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doi: 10.3934/naco.2021006
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Convergence of interval AOR method for linear interval equations

Jahnabi Chakravarty , Ashiho Athikho and  Manideepa Saha ,

Department of Mathematics, National Institute of Technology Meghalaya, Shillong, India-793003

* Corresponding author: Manideepa Saha

Received  June 2020 Revised  January 2021 Early access February 2021

A real interval vector/matrix is an array whose entries are real intervals. In this paper, we consider the real linear interval equations $ \bf{Ax} = \bf{b} $ with $ {{\bf{A}} }$, $ \bf{b} $ respectively, denote an interval matrix and an interval vector. The aim of the paper is to study the numerical solution of the linear interval equations for various classes of coefficient interval matrices. In particular, we study the convergence of interval AOR method when the coefficient interval matrix is either interval strictly diagonally dominant matrices, interval $ L $-matrices, interval $ M $-matrices, or interval $ H $-matrices.

Keywords: AOR method, convergence, linear interval equations.
Mathematics Subject Classification: Primary: 15A30, 65G40, 65H10; Secondary: 65G30.
Citation: Jahnabi Chakravarty, Ashiho Athikho, Manideepa Saha. Convergence of interval AOR method for linear interval equations. Numerical Algebra, Control & Optimization, doi: 10.3934/naco.2021006
References:
[1]

M. Allahdadi and H. M. Nehi, The optimal solution set of the interval linear programming problems, Optimization Letters, 7 (2013), 1893-1911.  doi: 10.1007/s11590-012-0530-4.  Google Scholar

[2]

A. Berman and R. J. Plemmons, Nonnegative Matrices in the Mathematical Science, SIAM, Philadelphia 1979. doi: 10.1137/1.9781611971262.  Google Scholar

[3]

L. Cvetković and H. Dragoslav, The AOR method for solving linear interval equations, Computing, 41 (1989), 359-364.  doi: 10.1007/BF02241224.  Google Scholar

[4]

M. T. Darvishi and P. Hessari, On convergence of the generalized AOR method for linear systems with diagonally dominant coefficient matrices, Applied Mathematics and Computation, 176 (2006), 128-133.  doi: 10.1016/j.amc.2005.09.051.  Google Scholar

[5]

M. Fiedler, J. Nedoma, J. Ramík, J. Rohn and K. Zimmermann, Linear Optimization Problems with Inexact Data, Springer, New York, 2006.  Google Scholar

[6]

A. Hadjidimos, Accelerated overrelaxation method, Mathematics of Computation, 32 (1978), 149-157.  doi: 10.2307/2006264.  Google Scholar

[7]

A. Hadjidimos, Successive overrelaxation (SOR) and related methods, Journal of Computational and Applied Mathematics, 123 (2000), 177-199.  doi: 10.1016/S0377-0427(00)00403-9.  Google Scholar

[8]

M. Hladík, Interval linear programming: A survey, Linear Programming: New Frontiers in Theory and Applications, Nova Science Publishers, New York, (2012), 85–120. Google Scholar

[9]

M. Hladík, New operator and method for solving real interval preconditioned interval linear equations, SIAM J. Numer. Anal., 52(1) (2014), 194-206.  doi: 10.1137/130914358.  Google Scholar

[10]

M. Hladík and J. Horáček, Interval linear programming techniques in constraint programming and global optimization, Constraint Programming and Decision Making, 539 (2014), 44-59.   Google Scholar

[11]

M. Hladík and I. Skalna, Relation between various methods for solving linear interval and parametric equations, Linear Alg. Appl., 574 (2019), 1-21.  doi: 10.1016/j.laa.2019.03.019.  Google Scholar

[12] R. A. Horn and C. R. Johnson, Matrix Analysis, Cambridge University Press, 1990.   Google Scholar
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L. Jaulin, M. Kieffer, O. Didrit and É. Walter, Applied Interval Analysis, Springer, London, 2001. doi: 10.1007/978-1-4471-0249-6.  Google Scholar

[15]

R. Kearfott and V. Kreinovich, Applications of Interval computations, Kluwer, Dordrecht, 1996. doi: 10.1007/978-1-4613-3440-8_1.  Google Scholar

