Picard splitting method and Picard CG method for solving the absolute value equation

Volume 10, Issue 7, pp 3643--3654 http://dx.doi.org/10.22436/jnsa.010.07.24

Publication Date: July 25, 2017 Submission Date: June 15, 2017

Download PDF

Download XML

• Export citations
  • CITATIONS & REFERENCES
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)
  • ARTICLE CITATION
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)
  • REFERENCES
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)

• 2074 Downloads
• 3399 Views

Authors

Chang-Qing Lv - College of Mathematics and Informatics, Fujian Key Laborotary of Mathematical Analysis and Applications, Fujian Normal University, Fuzhou, 350117, P. R. China. - School of Mathematics and Statistics, Zaozhuang University, Zaozhuang, 277160, P. R. China. Chang-Feng Ma - College of Mathematics and Informatics, Fujian Key Laborotary of Mathematical Analysis and Applications, Fujian Normal University, Fuzhou, 350117, P. R. China.

Abstract

In this paper, we combine matrix splitting iteration algorithms, such as, Jacobi, SSOR or SAOR algorithms with Picard method for solving absolute value equation. Then, we propose Picard CG for solving the absolute value equation. We discuss the convergence of those methods we proposed. At last, some examples are provided to illustrate the efficiency and validity of methods that we present.

Share and Cite

Keywords

• Absolute value equation
• Picard algorithm
• matrix splitting iteration method
• conjugate gradient method.

MSC

•  65H10
•  47H10

References

• [1] Z.-Z. Bai, X. Yang, On HSS-based iteration methods for weakly nonlinear systems, Appl. Numer. Math., 59 (2009), 2923–2936.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [2] R. W. Cottle, G. B. Dantzig, Complementary pivot theory of mathematical programming, Linear Algebra and Appl., 1 (1968), 103–125.
  • MathSciNet
  • Google Scholar


• [3] R.W. Cottle, J.-S. Pang, R. E. Stone, The linear complementarity problem, Computer Science and Scientific Computing, Academic Press, Inc., Boston, MA (1992)
  • Google Scholar


• [4] T.-X. Gu, X.-W. Xu, X.-P. Liu, H.-B. An, X.-D. Hang, Iterative methods and preconditioning techniques, (Chinese) Science Press, Beijing (2015)
  • Google Scholar


• [5] S.-L. Hu, Z.-H. Huang, A note on absolute value equations, Optim. Lett., 4 (2010), 417–424.
  • View Article
  • MathSciNet
  • Google Scholar


• [6] S. Ketabchi, H. Moosaei, An efficient method for optimal correcting of absolute value equations by minimal changes in the right hand side, Comput. Math. Appl., 64 (2012), 1882–1885.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [7] S. Ketabchi, H. Moosaei, Minimum norm solution to the absolute value equation in the convex case, J. Optim. Theory Appl., 152 (2012), 1080–1087.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [8] S. Ketabchi, H. Moosaei, S. Fallahi, Optimal error correction of the absolute value equation using a genetic algorithm, Math. Comput. Model., 57 (2013), 2339–2342.
  • View Article
  • Google Scholar


• [9] O. L. Mangasarian, Linear complementarity problems solvable by a single linear program, Math. Programming, 10 (1976), 263–270.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [10] O. L. Mangasarian, Absolute value equation solution via concave minimization, Optim. Lett., 1 (2007), 3–8.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [11] O. L. Mangasarian, Absolute value programming, Comput. Optim. Appl., 36 (2007), 43–53.
  • View Article
  • Google Scholar


• [12] O. L. Mangasarian, A generalized Newton method for absolute value equations, Optim. Lett., 3 (2009), 101–108.
  • View Article
  • Google Scholar


• [13] O. L. Mangasarian, Primal-dual bilinear programming solution of the absolute value equation, Optim. Lett., 6 (2012), 1527–1533.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [14] O. L. Mangasarian, Absolute value equation solution via dual complementarity, Optim. Lett., 7 (2013), 625–630.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [15] O. L. Mangasarian, Absolute value equation solution via linear programming, J. Optim. Theory Appl., 161 (2014), 870–876.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [16] O. L. Mangasarian, R. R. Meyer, Absolute value equations, Linear Algebra Appl., 419 (2006), 359–367.
  • View Article
  • Google Scholar


• [17] M. A. Noor, J. Iqbal, K. I. Noor, E. Al-Said, On an iterative method for solving absolute value equations, Optim. Lett., 6 (2012), 1027–1033.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [18] J. M. Ortega, W. C. Rheinboldt, Iterative solution of nonlinear equations in several variables, Academic Press, New York-London, (1970),
  • MathSciNet
  • Google Scholar


• [19] O. Prokopyev, On equivalent reformulations for absolute value equations, Comput. Optim. Appl., 44 (2009), 363–372.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [20] J. Rohn, A theorem of the alternatives for the equation \(Ax + B|x| = b\), Linear Multilinear Algebra, 52 (2004), 421–426.
  • View Article
  • MathSciNet
  • Google Scholar


• [21] J. Rohn, An algorithm for solving the absolute value equation, Electron. J. Linear Algebra, 18 (2009), 589–599.
  • View Article
  • MathSciNet
  • Google Scholar


• [22] J. Rohn, On unique solvability of the absolute value equation, Optim. Lett., 3 (2009), 603–606.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [23] J. Rohn, An algorithm for computing all solutions of an absolute value equation, Optim. Lett., 6 (2012), 851–856.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [24] J. Rohn, V. Hooshyarbakhsh, R. Farhadsefat, An iterative method for solving absolute value equations and sufficient conditions for unique solvability, Optim. Lett., 8 (2014), 35–44.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [25] D. K. Salkuyeh, The Picard-HSS iteration method for absolute value equations, Optim. Lett., 8 (2014), 2191–2202.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [26] A.-X. Wang, H.-J. Wang, Y.-K. Deng, Interval algorithm for absolute value equations, Cent. Eur. J. Math., 9 (2011), 1171–1184.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH