Studying a numerical stable and quadratic convergence method for solving a new class of absolute value equations.
Subject Areas :
Statistics
mozafar rostami
1
,
Taher Lotfi
2
,
Ali Berahmand
3
1 - Department of Applied Mathematics, Hamadan Branch, Islamic Azad University, Hamedan, Iran.
2 - Department of Applied Mathematics, Hamadan Branch, Islamic Azad University, Hamedan, Iran.
3 - Department of Applied Mathematics, Hamadan Branch, Islamic Azad University, Hamedan, Iran.
Received: 2021-02-16
Accepted : 2022-03-07
Published : 2022-10-23
Keywords:
معادلهی مقدار قدرمطلقی,
روش تکرار,
پایداری عددی,
مرتبهی همگرایی,
سیستمهای غیرخطی,
Abstract :
In this paper, a new class of absolute value equations is studied as follows:Ax-B|x|-b=o, ( B≠I, σ_"max" (|B|)<σ_"min" (A) ), This new class of absolute value equations, the single value absolute matrix B is less than the single value matrix A and the matrix B is not exclusively the identity matrix..Therfore the power of choice is wider than other methods of the absolute value equations and all matrices are arbitrary and this new class of absolute value equation is the NP hard problem..We solve this new class using a generalized Newton method and also convergence and numerical stability. Also, by testing the numerical examples of the efficiency and effectiveness of the solution method for the new class, it has been studied with other works that have been done including Lotfi and Zainali and Mangasarain and Khaksars method.Eceptthis new class and Lotfi and Zainali method are quadratic convergence, the rest methods are linear convergence.
References:
Caccetta, B. Qu, and G. Zhou. A globally and quadratically convergent method for absolute value equations. Computational Optimization and Applications, 48(1):45–58, 2011.
Cordero, J. L. Hueso, E. Martinez, and J. R. Torregrosa. A modified Newton-Jarratts composition. Numerical Algorithms, 55(1):87–99, 2010.
Cordero, T. Lotfi, K. Mahdiani, and J. R. Torregrosa. A stable family with high order of convergence for solving nonlinear equations. Applied Mathematics and Computation, 254:240–251, 2015.
Farhadsefat, T. Lotfi, and J. Rohn. A note on regularity and positive definiteness of interval matrices. Open Mathematics, 10(1):322–328, 2012.
K. Haghani. On generalized Traubs method for absolute value equations. Journal of Optimization Theory and Applications, 166(2):619–625, 2015.
J. Higham. Accuracy and stability of numerical algorithms. SIAM, 1996.
L. Hueso, E. Martinez, and J. R. Torregrosa. Modified Newtons method for systems of nonlinear equations with singular Jacobian. Journal of Computational and Applied Mathematics, 224(1):77–83, 2009.
Lotfi, P. Bakhtiari, A. Cordero, K. Mahdiani, and J. R. Torregrosa. Some new efficient multipoint iterative methods for solving nonlinear systems of equations. International Journal of Computer Mathematics, 92(9):1921–1934, 2015.
Lotfi, K. Mahdiani, P. Bakhtiari, and F. Soleymani. Constructing two-step iterative methods with and without memory. Computational Mathematics and Mathematical Physics, 55(2):183–193, 2015.
Mangasarian. A generalized newton method for absolute value equations. Optimization Letters, 3(1):101–108, 2009.
Mangasarian and R. Meyer. Absolute value equations. Linear Algebra and Its Applications, 419(2-3):359–367, 2006.
L. Mangasarian. Absolute value equation solution via concave minimization. Optimization Letters, 1(1):3–8, 2007.
L. Mangasarian. A hybrid algorithm for solving the absolute value equation. Optimization Letters, 9(7):1469–1474, 2015.
Prokopyev. On equivalent reformulations for absolute value equations. Computational Optimization and Applications, 44(3):363, 2009.
Rohn. On unique solvability of the absolute value equation. Optimization Letters, 3(4):603–606, 2009.
Rohn, V. Hooshyarbakhsh, and R. Farhadsefat. An iterative method for solving absolute value equations and sufficient conditions for unique solvability. Optimization Letters, 8(1):35–44, 2014.
Soleymani, T. Lotfi, and P. Bakhtiari. A multi-step class of iterative methods for nonlinear systems. Optimization Letters, 8(3):1001–1015, 2014.
Stoer and R. Bulirsch. Introduction to numerical analysis, volume 12. Springer Science & Business Media, 2013.
F. Traub. Iterative methods for the solution of equations, volume 312. American Mathematical Soc., 1982.
Wozniakowski. Numerical stability for solving nonlinear equations. Numerische Mathematik, 27(4):373–390, 1976.
Zainali and T. Lotfi. On developing a stable and quadratic convergent method for solving absolute value equation. Journal of Computational and Applied Mathematics, 330:742–747, 2018.
Lotfi; Y. Seif, An improved generalized Newton generalized method for absolute value equation, New Researches in Mathematics, 29 (7) 103-110, 2021
Wang, A., Cao, Y., Chen, J.-X., Modified Newton-Type iteration methods for generalized absolute value equations, J. Optimization Theory and Applications, 181(1), 216-230, 2019
Zheng, L. The Picard-HSS-SOR iteration method for absolute value equations. J Inequal Appl 2020, 258 2020.
Cao, Q. Shi, S. Zhu, A relaxed generaized Newton iteration method for generalized absolute value equation, AIMS Mathematics, 6(2), 1258-1275, 2021