[16]

W. Li and W. W. Sun, Modified Gauss-Seidel type methods and Jacobi type methods for Z-matrices, Linear Algebra and its Applications, 317 (2000), 227-240.  doi: 10.1016/S0024-3795(00)00140-3.  Google Scholar

[17]

G. Mayer, Interval Analysis, and Automatic Result Verification, Walter de Gruyter GmbH & Co KG, Vol(65), 2017. doi: 10.1515/9783110499469.  Google Scholar

[18]

R. E. Moore, Methods and Applications of Interval Analysis, SIAM, Philadelphia, PA, 1979.  Google Scholar

[19]

A. Neumaier, New techniques for the analysis of linear interval equations, Linear Algebra and its Applications, 58 (1984), 273-325.  doi: 10.1016/0024-3795(84)90217-9.  Google Scholar

[20] A. Neumaier, Interval Methods For Systems of Equations, Cambridge University Press, 37, 1990.   Google Scholar
[21]

L. Qingrong and and J. Zhiying, The SOR method for solving linear interval equations, Freiburger Intervall-Berichte, 87 (1987), 1-7.   Google Scholar

[22]

J. Rohn, Forty necessary and sufficient conditions for regularity of interval matrices: A survey, Electron. J. Linear Algebra, 18 (2009), 500-512.  doi: 10.13001/1081-3810.1327.  Google Scholar

[23]

J. Rohn and S. Shary, Interval matrices: regularity generates singularity, Linear Algebra and its Applications, 540 (2018), 149-159.  doi: 10.1016/j.laa.2017.11.020.  Google Scholar

[24]

S. M. Rump, INTLAB-INTerval LABoratory, In Developments in Reliable Computing(ed. Tibor Csendes), 77–104. Kluwer Academic Publishers, Dordrecht, 1999. http://www.ti3.tuhh.de/intlab. doi: 10.1007/978-94-017-1247-7.  Google Scholar

[25]

Y. Saad, Iterative Methods for Sparse Linear Systems, SIAM, 2003. doi: 10.1137/1.9780898718003.  Google Scholar

[26]

D. K. Salkuyeh, Generalized Jacobi and Gauss-Seidel methods for solving linear system of equations, Numer. Math. J. Chinese Univ. (English Ser.), 16 (2007), 164-170.   Google Scholar

[27]

R. S. Varga, Matrix Iterative Analysis, Prentice-Hall, Englewood Cliffs, NJ, 1962.  Google Scholar

show all references

References:
[1]

M. Allahdadi and H. M. Nehi, The optimal solution set of the interval linear programming problems, Optimization Letters, 7 (2013), 1893-1911.  doi: 10.1007/s11590-012-0530-4.  Google Scholar

[2]

A. Berman and R. J. Plemmons, Nonnegative Matrices in the Mathematical Science, SIAM, Philadelphia 1979. doi: 10.1137/1.9781611971262.  Google Scholar

[3]

L. Cvetković and H. Dragoslav, The AOR method for solving linear interval equations, Computing, 41 (1989), 359-364.  doi: 10.1007/BF02241224.  Google Scholar

[4]

M. T. Darvishi and P. Hessari, On convergence of the generalized AOR method for linear systems with diagonally dominant coefficient matrices, Applied Mathematics and Computation, 176 (2006), 128-133.  doi: 10.1016/j.amc.2005.09.051.  Google Scholar

[5]

M. Fiedler, J. Nedoma, J. Ramík, J. Rohn and K. Zimmermann, Linear Optimization Problems with Inexact Data, Springer, New York, 2006.  Google Scholar

[6]

A. Hadjidimos, Accelerated overrelaxation method, Mathematics of Computation, 32 (1978), 149-157.  doi: 10.2307/2006264.  Google Scholar

[7]

A. Hadjidimos, Successive overrelaxation (SOR) and related methods, Journal of Computational and Applied Mathematics, 123 (2000), 177-199.  doi: 10.1016/S0377-0427(00)00403-9.  Google Scholar

[8]

