Hostname: page-component-78c5997874-ndw9j Total loading time: 0 Render date: 2024-11-10T21:35:28.971Z Has data issue: false hasContentIssue false
  • English
  • Français

The Explicit Inverses of CUPL-Toeplitz and CUPL-Hankel Matrices

Part of: Basic linear algebra Special matrices

Published online by Cambridge University Press:  31 January 2017

Zhao-Lin Jiang ,
Xiao-Ting Chen  and
Jian-Min Wang
Zhao-Lin Jiang*
Affiliation:
Department of Mathematics, Linyi University, Linyi 276005, P. R. China
Xiao-Ting Chen*
Affiliation:
Department of Mathematics, Linyi University, Linyi 276005, P. R. China School of Mathematical Sciences, Shandong Normal University, Jinan 250014, P. R. China
Jian-Min Wang*
Affiliation:
Department of Mathematics, Linyi University, Linyi 276005, P. R. China
*
*Corresponding author. Email addresses:jzh1208@sina.com (Z.-L. Jiang), chenxt0723@163.com (X.- T. Chen), wjm0818@163.com (J.-M. Wang)
*Corresponding author. Email addresses:jzh1208@sina.com (Z.-L. Jiang), chenxt0723@163.com (X.- T. Chen), wjm0818@163.com (J.-M. Wang)
*Corresponding author. Email addresses:jzh1208@sina.com (Z.-L. Jiang), chenxt0723@163.com (X.- T. Chen), wjm0818@163.com (J.-M. Wang)
Get access

Abstract

In this paper, we consider two innovative structured matrices, CUPL-Toeplitz matrix and CUPL-Hankel matrix. The inverses of CUPL-Toeplitz and CUPL-Hankel matrices can be expressed by the Gohberg-Heinig type formulas, and the stability of the inverse matrices is verified in terms of 1-, ∞- and 2-norms, respectively. In addition, two algorithms for the inverses of CUPL-Toeplitz and CUPL-Hankel matrices are given and examples are provided to verify the feasibility of these algorithms.

