{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,5,14]],"date-time":"2025-05-14T04:47:34Z","timestamp":1747198054834,"version":"3.40.5"},"reference-count":23,"publisher":"Walter de Gruyter GmbH","issue":"3","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2022,7,1]]},"abstract":"<jats:title>Abstract<\/jats:title>\n               <jats:p>We consider a class of iterative methods based on block splitting (BBS) to solve absolute value equations <jats:inline-formula>\n                     <jats:alternatives>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                           <m:mrow>\n                              <m:mrow>\n                                 <m:mrow>\n                                    <m:mi>A<\/m:mi>\n                                    <m:mo>\u2062<\/m:mo>\n                                    <m:mi>x<\/m:mi>\n                                 <\/m:mrow>\n                                 <m:mo>-<\/m:mo>\n                                 <m:mrow>\n                                    <m:mo fence=\"true\" stretchy=\"false\">|<\/m:mo>\n                                    <m:mi>x<\/m:mi>\n                                    <m:mo fence=\"true\" stretchy=\"false\">|<\/m:mo>\n                                 <\/m:mrow>\n                              <\/m:mrow>\n                              <m:mo>=<\/m:mo>\n                              <m:mi>b<\/m:mi>\n                           <\/m:mrow>\n                        <\/m:math>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/j_cmam-2022-0020_ineq_0001.png\"\/>\n                        <jats:tex-math>Ax-\\lvert x\\rvert=b<\/jats:tex-math>\n                     <\/jats:alternatives>\n                  <\/jats:inline-formula>.\nRecently, several works were devoted to deriving sufficient conditions for the convergence of iterative methods of this type under certain assumptions including <jats:inline-formula>\n                     <jats:alternatives>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                           <m:mrow>\n                              <m:mi>\u03bd<\/m:mi>\n                              <m:mo>:=<\/m:mo>\n                              <m:mrow>\n                                 <m:mo fence=\"true\" stretchy=\"false\">\u2225<\/m:mo>\n                                 <m:msup>\n                                    <m:mi>A<\/m:mi>\n                                    <m:mrow>\n                                       <m:mo>-<\/m:mo>\n                                       <m:mn>1<\/m:mn>\n                                    <\/m:mrow>\n                                 <\/m:msup>\n                                 <m:mo fence=\"true\" stretchy=\"false\">\u2225<\/m:mo>\n                              <\/m:mrow>\n                              <m:mo>&lt;<\/m:mo>\n                              <m:mn>1<\/m:mn>\n                           <\/m:mrow>\n                        <\/m:math>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/j_cmam-2022-0020_ineq_0002.png\"\/>\n                        <jats:tex-math>\\nu:=\\lVert A^{-1}\\rVert&lt;1<\/jats:tex-math>\n                     <\/jats:alternatives>\n                  <\/jats:inline-formula>.\nHowever, the BBS-type iterative methods tend to converge slowly when \ud835\udf08 is very close to one (i.e., <jats:inline-formula>\n                     <jats:alternatives>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                           <m:mrow>\n                              <m:mi>\u03bd<\/m:mi>\n                              <m:mo>\u2248<\/m:mo>\n                              <m:mn>1<\/m:mn>\n                           <\/m:mrow>\n                        <\/m:math>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/j_cmam-2022-0020_ineq_0003.png\"\/>\n                        <jats:tex-math>\\nu\\approx 1<\/jats:tex-math>\n                     <\/jats:alternatives>\n                  <\/jats:inline-formula>).\nIn this paper, using an auxiliary matrix, we develop a new approach by first rewriting the main problem into a new equivalent block system having shifted <jats:inline-formula>\n                     <jats:alternatives>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                           <m:mrow>\n                              <m:mo stretchy=\"false\">(<\/m:mo>\n                              <m:mn>1<\/m:mn>\n                              <m:mo>,<\/m:mo>\n                              <m:mn>1<\/m:mn>\n                              <m:mo stretchy=\"false\">)<\/m:mo>\n                           <\/m:mrow>\n                        <\/m:math>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/j_cmam-2022-0020_ineq_0004.png\"\/>\n                        <jats:tex-math>(1,1)<\/jats:tex-math>\n                     <\/jats:alternatives>\n                  <\/jats:inline-formula>-block and then constructing a fixed point iteration.\nThe exploited strategy can significantly improve the convergence speed of the BBS-type iterative methods when <jats:inline-formula>\n                     <jats:alternatives>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                           <m:mrow>\n                              <m:mi>\u03bd<\/m:mi>\n                              <m:mo>\u2248<\/m:mo>\n                              <m:mn>1<\/m:mn>\n                           <\/m:mrow>\n                        <\/m:math>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/j_cmam-2022-0020_ineq_0003.png\"\/>\n                        <jats:tex-math>\\nu\\approx 1<\/jats:tex-math>\n                     <\/jats:alternatives>\n                  <\/jats:inline-formula>.\nNumerical experiments are reported to demonstrate the superiority of the new modified iterative scheme over the existing original form of BBS-type methods in the literature.<\/jats:p>","DOI":"10.1515\/cmam-2022-0020","type":"journal-article","created":{"date-parts":[[2022,5,25]],"date-time":"2022-05-25T20:28:35Z","timestamp":1653510515000},"page":"663-673","source":"Crossref","is-referenced-by-count":1,"title":["An Improvement on a Class of Fixed Point Iterative Methods for Solving Absolute Value Equations"],"prefix":"10.1515","volume":"22","author":[{"given":"Nafiseh Nasseri","family":"Shams","sequence":"first","affiliation":[{"name":"Department of Mathematics , Shiraz University of Technology , Shiraz , Iran"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9050-3506","authenticated-orcid":false,"given":"Fatemeh Panjeh Ali","family":"Beik","sequence":"additional","affiliation":[{"name":"Department of Mathematics , Vali-e-Asr University of Rafsanjan , PO Box 518 , Rafsanjan , Iran"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"374","published-online":{"date-parts":[[2022,5,26]]},"reference":[{"key":"2023033112440915969_j_cmam-2022-0020_ref_001","doi-asserted-by":"crossref","unstructured":"L. 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