{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T14:42:29Z","timestamp":1776782549573,"version":"3.51.2"},"reference-count":27,"publisher":"American Mathematical Society (AMS)","issue":"246","license":[{"start":{"date-parts":[[2004,11,24]],"date-time":"2004-11-24T00:00:00Z","timestamp":1101254400000},"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                    The so-called parallel multisplitting nonstationary iterative Model\u00a0A was introduced by Bru, Elsner, and Neumann [\n                    <italic>Linear Algebra and its Applications<\/italic>\n                    103:175-192 (1988)] for solving a nonsingular linear system\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper A bold-italic x equals bold-italic b\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>A<\/mml:mi>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mi mathvariant=\"bold-italic\">x<\/mml:mi>\n                            <\/mml:mrow>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mi mathvariant=\"bold-italic\">b<\/mml:mi>\n                            <\/mml:mrow>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">A\\mathbfit {x} = \\mathbfit {b}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    using a weak nonnegative multisplitting of the first type. In this paper new results are introduced when\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                    is a monotone matrix using a weak nonnegative multisplitting of the second type and when\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                    is a symmetric positive definite matrix using a\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper P\">\n                        <mml:semantics>\n                          <mml:mi>P<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">P<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    -regular multisplitting. Also, nonstationary alternating iterative methods are studied. Finally, combining Model\u00a0A and alternating iterative methods, two new models of parallel multisplitting nonstationary iterations are introduced. When matrix\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                    is monotone and the multisplittings are weak nonnegative of the first or of the second type, both models lead to convergent schemes. Also, when matrix\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                    is symmetric positive definite and the multisplittings are\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper P\">\n                        <mml:semantics>\n                          <mml:mi>P<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">P<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    -regular, the schemes are also convergent.\n                  <\/p>","DOI":"10.1090\/s0025-5718-03-01607-7","type":"journal-article","created":{"date-parts":[[2004,4,21]],"date-time":"2004-04-21T12:29:08Z","timestamp":1082550548000},"page":"691-717","source":"Crossref","is-referenced-by-count":1,"title":["Sequential and parallel synchronous alternating iterative methods"],"prefix":"10.1090","volume":"73","author":[{"given":"Joan-Josep","family":"Climent","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Carmen","family":"Perea","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Leandro","family":"Tortosa","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Antonio","family":"Zamora","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"14","published-online":{"date-parts":[[2003,11,24]]},"reference":[{"key":"1","series-title":"Classics in Applied Mathematics","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1137\/1.9781611971262","volume-title":"Nonnegative matrices in the mathematical sciences","volume":"9","author":"Berman, Abraham","year":"1994","ISBN":"https:\/\/id.crossref.org\/isbn\/0898713218"},{"issue":"3","key":"2","doi-asserted-by":"publisher","first-page":"309","DOI":"10.1007\/s002110050265","article-title":"Existence and uniqueness of splittings for stationary iterative methods with applications to alternating methods","volume":"76","author":"Benzi, Michele","year":"1997","journal-title":"Numer. 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