Minimum Residual BAS Iteration Method for Solving the System of Absolute Value Equations

doi:10.1007/s42967-024-00403-z

Communications on Applied Mathematics and Computation ›› 2025, Vol. 7 ›› Issue (5): 1815-1825.doi: 10.1007/s42967-024-00403-z

• ORIGINAL PAPERS • Previous Articles

Minimum Residual BAS Iteration Method for Solving the System of Absolute Value Equations

Yan-Xia Dai¹, Ren-Yi Yan², Ai-Li Yang¹

1. School of Mathematics and Statistics, Hainan Normal University, Haikou, 571158, Hainan, China;
2. School of Economics and Management, Hainan Normal University, Haikou, 571158, Hainan, China

Received:2023-10-10 Revised:2024-02-26 Accepted:2024-03-22 Online:2024-06-13 Published:2024-06-13
Contact: Ren-Yi Yan,E-mail:920188@hainnu.edu.cn;Ai-Li Yang,E-mail:yangaili@hainnu.edu.cn E-mail:920188@hainnu.edu.cn;yangaili@hainnu.edu.cn
Supported by:
This work is supported by the National Natural Science Foundation of China [Grant no. 12161030] and the Natural Science Foundation of Hainan Province China [Grant nos. 523MS039 and 121MS030].

Abstract

Abstract: In this work, by applying the minimum residual technique to the block-diagonal and anti-block-diagonal splitting (BAS) iteration scheme, an iteration method named minimum residual BAS (MRBAS) is proposed to solve a two-by-two block system of nonlinear equations arising from the reformulation of the system of absolute value equations (AVEs). The theoretical analysis shows that the MRBAS iteration method is convergent under suitable conditions. Numerical results demonstrate the feasibility and the effectiveness of the MRBAS iteration method.

Key words: Absolute value equations (AVEs), Block-diagonal and anti-block-diagonal splitting (BAS), Minimum residual, Minimum residual BAS (MRBAS) iteration, Convergence analysis

Yan-Xia Dai, Ren-Yi Yan, Ai-Li Yang. Minimum Residual BAS Iteration Method for Solving the System of Absolute Value Equations[J]. Communications on Applied Mathematics and Computation, 2025, 7(5): 1815-1825.

TrendMD

References

[1] Bai, Z.-Z.: Parallel multisplitting two-stage iterative methods for large sparse systems of weakly nonlinear equations. Numer. Algorithms 15(3/4), 347-372 (1997)
[2] Bai, Z.-Z.: A class of two-stage iterative methods for systems of weakly nonlinear equations. Numer. Algorithms 14, 295-319 (1997)
[3] Bai, Z.-Z.: Modulus-based matrix splitting iteration methods for linear complementarity problems. Numer. Linear Algebra Appl. 17, 917-933 (2010)
[4] Bai, Z.-Z., Benzi, M., Chen, F.: Modified HSS iteration methods for a class of complex symmetric linear systems. Computing 87, 93-111 (2010)
[5] Bai, Z.-Z., Golub, G.H., Ng, M.K.: Hermitian and skew-Hermitian splitting methods for non-Hermitian positive definite linear systems. SIAM J. Matrix Anal. Appl. 24, 603-626 (2003)
[6] Bai, Z.-Z., Migallón, V., Penadés, J., Szyld, D.B.: Block and asynchronous two-stage methods for mildly nonlinear systems. Numer. Math. 82, 1-20 (1999)
[7] Bai, Z.-Z., Pan, J.-Y.: Matrix Analysis and Computations. SIAM, Philadelphia (2021)
[8] Bai, Z.-Z., Yang, X.: On HSS-based iteration methods for weakly nonlinear systems. Appl. Numer. Math. 59, 2923-2936 (2009)
[9] Bai, Z.-Z., Zhang, L.-L.: Modulus-based synchronous multisplitting iteration methods for linear complementarity problems. Numer. Linear Algebra Appl. 20, 425-439 (2013)
[10] Caccetta, L., Qu, B., Zhou, G.-L.: A globally and quadratically convergent method for absolute value equations. Comput. Optim. Appl. 48, 45-58 (2011)
[11] Cao, Y., Shi, Q., Zhu, S.-L.: A relaxed generalized Newton iteration method for generalized absolute value equations. AIMS Math. 6, 1258-1275 (2021)
[12] Cottle, R.W., Pang, J.-S., Stone, R.E.: The Linear Complementarity Problem. SIAM, Philadelphia (2009)
[13] Dehghan, M., Shirilord, A.: Matrix multisplitting Picard-iterative method for solving generalized absolute value matrix equation. Appl. Numer. Math. 158, 425-438 (2020)
[14] Gutknecht, M.H., Rozložník, M.: Residual smoothing techniques: do they improve the limiting accuracy of iterative solvers? BIT Numer. Math. 41, 86-114 (2001)
[15] Haghani, F.K.: On generalized Traub’s method for absolute value equations. J. Optim. Theory Appl. 166, 619-625 (2015)
[16] Horn, R.A., Johnson, C.R.: Matrix Analysis. Cambridge University Press, Cambridge (2012)
[17] Ke, Y.-F.: The new iteration algorithm for absolute value equation. Appl. Math. Lett. 99, 105990 (2020)
[18] Ke, Y.-F., Ma, C.-F.: SOR-like iteration method for solving absolute value equations. Appl. Math. Comput. 311, 195-202 (2017)
[19] Ketabchi, S., Moosaei, H.: An efficient method for optimal correcting of absolute value equations by minimal changes in the right hand side. Comput. Math. Appl. 64, 1882-1885 (2012)
[20] Li, C.-X.: A modified generalized Newton method for absolute value equations. J. Optim. Theory Appl. 170, 1055-1059 (2016)
[21] Li, C.-X., Wu, S.-L.: Block-diagonal and anti-block-diagonal splitting iteration method for absolute value equation. In: Song, H., Jiang, D. (eds.) Simulation Tools and Techniques, vol. 369, pp. 572-581. Springer, Cham (2021)
[22] Mangasarian, O.L.: Absolute value programming. Comput. Optim. Appl. 36, 43-53 (2007)
[23] Mangasarian, O.L.: A generalized Newton method for absolute value equations. Optim. Lett. 3, 101-108 (2009)
[24] Mangasarian, O.L.: A hybrid algorithm for solving the absolute value equation. Optim. Lett. 9, 1469-1474 (2015)
[25] Mangasarian, O.L., Meyer, R.R.: Absolute value equations. Linear Algebra Appl. 419, 359-367 (2006)
[26] Miao, S.-X., Xiong, X.-T., Wen, J.: On Picard-SHSS iteration method for absolute value equation. AIMS Math. 6, 1743-1753 (2021)
[27] Ortega, J.M., Rheinboldt, W.C.: Iterative Solution of Nonlinear Equations in Several Variables. SIAM, Philadelphia (2000)
[28] Prokopyev, O.: On equivalent reformulations for absolute value equations. Comput. Optim. Appl. 44, 363-372 (2009)
[29] Rohn, J.: A theorem of the alternatives for the equation

