{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T15:22:10Z","timestamp":1776784930906,"version":"3.51.2"},"reference-count":31,"publisher":"American Mathematical Society (AMS)","issue":"253","license":[{"start":{"date-parts":[[2006,7,25]],"date-time":"2006-07-25T00:00:00Z","timestamp":1153785600000},"content-version":"am","delay-in-days":365,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>\n                    We present and analyze homotopic (continuation) residual correction algorithms for the computation of matrix inverses. For complex indefinite Hermitian input matrices, our homotopic methods substantially accelerate the known nonhomotopic algorithms. Unlike the nonhomotopic case our algorithms require no pre-estimation of the smallest singular value of an input matrix. Furthermore, we guarantee rapid convergence to the inverses of well-conditioned structured matrices even where no good initial approximation is available. In particular we yield the inverse of a well-conditioned\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"n times n\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>n<\/mml:mi>\n                            <mml:mo>\n                              \u00d7\n                              \n                            <\/mml:mo>\n                            <mml:mi>n<\/mml:mi>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">n \\times n<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    matrix with a structure of Toeplitz\/Hankel type in\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper O left-parenthesis n log cubed n right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>O<\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>n<\/mml:mi>\n                            <mml:msup>\n                              <mml:mi>log<\/mml:mi>\n                              <mml:mn>3<\/mml:mn>\n                            <\/mml:msup>\n                            <mml:mo>\n                              \u2061\n                              \n                            <\/mml:mo>\n                            <mml:mi>n<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">O(n\\log ^3n)<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    flops. For a large class of input matrices, our methods can be extended to computing numerically the generalized inverses. Our numerical experiments confirm the validity of our analysis and the efficiency of the presented algorithms for well-conditioned input matrices and furnished us with the proper values of the parameters that define our algorithms.\n                  <\/p>","DOI":"10.1090\/s0025-5718-05-01771-0","type":"journal-article","created":{"date-parts":[[2005,11,16]],"date-time":"2005-11-16T10:22:35Z","timestamp":1132136555000},"page":"345-368","source":"Crossref","is-referenced-by-count":10,"title":["Homotopic residual correction processes"],"prefix":"10.1090","volume":"75","author":[{"given":"V.","family":"Pan","sequence":"first","affiliation":[]},{"given":"M.","family":"Kunin","sequence":"additional","affiliation":[]},{"given":"R.","family":"Rosholt","sequence":"additional","affiliation":[]},{"given":"H.","family":"Kodal","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2005,7,25]]},"reference":[{"issue":"2","key":"1","doi-asserted-by":"publisher","first-page":"349","DOI":"10.1137\/0906025","article-title":"Stability of methods for solving Toeplitz systems of equations","volume":"6","author":"Bunch, James R.","year":"1985","journal-title":"SIAM J. Sci. Statist. Comput.","ISSN":"https:\/\/id.crossref.org\/issn\/0196-5204","issn-type":"print"},{"key":"2","doi-asserted-by":"crossref","unstructured":"[B-I66] A. Ben-Israel, A Note on Iterative Method for Generalized Inversion of Matrices, Math. Comp., 20, 439\u2013440, 1966.","DOI":"10.1090\/S0025-5718-66-99922-4"},{"key":"3","isbn-type":"print","doi-asserted-by":"publisher","first-page":"215","DOI":"10.1090\/conm\/281\/04659","article-title":"Approximate displacement rank and applications","author":"Bini, Dario Andrea","year":"2001","ISBN":"https:\/\/id.crossref.org\/isbn\/0821820923"},{"issue":"6","key":"4","doi-asserted-by":"publisher","first-page":"899","DOI":"10.1137\/0908073","article-title":"Fast parallel algorithms for \ud835\udc44\ud835\udc45 and triangular factorization","volume":"8","author":"Chun, J.","year":"1987","journal-title":"SIAM J. Sci. Statist. 