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Our analysis of the Schwarz preconditioner for elliptic eigenvalue problems demonstrates that RAP achieves a convergence rate of <jats:inline-formula>\n              <jats:alternatives>\n                <jats:tex-math>$$1-C\\kappa ^{-1\/2}$$<\/jats:tex-math>\n                <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mn>1<\/mml:mn>\n                    <mml:mo>-<\/mml:mo>\n                    <mml:mi>C<\/mml:mi>\n                    <mml:msup>\n                      <mml:mi>\u03ba<\/mml:mi>\n                      <mml:mrow>\n                        <mml:mo>-<\/mml:mo>\n                        <mml:mn>1<\/mml:mn>\n                        <mml:mo>\/<\/mml:mo>\n                        <mml:mn>2<\/mml:mn>\n                      <\/mml:mrow>\n                    <\/mml:msup>\n                  <\/mml:mrow>\n                <\/mml:math>\n              <\/jats:alternatives>\n            <\/jats:inline-formula>, which is an improvement over the preconditioned steepest descent method\u2019s rate of <jats:inline-formula>\n              <jats:alternatives>\n                <jats:tex-math>$$1-C\\kappa ^{-1}$$<\/jats:tex-math>\n                <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mn>1<\/mml:mn>\n                    <mml:mo>-<\/mml:mo>\n                    <mml:mi>C<\/mml:mi>\n                    <mml:msup>\n                      <mml:mi>\u03ba<\/mml:mi>\n                      <mml:mrow>\n                        <mml:mo>-<\/mml:mo>\n                        <mml:mn>1<\/mml:mn>\n                      <\/mml:mrow>\n                    <\/mml:msup>\n                  <\/mml:mrow>\n                <\/mml:math>\n              <\/jats:alternatives>\n            <\/jats:inline-formula>. The exponent in <jats:inline-formula>\n              <jats:alternatives>\n                <jats:tex-math>$$\\kappa ^{-1\/2}$$<\/jats:tex-math>\n                <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:msup>\n                    <mml:mi>\u03ba<\/mml:mi>\n                    <mml:mrow>\n                      <mml:mo>-<\/mml:mo>\n                      <mml:mn>1<\/mml:mn>\n                      <mml:mo>\/<\/mml:mo>\n                      <mml:mn>2<\/mml:mn>\n                    <\/mml:mrow>\n                  <\/mml:msup>\n                <\/mml:math>\n              <\/jats:alternatives>\n            <\/jats:inline-formula> is sharp, and numerical experiments confirm our theoretical findings.<\/jats:p>","DOI":"10.1007\/s00211-025-01451-0","type":"journal-article","created":{"date-parts":[[2025,1,20]],"date-time":"2025-01-20T15:20:15Z","timestamp":1737386415000},"page":"307-354","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Riemannian acceleration with preconditioning for symmetric eigenvalue problems"],"prefix":"10.1007","volume":"157","author":[{"given":"Nian","family":"Shao","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Wenbin","family":"Chen","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2025,1,20]]},"reference":[{"key":"1451_CR1","doi-asserted-by":"publisher","DOI":"10.1515\/9781400830244","volume-title":"Optimization Algorithms on Matrix Manifolds","author":"P-A Absil","year":"2009","unstructured":"Absil, P.-A., Mahony, R., Sepulchre, R.: Optimization Algorithms on Matrix Manifolds. 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