Infinity Norm Bounds for the Inverse of

{\text{SDD}}_1

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\textrm{SDD}_1$$\end{document}

-Type Matrices with Applications

Abstract

A new subclass of H-matrices named

{\text{SDD}}_1

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\textrm{SDD}_1$$\end{document}

-type matrices is introduced. The relationships between

{\text{SDD}}_1

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\textrm{SDD}_1$$\end{document}

-type matrices and other subclasses of H-matrices are studied. Moreover, the infinite norm bounds for the inverse of

{\text{SDD}}_1

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\textrm{SDD}_1$$\end{document}

-type matrices are provided. As applications, error bounds of the linear complementarity problems (LCPs) for

{\text{SDD}}_1

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\textrm{SDD}_1$$\end{document}

-type matrices and strictly diagonally dominant (

\text{SDD}

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\textrm{SDD}$$\end{document}

) matrices strictly diagonally dominant (are also presented, which improve some existing bounds. Numerical examples are presented to demonstrate the effectiveness of the obtained results.

{\text{SDD}}_1

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-type matrices / Strictly diagonally dominant (SDD) matrices / Infinity norm / Linear complementarity problems (LCPs) / Error bounds / 15A48 / 90C33 / 65G50 / 65F05

References

[1] Berman A, Plemmons RJ. Nonnegative Matrices in the Mathematical Sciences. 1994, Philadelphia, Society for Industrial and Applied Mathematics.

[2] Chen XJ, Xiang SH. Computation of error bounds for P\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$P$$\end{document}-matrix linear complementarity problems. Math. Program.. 2006, 106(3): 513-525.

[3] Chen, X.Y., Li, Y.T., Liu, L., Wang, Y.Q.: Infinity norm upper bounds for the inverse of SDD1\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\text{SDD}_1$$\end{document} matrices. AIMS Math. 7(5), 8847–8860 (2022)

[4] Cvetković D, Cvetković L, Li CQ. CKV-type matrices with applications. Linear Algebra Appl.. 2021, 608(1): 158-184.

[5] Cvetković, L.: H\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$H$$\end{document}-matrix theory vs. eigenvalue localization. Numer. Algorithms 42(3/4), 229–245 (2006)

[6] Cvetković L, Kostić V, Bru R, Pedroche F. A simple generalization of Geršgorin’s theorem. Adv. Comput. Math.. 2011, 35: 271-280.

[7] Dai, P.F., Li, J.P., Zhao, S.Y.: Infinity norm bounds for the inverse for GSDD1\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\text{ GSDD}_1$$\end{document} matrices using scaling matrices. Comput. Appl. Math. 42, 1–21 (2023)

[8] García-Esnaola M, Peña JM. A comparison of error bounds for linear complementarity problems of H\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$H$$\end{document}-matrices. Linear Algebra Appl.. 2010, 433(5): 956-964.

[9] Geng, Y.J., Sun, D.S.: Error bounds for linear complementarity problems of strong SDD1\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\text{ SDD}_1$$\end{document} matrices and strong SDD1\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\text{ SDD}_1$$\end{document}-B\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$B$$\end{document} matrices. AIMS Math. 8(11), 27052–27064 (2023)

[10] Kolotilina LY . On Dashnic-Zusmanovich (DZ) and Dashnic-Zusmanovich type (DZT) matrices and their inverses. J. Math. Sci.. 2019, 240(6): 799-812.

[11] Kolotilina LY. A new subclass of the class of nonsingular H\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$H$$\end{document}-matrices and related inclusion sets for eigenvalues and singular values. J. Math. Sci.. 2019, 240(6): 813-821.

[12] Kolotilina LY. Some bounds for inverses involving matrix sparsity pattern. J. Math. Sci.. 2020, 249(2): 242-255.

[13] Li, C.Q., Cvetković, L., Wei, Y.M., Zhao, J.X.: An infinity norm bound for the inverse of Dashnic-Zusmanovich type matrices with applications. Linear Algebra Appl. 565, 99-122 (2019)

[14] Liu JZ, Zhang J, Liu Y. The Schur complement of strictly doubly diagonally dominant matrices and its application. Linear Algebra Appl.. 2012, 437(1): 168-183.

[15] Peña JM. Diagonal dominance, Schur complements and some classes of H\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$H$$\end{document}-matrices and P\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$P$$\end{document}-matrices. Adv. Comput. Math.. 2011, 35: 357-373.

[16] Varah JM. A lower bound for the smallest singular value of a matrix. Linear Algebra Appl.. 1975, 111(1): 3-5.

[17] Varga RS. Matrix Iterative Analysis. 2000, Berlin, Springer.

[18] Wang F, Yan WW, Zhao YX, Zhao PC. New error bounds for linear complementarity problems for BS\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$B^S$$\end{document}-matrices. Comput. Appl. Math.. 2023, 42(226): 1-17.

[19] Wang, X.D., Wang, F.: Infinity norm upper bounds for the inverse of SDDk\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\text{ SDD}_k$$\end{document} matrices. AIMS Math. 8(10), 24999–25016 (2023)

[20] Wang, Y.H., Song, X.N., Gao, L.: An infinity norm bound for the inverse of strong SDD1\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\text{ SDD}_1$$\end{document} matrices with applications. Jpn. J. Ind. Appl. Math. 40, 1287–1304 (2023)

[21] Wang, Z.F., Li, C.Q., Li, Y.T.: Infimum of error bounds for linear complementarity problems of Σ\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\Sigma $$\end{document}-SDD\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\text{ SDD }$$\end{document} and Σ1\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\Sigma _1$$\end{document}-SSD\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\text{ SSD }$$\end{document} matrices. Linear Algebra Appl. 581(1), 285–303 (2019)

[22] Zhao, Y.X., Liu, L.L., Wang, F.: Error bounds for linear complementarity problems of SDD1\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$ \text{ SDD}_1$$\end{document} matrices and SDD1\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$ \text{ SDD}_1$$\end{document}-B\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$ B $$\end{document} matrices. AIMS Math. 7, 11862–11878 (2022)

Funding

Guizhou Provincial Science and Technology Department(20191161)

Guizhou Provincial Youth Science and Technology Talents Growth Project(QJJ2023012)

RIGHTS & PERMISSIONS

Shanghai University