\begin{document}$ Ax+B|x|=b $\end{document}

. Linear Multilinear Algebra 52, 421-426 (2004)
[30] Rohn, J.: An algorithm for solving the absolute value equations. Electron. J. Linear Algebra 18, 589-599 (2009)
[31] Rohn, J., Hooshyarbakhsh, V., Farhadsefat, R.: An iterative method for solving absolute value equations and sufficient conditions for unique solvability. Optim. Lett. 8, 35-44 (2014)
[32] Salkuyeh, D.K.: The Picard-HSS iteration method for absolute value equations. Optim. Lett. 8, 2191-2202 (2014)
[33] Varga, R.S.: Matrix Iterative Analysis. Springer, Berlin (2000)
[34] Walker, H.F.: Residual smoothing and peak/plateau behavior in Krylov subspace methods. Appl. Numer. Math. 19, 279-286 (1995)
[35] Wu, S.-L., Guo, P.: Modulus-based matrix splitting algorithms for the quasi-complementarity problems. Appl. Numer. Math. 132, 127-137 (2018)
[36] Yang, A.-L., Wu, Y.-J., Cao, Y.: Minimum residual Hermitian and skew-Hermitian splitting iteration method for non-Hermitian positive definite linear systems. BIT Numer. Math. 59, 299-319 (2019)
[37] Young, D.M.: Iterative Solution of Large Linear Systems. Academic Press, New York (1971)
[38] Yu, D.-M., Chen, C.-R., Han, D.-R.: A modified fixed point iteration method for solving the system of absolute value equations. Optimization 71, 449-461 (2022)
[39] Zhang, J.-L., Zhang, G.-F., Liang, Z.-Z.: A modified generalized SOR-like method for solving an absolute value equation. Linear Multilinear Algebra 71, 1578-1595 (2023)

Minimum Residual BAS Iteration Method for Solving the System of Absolute Value Equations

Knowledge

Abstract

Cite this article

share this article

References

Related Articles 6

Recommended Articles

Metrics

Comments

[1]	Xu Li, Jian-Sheng Feng. Parameterized QHSS Iteration Method and Its Variants for Non-Hermitian Positive Definite Linear Systems of Strong Skew-Hermitian Parts [J]. Communications on Applied Mathematics and Computation, 2025, 7(5): 1665-1683.
[2]	Zhao-Zheng Liang, Jun-Lin Tian, Hong-Yi Wan. A Note on Chebyshev Accelerated PMHSS Iteration Method for Block Two-by-Two Linear Systems [J]. Communications on Applied Mathematics and Computation, 2025, 7(4): 1242-1263.
[3]	Gang Bao, Peijun Li, Xiaokai Yuan. Convergence of the PML Method for the Biharmonic Wave Scattering Problem in Periodic Structures [J]. Communications on Applied Mathematics and Computation, 2025, 7(3): 1122-1145.
[4]	Yuan Xu, Qiang Zhang. Superconvergence Analysis of the Runge-Kutta Discontinuous Galerkin Method with Upwind-Biased Numerical Flux for Two-Dimensional Linear Hyperbolic Equation [J]. Communications on Applied Mathematics and Computation, 2022, 4(1): 319-352.
[5]	Qingqing Wu, Zhongshu Wu, Xiaoyan Zeng. A Jacobi Spectral Collocation Method for Solving Fractional Integro-Differential Equations [J]. Communications on Applied Mathematics and Computation, 2021, 3(3): 509-526.
[6]	Somayeh Yeganeh, Reza Mokhtari, Jan S. Hesthaven. A Local Discontinuous Galerkin Method for Two-Dimensional Time Fractional Difusion Equations [J]. Communications on Applied Mathematics and Computation, 2020, 2(4): 689-709.