An improved Schulz-type iterative method for matrix inversion with application
In this paper, we present an algorithm which could be considered an improvement to the well-known Schulz iteration for finding the inverse of a square matrix iteratively. The convergence of the proposed method is proved and its computational complexity is analysed. The extension of the scheme to gen...
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| Published in | Transactions of the Institute of Measurement and Control Vol. 36; no. 8; pp. 983 - 991 |
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| Main Authors | , |
| Format | Journal Article |
| Language | English |
| Published |
London, England
SAGE Publications
01.12.2014
Sage Publications Ltd |
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| Abstract | In this paper, we present an algorithm which could be considered an improvement to the well-known Schulz iteration for finding the inverse of a square matrix iteratively. The convergence of the proposed method is proved and its computational complexity is analysed. The extension of the scheme to generalized outer inverses will be treated. In order to validate the new scheme, we apply it to large sparse matrices alongside the application to preconditioning of practical problems. |
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| AbstractList | In this paper, we present an algorithm which could be considered an improvement to the well-known Schulz iteration for finding the inverse of a square matrix iteratively. The convergence of the proposed method is proved and its computational complexity is analysed. The extension of the scheme to generalized outer inverses will be treated. In order to validate the new scheme, we apply it to large sparse matrices alongside the application to preconditioning of practical problems. In this paper, we present an algorithm which could be considered an improvement to the well-known Schulz iteration for finding the inverse of a square matrix iteratively. The convergence of the proposed method is proved and its computational complexity is analysed. The extension of the scheme to generalized outer inverses will be treated. In order to validate the new scheme, we apply it to large sparse matrices alongside the application to preconditioning of practical problems. [PUBLICATION ABSTRACT] |
| Author | Soleymani, F Haghani, F Khaksar |
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| CitedBy_id | crossref_primary_10_1109_TCYB_2018_2818747 crossref_primary_10_1109_ACCESS_2020_2979371 crossref_primary_10_1016_j_amc_2015_01_045 crossref_primary_10_1016_j_tcs_2016_07_024 crossref_primary_10_1155_2018_8973867 crossref_primary_10_3390_fractalfract7010032 crossref_primary_10_1080_00207160_2021_1881498 crossref_primary_10_1002_mma_5782 crossref_primary_10_3390_math11071738 crossref_primary_10_1007_s10957_015_0712_1 crossref_primary_10_1016_j_aej_2020_09_059 crossref_primary_10_1002_mma_8855 crossref_primary_10_1016_j_amc_2021_126720 crossref_primary_10_1109_TVT_2022_3194870 crossref_primary_10_1080_03081087_2022_2088673 crossref_primary_10_3390_math12111772 crossref_primary_10_1109_TVT_2018_2874811 |
| Cites_doi | 10.1037/h0070888 10.1016/j.amc.2013.07.039 10.1177/014233129601800405 10.1177/0142331208095777 10.1016/j.cam.2013.12.033 10.1117/12.942056 10.1155/2013/708647 10.1137/0914010 10.1142/9789812836021_0024 10.1137/1.9780898718003 10.1016/S0096-3003(02)00238-2 10.1002/1099-1506(200010/12)7:7/8<527::AID-NLA211>3.0.CO;2-O 10.1016/j.amc.2011.05.036 10.1002/zamm.19330130111 10.1007/978-1-4612-0129-8 10.4134/CKMS.2013.28.2.407 10.1002/nla.1890. 10.1137/S0895479899345571 |
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| Keywords | outer inverse Sparse matrices approximate inverse preconditioner |
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| References | El-Sayed, Ran 2001; 23 Soleymani 2013b 2013 Stanimirović, Soleymani 2014; 263 Li, Huang, Zhang 2011; 218 Whalley 1996; 18 Grosz 2000; 7 Sen, Prabhu 1976; 8 Schulz 1933; 13 Soleymani, Stanimirović 2013 Soleymani, Stanimirović, Ullah 2013; 222 Soleymani 2013a; 28 Davis, Hu 2011; 38 Frontini, Sormani 2003; 140 Alpert, Beylkin, Coifman 1993; 14 Zhou, Yan 2008; 30 Hotelling 1933; 24 bibr10-0142331214528350 bibr23-0142331214528350 Sen SK (bibr15-0142331214528350) 1976; 8 bibr24-0142331214528350 bibr6-0142331214528350 bibr11-0142331214528350 bibr5-0142331214528350 bibr19-0142331214528350 bibr1-0142331214528350 Davis TA (bibr2-0142331214528350) 2011; 38 bibr9-0142331214528350 Trott M (bibr22-0142331214528350) 2006 bibr14-0142331214528350 bibr4-0142331214528350 bibr21-0142331214528350 Householder AS (bibr7-0142331214528350) 1964 bibr17-0142331214528350 bibr13-0142331214528350 Krishnamurthy EV (bibr8-0142331214528350) 1986 bibr3-0142331214528350 Soleymani F (bibr18-0142331214528350) 2013 bibr20-0142331214528350 bibr12-0142331214528350 bibr25-0142331214528350 bibr16-0142331214528350 |