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K.","year":"1963"},{"key":"7","series-title":"Johns Hopkins Series in the Mathematical Sciences","isbn-type":"print","volume-title":"Matrix computations","volume":"3","author":"Golub, Gene H.","year":"1989","ISBN":"https:\/\/id.crossref.org\/isbn\/0801837723","edition":"2"},{"issue":"3","key":"8","doi-asserted-by":"publisher","first-page":"629","DOI":"10.1137\/0614044","article-title":"On the Bezoutian structure of the Moore-Penrose inverses of Hankel matrices","volume":"14","author":"Heinig, Georg","year":"1993","journal-title":"SIAM J. Matrix Anal. Appl.","ISSN":"https:\/\/id.crossref.org\/issn\/0895-4798","issn-type":"print"},{"issue":"2","key":"9","doi-asserted-by":"publisher","first-page":"418","DOI":"10.1137\/S0895479892225853","article-title":"Moore-Penrose inversion of square Toeplitz matrices","volume":"15","author":"Heinig, Georg","year":"1994","journal-title":"SIAM J. Matrix Anal. Appl.","ISSN":"https:\/\/id.crossref.org\/issn\/0895-4798","issn-type":"print"},{"key":"10","volume-title":"Analysis of numerical methods","author":"Isaacson, Eugene","year":"1966"},{"issue":"2","key":"11","doi-asserted-by":"publisher","first-page":"395","DOI":"10.1016\/0022-247X(79)90124-0","article-title":"Displacement ranks of matrices and linear equations","volume":"68","author":"Kailath, Thomas","year":"1979","journal-title":"J. Math. Anal. Appl.","ISSN":"https:\/\/id.crossref.org\/issn\/0022-247X","issn-type":"print"},{"key":"12","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1137\/1.9781611971354","volume-title":"Fast reliable algorithms for matrices with structure","year":"1999","ISBN":"https:\/\/id.crossref.org\/isbn\/0898714311"},{"issue":"191","key":"13","doi-asserted-by":"publisher","first-page":"179","DOI":"10.2307\/2008798","article-title":"On computations with dense structured matrices","volume":"55","author":"Pan, Victor","year":"1990","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"1","key":"14","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1016\/0885-064X(92)90031-6","article-title":"Parallel solution of Toeplitzlike linear systems","volume":"8","author":"Pan, Victor","year":"1992","journal-title":"J. Complexity","ISSN":"https:\/\/id.crossref.org\/issn\/0885-064X","issn-type":"print"},{"key":"15","unstructured":"[P92b] V. Y. Pan, Can We Utilize the Cancelation of the Most Significant Digits? Tech. Report TR-92-061, The International Computer Science Institute, Berkeley, California, 1992."},{"issue":"1","key":"16","doi-asserted-by":"publisher","first-page":"118","DOI":"10.1137\/0614010","article-title":"Decreasing the displacement rank of a matrix","volume":"14","author":"Pan, Victor","year":"1993","journal-title":"SIAM J. Matrix Anal. Appl.","ISSN":"https:\/\/id.crossref.org\/issn\/0895-4798","issn-type":"print"},{"key":"17","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-0129-8","volume-title":"Structured matrices and polynomials","author":"Pan, Victor Y.","year":"2001","ISBN":"https:\/\/id.crossref.org\/isbn\/0817642404"},{"key":"18","doi-asserted-by":"crossref","unstructured":"[P01b] V. Y. Pan, A Homotopic Residual Correction Process, Proceedings of the Second Conference on Numerical Analysis and Applications (P. Yalamov, editor), Lecture Notes in Computer Science, 1988, 644\u2013649, Springer, Berlin, 2001.","DOI":"10.1007\/3-540-45262-1_76"},{"key":"19","unstructured":"[Pa] V. Y. Pan, A Homotopic\/Factorization Process for Toeplitz-like Matrices with Newton\u2019s\/Conjugate Gradient Stages, Technical Report TR 2004014, Ph.D. Program in Computer Science, Graduate Center, City University of New York, New York, 2004."},{"key":"20","isbn-type":"print","first-page":"189","article-title":"Newton\u2019s iteration for structured matrices","author":"Pan, Victor Y.","year":"1999","ISBN":"https:\/\/id.crossref.org\/isbn\/0898714311"},{"key":"21","unstructured":"[PKRC02] V. Y. Pan, M. Kunin, R. Rosholt, H. Cebecio\u011flu, Residual Correction Algorithms for General and Structured Matrices, Technical Report TR 2002020, Ph.D. Program in Computer Science, Graduate Center, City University of New York, New York, 2002."},{"key":"22","unstructured":"[PMRTY] V. Y. Pan, B. Murphy, R. E. Rosholt, Y. Tang, X. Yan, Additive Preconditioning in Matrix Computations, Technical Report TR 2005009, Ph.D. Program in Computer Science, Graduate Center, City University of New York, New York, 2005."},{"key":"23","unstructured":"[PR01] V. Y. Pan, Y. Rami, Newton\u2019s Iteration for the Inversion of Structured Matrices, Advances in the Theory of Computational Mathematics, Vol. 4: Structured Matrices: Recent Developments in Theory and Computation, (edited by D. A. Bini, E. Tyrtyshnikov and P. Yalamov), 79\u201390, Nova Science Publishers, Huntington, New York, 2001."},{"key":"24","doi-asserted-by":"publisher","first-page":"233","DOI":"10.1016\/S0024-3795(01)00336-6","article-title":"Structured matrices and Newton\u2019s iteration: unified approach","volume":"343\/344","author":"Pan, Victor Y.","year":"2002","journal-title":"Linear Algebra Appl.","ISSN":"https:\/\/id.crossref.org\/issn\/0024-3795","issn-type":"print"},{"issue":"5","key":"25","doi-asserted-by":"publisher","first-page":"1109","DOI":"10.1137\/0912058","article-title":"An improved Newton iteration for the generalized inverse of a matrix, with applications","volume":"12","author":"Pan, Victor","year":"1991","journal-title":"SIAM J. Sci. Statist. 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