{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,5,14]],"date-time":"2025-05-14T02:48:47Z","timestamp":1747190927682,"version":"3.40.5"},"reference-count":41,"publisher":"Wiley","license":[{"start":{"date-parts":[[2022,3,2]],"date-time":"2022-03-02T00:00:00Z","timestamp":1646179200000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Computational and Mathematical Methods"],"published-print":{"date-parts":[[2022,3,2]]},"abstract":"<jats:p>In this paper, we derive upper bounds that characterize the rate of convergence of the SOR method for solving a linear system of the form <jats:inline-formula>\n                     <a:math xmlns:a=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M1\">\n                        <a:mi>G<\/a:mi>\n                        <a:mi>x<\/a:mi>\n                        <a:mo>=<\/a:mo>\n                        <a:mi>b<\/a:mi>\n                     <\/a:math>\n                  <\/jats:inline-formula>, where <jats:inline-formula>\n                     <c:math xmlns:c=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M2\">\n                        <c:mi>G<\/c:mi>\n                     <\/c:math>\n                  <\/jats:inline-formula> is a real symmetric positive semidefinite <jats:inline-formula>\n                     <e:math xmlns:e=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M3\">\n                        <e:mi>n<\/e:mi>\n                        <e:mo>\u00d7<\/e:mo>\n                        <e:mi>n<\/e:mi>\n                     <\/e:math>\n                  <\/jats:inline-formula> matrix. The bounds are given in terms of the condition number of <jats:inline-formula>\n                     <g:math xmlns:g=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M4\">\n                        <g:mi>G<\/g:mi>\n                     <\/g:math>\n                  <\/jats:inline-formula>, which is the ratio <jats:inline-formula>\n                     <i:math xmlns:i=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M5\">\n                        <i:mi>\u03ba<\/i:mi>\n                        <i:mo>=<\/i:mo>\n                        <i:mi>\u03b1<\/i:mi>\n                        <i:mo>\/<\/i:mo>\n                        <i:mi>\u03b2<\/i:mi>\n                     <\/i:math>\n                  <\/jats:inline-formula>, where <jats:inline-formula>\n                     <k:math xmlns:k=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M6\">\n                        <k:mi>\u03b1<\/k:mi>\n                     <\/k:math>\n                  <\/jats:inline-formula> is the largest eigenvalue of <jats:inline-formula>\n                     <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M7\">\n                        <m:mi>G<\/m:mi>\n                     <\/m:math>\n                  <\/jats:inline-formula> and <jats:inline-formula>\n                     <o:math xmlns:o=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M8\">\n                        <o:mi>\u03b2<\/o:mi>\n                     <\/o:math>\n                  <\/jats:inline-formula> is the smallest nonzero eigenvalue of <jats:inline-formula>\n                     <q:math xmlns:q=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M9\">\n                        <q:mi>G<\/q:mi>\n                     <\/q:math>\n                  <\/jats:inline-formula>. Let <jats:inline-formula>\n                     <s:math xmlns:s=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M10\">\n                        <s:mi>H<\/s:mi>\n                     <\/s:math>\n                  <\/jats:inline-formula> denote the related iteration matrix. Then, since <jats:inline-formula>\n                     <u:math xmlns:u=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M11\">\n                        <u:mi>G<\/u:mi>\n                     <\/u:math>\n                  <\/jats:inline-formula> has a zero eigenvalue, the spectral radius of <jats:inline-formula>\n                     <w:math xmlns:w=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M12\">\n                        <w:mi>H<\/w:mi>\n                     <\/w:math>\n                  <\/jats:inline-formula> equals 1, and the rate of convergence is determined by the size of <jats:inline-formula>\n                     <y:math xmlns:y=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M13\">\n                        <y:mi>\u03b7<\/y:mi>\n                     <\/y:math>\n                  <\/jats:inline-formula>, the largest eigenvalue of <jats:inline-formula>\n                     <ab:math xmlns:ab=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M14\">\n                        <ab:mi>H<\/ab:mi>\n                     <\/ab:math>\n                  <\/jats:inline-formula> whose modulus differs from 1. The bound has the form <jats:inline-formula>\n                     <cb:math xmlns:cb=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M15\">\n                        <cb:msup>\n                           <cb:mrow>\n                              <cb:mfenced open=\"|\" close=\"|\">\n                                 <cb:mrow>\n                                    <cb:mi>\u03b7<\/cb:mi>\n                                 <\/cb:mrow>\n                              <\/cb:mfenced>\n                           <\/cb:mrow>\n                           <cb:mrow>\n                              <cb:mn>2<\/cb:mn>\n                           <\/cb:mrow>\n                        <\/cb:msup>\n                        <cb:mo>\u2264<\/cb:mo>\n                        <cb:mn>1<\/cb:mn>\n                        <cb:mo>\u2212<\/cb:mo>\n                        <cb:mn>1<\/cb:mn>\n                        <cb:mo>\/<\/cb:mo>\n                        <cb:mfenced open=\"(\" close=\")\">\n                           <cb:mrow>\n                              <cb:mi>\u03ba<\/cb:mi>\n                              <cb:mi>c<\/cb:mi>\n                           <\/cb:mrow>\n                        <\/cb:mfenced>\n                     <\/cb:math>\n                  <\/jats:inline-formula>, where <jats:inline-formula>\n                     <ib:math xmlns:ib=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M16\">\n                        <ib:mi>c<\/ib:mi>\n                        <ib:mo>=<\/ib:mo>\n                        <ib:mn>2<\/ib:mn>\n                        <ib:mo>+<\/ib:mo>\n                        <ib:msub>\n                           <ib:mrow>\n                              <ib:mi mathvariant=\"normal\">log<\/ib:mi>\n                           <\/ib:mrow>\n                           <ib:mrow>\n                              <ib:mn>2<\/ib:mn>\n                           <\/ib:mrow>\n                        <\/ib:msub>\n                        <ib:mi>n<\/ib:mi>\n                        <ib:mo>.<\/ib:mo>\n                     <\/ib:math>\n                  <\/jats:inline-formula> The main consequence from this bound is that small condition number forces fast convergence while large condition number allows slow convergence.<\/jats:p>","DOI":"10.1155\/2022\/6143444","type":"journal-article","created":{"date-parts":[[2022,3,2]],"date-time":"2022-03-02T22:39:28Z","timestamp":1646260768000},"page":"1-8","source":"Crossref","is-referenced-by-count":2,"title":["The Rate of Convergence of the SOR Method in the Positive Semidefinite Case"],"prefix":"10.1155","volume":"2022","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-6403-1020","authenticated-orcid":true,"given":"Achiya","family":"Dax","sequence":"first","affiliation":[{"name":"Hydrological Service, P.O.B. 36118, Jerusalem 91360, Israel"}]}],"member":"311","reference":[{"key":"1","doi-asserted-by":"crossref","DOI":"10.1017\/CBO9780511624100","volume-title":"Iterative Solution Methods","author":"O. 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