{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T15:03:02Z","timestamp":1776783782775,"version":"3.51.2"},"reference-count":15,"publisher":"American Mathematical Society (AMS)","issue":"249","license":[{"start":{"date-parts":[[2005,1,27]],"date-time":"2005-01-27T00:00:00Z","timestamp":1106784000000},"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 two matrix iterations\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper X Subscript k plus 1 Baseline equals upper I minus-or-plus upper A Superscript asterisk Baseline upper X Subscript k Superscript negative 1 Baseline upper A\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:msub>\n                              <mml:mi>X<\/mml:mi>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi>k<\/mml:mi>\n                                <mml:mo>+<\/mml:mo>\n                                <mml:mn>1<\/mml:mn>\n                              <\/mml:mrow>\n                            <\/mml:msub>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:mi>I<\/mml:mi>\n                            <mml:mo>\n                              \u2213\n                              \n                            <\/mml:mo>\n                            <mml:msup>\n                              <mml:mi>A<\/mml:mi>\n                              <mml:mo>\n                                \u2217\n                                \n                              <\/mml:mo>\n                            <\/mml:msup>\n                            <mml:msubsup>\n                              <mml:mi>X<\/mml:mi>\n                              <mml:mi>k<\/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:msubsup>\n                            <mml:mi>A<\/mml:mi>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">X_{k+1}=I\\mp A^*X_k^{-1}A<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    are known to converge linearly to a positive definite solution of the matrix equations\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper X plus-or-minus upper A Superscript asterisk Baseline upper X Superscript negative 1 Baseline upper A equals upper I\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>X<\/mml:mi>\n                            <mml:mo>\n                              \u00b1\n                              \n                            <\/mml:mo>\n                            <mml:msup>\n                              <mml:mi>A<\/mml:mi>\n                              <mml:mo>\n                                \u2217\n                                \n                              <\/mml:mo>\n                            <\/mml:msup>\n                            <mml:msup>\n                              <mml:mi>X<\/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:mi>A<\/mml:mi>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:mi>I<\/mml:mi>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">X\\pm A^*X^{-1}A=I<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    , respectively, for known choices of\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper X 0\">\n                        <mml:semantics>\n                          <mml:msub>\n                            <mml:mi>X<\/mml:mi>\n                            <mml:mn>0<\/mml:mn>\n                          <\/mml:msub>\n                          <mml:annotation encoding=\"application\/x-tex\">X_0<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    and under certain restrictions on\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 convergence for previously suggested starting matrices\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper X 0\">\n                        <mml:semantics>\n                          <mml:msub>\n                            <mml:mi>X<\/mml:mi>\n                            <mml:mn>0<\/mml:mn>\n                          <\/mml:msub>\n                          <mml:annotation encoding=\"application\/x-tex\">X_0<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    is generally very slow. This paper explores different initial choices of\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper X 0\">\n                        <mml:semantics>\n                          <mml:msub>\n                            <mml:mi>X<\/mml:mi>\n                            <mml:mn>0<\/mml:mn>\n                          <\/mml:msub>\n                          <mml:annotation encoding=\"application\/x-tex\">X_0<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    in both iterations that depend on the extreme singular values of\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 lead to much more rapid convergence. Further, the paper offers a new algorithm for solving the minus sign equation and explores mixed algorithms that use Newton\u2019s method in part.\n                  <\/p>","DOI":"10.1090\/s0025-5718-04-01636-9","type":"journal-article","created":{"date-parts":[[2004,9,20]],"date-time":"2004-09-20T09:57:53Z","timestamp":1095674273000},"page":"263-278","source":"Crossref","is-referenced-by-count":65,"title":["Improved methods and starting values to solve the matrix equations \ud835\udc4b\u00b1\ud835\udc34*\ud835\udc4b\u207b\u00b9\ud835\udc34=\ud835\udc3c iteratively"],"prefix":"10.1090","volume":"74","author":[{"given":"Ivan","family":"Ivanov","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Vejdi","family":"Hasanov","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Frank","family":"Uhlig","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"14","published-online":{"date-parts":[[2004,1,27]]},"reference":[{"key":"1","doi-asserted-by":"publisher","first-page":"53","DOI":"10.1016\/0024-3795(90)90005-W","article-title":"Positive solutions to \ud835\udc4b=\ud835\udc34-\ud835\udc35\ud835\udc4b\u207b\u00b9\ud835\udc35*","volume":"134","author":"Anderson, W. N., Jr.","year":"1990","journal-title":"Linear Algebra Appl.","ISSN":"https:\/\/id.crossref.org\/issn\/0024-3795","issn-type":"print"},{"key":"2","doi-asserted-by":"publisher","first-page":"627","DOI":"10.1137\/0707049","article-title":"On direct methods for solving Poisson\u2019s equations","volume":"7","author":"Buzbee, B. L.","year":"1970","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"key":"3","unstructured":"S. M. El-Sayed, Theorems for the Existence and Computing of Positive Definite Solutions for Two Nonlinear Matrix Equation, Proc. of 25^{\ud835\udc61\u210e} Spring Conference of the Union of Bulgarian Mathematicians, Kazanlak, 1996, pp.155-161, (in Bulgarian)."},{"issue":"3","key":"4","doi-asserted-by":"publisher","first-page":"632","DOI":"10.1137\/S0895479899345571","article-title":"On an iteration method for solving a class of nonlinear matrix equations","volume":"23","author":"El-Sayed, Salah M.","year":"2001","journal-title":"SIAM J. Matrix Anal. 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