M. Hladík, Interval linear programming: A survey, Linear Programming: New Frontiers in Theory and Applications, Nova Science Publishers, New York, (2012), 85–120. Google Scholar

[9]

M. Hladík, New operator and method for solving real interval preconditioned interval linear equations, SIAM J. Numer. Anal., 52(1) (2014), 194-206.  doi: 10.1137/130914358.  Google Scholar

[10]

M. Hladík and J. Horáček, Interval linear programming techniques in constraint programming and global optimization, Constraint Programming and Decision Making, 539 (2014), 44-59.   Google Scholar

[11]

M. Hladík and I. Skalna, Relation between various methods for solving linear interval and parametric equations, Linear Alg. Appl., 574 (2019), 1-21.  doi: 10.1016/j.laa.2019.03.019.  Google Scholar

[12] R. A. Horn and C. R. Johnson, Matrix Analysis, Cambridge University Press, 1990.   Google Scholar
[13] R. A. Horn and C. R. Johnson, Topics in Matrix Analysis, Cambridge University Press, 1994.   Google Scholar
[14]

L. Jaulin, M. Kieffer, O. Didrit and É. Walter, Applied Interval Analysis, Springer, London, 2001. doi: 10.1007/978-1-4471-0249-6.  Google Scholar

[15]

R. Kearfott and V. Kreinovich, Applications of Interval computations, Kluwer, Dordrecht, 1996. doi: 10.1007/978-1-4613-3440-8_1.  Google Scholar

[16]

W. Li and W. W. Sun, Modified Gauss-Seidel type methods and Jacobi type methods for Z-matrices, Linear Algebra and its Applications, 317 (2000), 227-240.  doi: 10.1016/S0024-3795(00)00140-3.  Google Scholar

[17]

G. Mayer, Interval Analysis, and Automatic Result Verification, Walter de Gruyter GmbH & Co KG, Vol(65), 2017. doi: 10.1515/9783110499469.  Google Scholar

[18]

R. E. Moore, Methods and Applications of Interval Analysis, SIAM, Philadelphia, PA, 1979.  Google Scholar

[19]

A. Neumaier, New techniques for the analysis of linear interval equations, Linear Algebra and its Applications, 58 (1984), 273-325.  doi: 10.1016/0024-3795(84)90217-9.  Google Scholar

[20] A. Neumaier, Interval Methods For Systems of Equations, Cambridge University Press, 37, 1990.   Google Scholar
[21]

L. Qingrong and and J. Zhiying, The SOR method for solving linear interval equations, Freiburger Intervall-Berichte, 87 (1987), 1-7.   Google Scholar

[22]

J. Rohn, Forty necessary and sufficient conditions for regularity of interval matrices: A survey, Electron. J. Linear Algebra, 18 (2009), 500-512.  doi: 10.13001/1081-3810.1327.  Google Scholar

[23]

J. Rohn and S. Shary, Interval matrices: regularity generates singularity, Linear Algebra and its Applications, 540 (2018), 149-159.  doi: 10.1016/j.laa.2017.11.020.  Google Scholar

[24]

S. M. Rump, INTLAB-INTerval LABoratory, In Developments in Reliable Computing(ed. Tibor Csendes), 77–104. Kluwer Academic Publishers, Dordrecht, 1999. http://www.ti3.tuhh.de/intlab. doi: 10.1007/978-94-017-1247-7.  Google Scholar

[25]

Y. Saad, Iterative Methods for Sparse Linear Systems, SIAM, 2003. doi: 10.1137/1.9780898718003.  Google Scholar

[26]

D. K. Salkuyeh, Generalized Jacobi and Gauss-Seidel methods for solving linear system of equations, Numer. Math. J. Chinese Univ. (English Ser.), 16 (2007), 164-170.   Google Scholar

[27]

R. S. Varga, Matrix Iterative Analysis, Prentice-Hall, Englewood Cliffs, NJ, 1962.  Google Scholar

Figure 1.1.  Solution set of linear interval equation
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Figure 4.1.  Solution set $ \sum(\textbf{A},b). $
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Figure 4.2.  Solution set $ \sum(\textbf{A},b). $
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