Keywords

CUPL-Toeplitz matrixCUPL-Hankel matrixinversestabilityalgorithm

MSC classification

Secondary: 15A09: Matrix inversion, generalized inverses 15B05: Toeplitz, Cauchy, and related matrices
Type
Research Article
Information
East Asian Journal on Applied Mathematics , Volume 7 , Issue 1 , February 2017 , pp. 38 - 54
Copyright
Copyright © Global-Science Press 2017 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Bai, Z.-Z., Li, G.-Q. and Lu, L.-Z., Combinative preconditioners of modified incomplete Cholesky factorization and Sherman-Morrison-Woodbury update for self-adjoint elliptic Dirichlet-periodic boundary value problems, J. Comput. Math., 22 (2004), pp. 833–856.Google Scholar
[2] Bai, Z.-Z. and Ren, Z.-R., Block-triangular preconditioning methods for linear third-order ordinary differential equations based on reduced-order sinc discretizations, Japan J. Industr. Appl. Math., 30 (2013), pp. 511–527.CrossRefGoogle Scholar
[3] Bai, Z.-Z., Chan, R.-H. and Ren, Z.-R., On sinc discretization and banded preconditioning for linear third-order ordinary differential equations, Numer. Linear Algebra Appl., 18 (2011), pp. 471–497.Google Scholar
[4] Bai, Z.-Z., Chan, R.-H. and Ren, Z.-R., On order-reducible sinc discretizations and block-diagonal preconditioning methods for linear third-order ordinary differential equations, Numer. Linear Algebra Appl., 21 (2014), pp. 108–135.CrossRefGoogle Scholar
[5] Bai, Z.-Z., Huang, Y.-M. and Ng, M. K., On preconditioned iterative methods for certain time-dependent partial differential equations, SIAM J. Numer. Anal., 47 (2009), pp. 1019–1037.CrossRefGoogle Scholar
[6] Bai, Z.-Z., Huang, Y.-M. and Ng, M. K., On preconditioned iterative methods for Burgers equations, SIAM J. Sci. Comput., 29 (2007), pp. 415–439.CrossRefGoogle Scholar
[7] Bai, Z.-Z. and Ng, M. K., Preconditioners for nonsymmetric block Toeplitz-like-plus-diagonal linear systems, Numer. Math., 96 (2003), pp. 197–220.Google Scholar
[8] Grenander, U. and Szego, G., Toeplitz Forms and Their Applications, Univ. of California Press, Berkeley, 1958.CrossRefGoogle Scholar
[9] Jiang, Z.-L. and Wang, D.-D., Explicit group inverse of an innovative patterned matrix, Appl. Math. Comput., 274 (2016), pp. 220–228.Google Scholar
[10] Trench, W. F., An algorithm for the inversion of finite Toeplitz matrices, J. Soc. Indust. Appl. Math., 12(3), (1964), pp. 515–522.CrossRefGoogle Scholar
[11] Gohberg, I. and Semencul, A., On the inversion of finite Toeplitz matrices and their continuous analogues (in Russian), Mat. Issled., 2 (1972), pp. 201–233.Google Scholar
[12] Zohar, S., Toeplitz Matrix Inversion: The Algorithm of W. F. Trench, J. ACM, 16(4), (1969), pp. 592–601.Google Scholar
[13] Gohberg, I. and Krupnik, N, A formula for the inversion of finite Toeolitz matrices, Mat. Issled., 2 (1972), pp. 272–283, 295(In Russian).Google Scholar
[14] Heinig, G. and Rost, K., Algebraic Methods for Toeplitz-Like Matrices and Operators, Akademie-Verlag, Berlin, 13, 1984.Google Scholar
[15] Artzi, A. B. and Shalom, T., On inversion of Toeplitz and close to Toeplitz matrices, Linear Algebra Appl., 75(1), (1986), pp. 173–192.Google Scholar
[16] Labahn, G. and Shalom, T., Inversion of Toeplitz matrices with only two standard equations, Linear Algebra Appl., 175(1), (1992), pp. 143–158.Google Scholar
[17] Ng, M. K., Rost, K. and Wen, Y. W., On inversion of Toeplitz matrices, Linear Algebra Appl., 348 (2002), pp. 145–151.CrossRefGoogle Scholar
[18] Heinig, G., On the reconstruction of Toeplitz matrix inverses from columns, Linear Algebra Appl., 350 (2002), pp. 199–212.Google Scholar
[19] Lv, X.-G. and Huang, T.-Z., A note on inversion of Toeplitz matrices, Appl. Math. Lett., 20 (2007), pp. 1189–1193.Google Scholar
[20] Labahn, G. and Shalom, T., Inversion of Toeplitz structured matrices using only standard equations, Linear Algebra Appl., 207 (1994), pp. 49–70.Google Scholar
[21] Jiang, Z.-L. and Chen, J.-X., The explicit inverse of nonsingular conjugate-Toeplitz and conjugate-Hankel matrices, J. Appl. Math. Comput., (2015), pp. 1–16.Google Scholar
[22] Wen, Y.-W., Ng, M. K. and Ching, W.-K., A note on the stability of Toeplitz matrix inversion formulas, Appl. Math. Lett., 17(8), (2004), pp. 903–907.CrossRefGoogle Scholar
[23] Gutknecht, M. H. and Hochbruck, M., The stability of inversion formulas for Toeplitz matrices, Linear Algebra Appl., 223-224, (1995), pp. 307–324.Google Scholar
[24] Xie, P.-P. and Wei, Y.-M., The stability of formulae of the Gohberg-Semencul-Trench type for Moore-Penrose and group inverses of Toeplitz matrices, Linear Algebra Appl., 498 (2015), pp. 117–135.Google Scholar
[25] Chillag, D., Regular representations of semisimple algebras, separable field extensions, group characters, generalized circulants, and generalized cyclic codes, Linear Algebra Appl., 218 (1995), pp. 147–183.Google Scholar
[26] Jiang, Z.-L. and Xu, Z.-B., Efficient algorithm for finding the inverse and the group inverse of FLS r-circulant matrix, J. Appl. Math. Comput., 18 (2005), pp. 45–57.Google Scholar