| References_xml | – year: 2013b 2013 article-title: A fast convergent iterative solver for approximate inverse of matrices publication-title: Numerical Linear Algebra with Applications contributor: fullname: Soleymani – volume: 7 start-page: 527 year: 2000 end-page: 541 article-title: Preconditioning by incomplete block elimination publication-title: Numerical Linear Algebra with Applications contributor: fullname: Grosz – volume: 18 start-page: 202 year: 1996 end-page: 211 article-title: Matrix quadratic models with variable parameters publication-title: Transactions of the Institute of Measurement and Control contributor: fullname: Whalley – volume: 218 start-page: 260 year: 2011 end-page: 270 article-title: Chebyshev-type methods and preconditioning techniques publication-title: Applied Mathematics and Computation contributor: fullname: Zhang – volume: 38 start-page: 1 year: 2011 end-page: 28 article-title: The University of Florida sparse matrix collection publication-title: ACM Transactions on Mathematical Software contributor: fullname: Hu – volume: 23 start-page: 632 year: 2001 end-page: 645 article-title: On an iteration method of solving a class of nonlinear matrix equations publication-title: SIAM Journal on Matrix Analysis and Applications contributor: fullname: Ran – volume: 263 start-page: 236 year: 2014 end-page: 245 article-title: A class of numerical algorithms for computing outer inverses publication-title: Journal of Computational and Applied Mathematics contributor: fullname: Soleymani – volume: 28 start-page: 407 year: 2013a end-page: 418 article-title: On a fast iterative method for approximate inverse of matrices publication-title: Communication of the Korean Mathematical Society contributor: fullname: Soleymani – volume: 140 start-page: 419 year: 2003 end-page: 426 article-title: Some variant of Newton’s method with third-order convergence publication-title: Applied Mathematics and Computation contributor: fullname: Sormani – start-page: 708647 year: 2013 article-title: A higher order iterative method for computing the Drazin inverse publication-title: The Scientific World Journal contributor: fullname: Stanimirović – volume: 30 start-page: 397 year: 2008 end-page: 426 article-title: Solutions to right co-prime factorizations and generalized Sylvester matrix equations publication-title: Transactions of the Institute of Measurement and Control contributor: fullname: Yan – volume: 222 start-page: 365 year: 2013 end-page: 371 article-title: On an accelerated iterative method for weighted Moore-Penrose inverse publication-title: Applied Mathematics and Computation contributor: fullname: Ullah – volume: 8 start-page: 748 year: 1976 end-page: 753 article-title: Optimal iterative schemes for computing Moore-Penrose matrix inverse publication-title: International Journal of Systems Science contributor: fullname: Prabhu – volume: 24 start-page: 498 year: 1933 end-page: 520 article-title: Analysis of a complex statistical variable into principal components publication-title: Journal of Educational Psychology contributor: fullname: Hotelling – volume: 14 start-page: 159 year: 1993 end-page: 184 article-title: Wavelet-like bases for the fast solution of second-kind integral equations publication-title: SIAM Journal on Scientific Computing contributor: fullname: Coifman – volume: 13 start-page: 57 year: 1933 end-page: 59 article-title: Iterative Berechnung der Reziproken matrix publication-title: Zeitschrift für Angewandte Mathematik und Mechanik contributor: fullname: Schulz – volume-title: Numerical Algorithms – Computations in Science and Engineering year: 1986 ident: bibr8-0142331214528350 contributor: fullname: Krishnamurthy EV – ident: bibr6-0142331214528350 doi: 10.1037/h0070888 – volume: 38 start-page: 1 year: 2011 ident: bibr2-0142331214528350 publication-title: ACM Transactions on Mathematical Software contributor: fullname: Davis TA – ident: bibr19-0142331214528350 doi: 10.1016/j.amc.2013.07.039 – volume: 8 start-page: 748 year: 1976 ident: bibr15-0142331214528350 publication-title: International Journal of Systems Science contributor: fullname: Sen SK – ident: bibr24-0142331214528350 doi: 10.1177/014233129601800405 – ident: bibr25-0142331214528350 doi: 10.1177/0142331208095777 – ident: bibr20-0142331214528350 doi: 10.1016/j.cam.2013.12.033 – ident: bibr23-0142331214528350 doi: 10.1117/12.942056 – start-page: 708647 year: 2013 ident: bibr18-0142331214528350 publication-title: The Scientific World Journal doi: 10.1155/2013/708647 contributor: fullname: Soleymani F – volume-title: The Theory of Matrices in Numerical Analysis year: 1964 ident: bibr7-0142331214528350 contributor: fullname: Householder AS – ident: bibr1-0142331214528350 doi: 10.1137/0914010 – ident: bibr11-0142331214528350 doi: 10.1142/9789812836021_0024 – ident: bibr13-0142331214528350 doi: 10.1137/1.9780898718003 – ident: bibr4-0142331214528350 doi: 10.1016/S0096-3003(02)00238-2 – ident: bibr5-0142331214528350 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| SubjectTerms | Algorithms Computation Convergence Inverse Inversions Iterative methods Matrix Preconditioning Sparse matrices |
| Title | An improved Schulz-type iterative method for matrix inversion with application |
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