24","author":"Neumann, M.","year":"1984","journal-title":"BIT","ISSN":"https:\/\/id.crossref.org\/issn\/0006-3835","issn-type":"print"},{"issue":"1","key":"6","doi-asserted-by":"publisher","first-page":"75","DOI":"10.1016\/0024-3795(77)90081-7","article-title":"The inverse \ud835\udc40-matrix problem","volume":"18","author":"Willoughby, R. A.","year":"1977","journal-title":"Linear Algebra Appl.","ISSN":"https:\/\/id.crossref.org\/issn\/0024-3795","issn-type":"print"},{"key":"7","doi-asserted-by":"publisher","first-page":"69","DOI":"10.2307\/1267494","article-title":"A time series approach to numerical differentiation","volume":"16","author":"Anderssen, R. S.","year":"1974","journal-title":"Technometrics","ISSN":"https:\/\/id.crossref.org\/issn\/0040-1706","issn-type":"print"},{"key":"8","doi-asserted-by":"publisher","first-page":"569","DOI":"10.1063\/1.1665622","article-title":"Correlation inequalities for coupled oscillators","volume":"12","author":"Leff, Harvey S.","year":"1971","journal-title":"J. Mathematical Phys.","ISSN":"https:\/\/id.crossref.org\/issn\/0022-2488","issn-type":"print"},{"key":"9","isbn-type":"print","volume-title":"Accuracy and stability of numerical algorithms","author":"Higham, Nicholas J.","year":"1996","ISBN":"https:\/\/id.crossref.org\/isbn\/0898713552"}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.ams.org\/mcom\/2004-73-246\/S0025-5718-03-01591-6\/S0025-5718-03-01591-6.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/2004-73-246\/S0025-5718-03-01591-6\/S0025-5718-03-01591-6.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T13:43:47Z","timestamp":1776779027000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2004-73-246\/S0025-5718-03-01591-6\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2003,10,17]]},"references-count":9,"journal-issue":{"issue":"246","published-print":{"date-parts":[[2004,4]]}},"alternative-id":["S0025-5718-03-01591-6"],"URL":"https:\/\/doi.org\/10.1090\/s0025-5718-03-01591-6","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":[[2003,10,17]]}}}