Figure 1.1.
Solution set of linear interval equation
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June
2022, 12(2): 293-308.
doi: 10.3934/naco.2021006
Convergence of interval AOR method for linear interval equations
Department of Mathematics, National Institute of Technology Meghalaya, Shillong, India-793003 |
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.
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 and Optimization,
2022, 12
(2)
: 293-308.
doi: 10.3934/naco.2021006
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References:
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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. |
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A. Berman and R. J. Plemmons, Nonnegative Matrices in the Mathematical Science, SIAM, Philadelphia 1979.
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L. Cvetković and H. Dragoslav,
The AOR method for solving linear interval equations, Computing, 41 (1989), 359-364.
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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. |
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M. Fiedler, J. Nedoma, J. Ramík, J. Rohn and K. Zimmermann, Linear Optimization Problems with Inexact Data, Springer, New York, 2006. |
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A. Hadjidimos,
Accelerated overrelaxation method, Mathematics of Computation, 32 (1978), 149-157.
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A. Hadjidimos,
Successive overrelaxation (SOR) and related methods, Journal of Computational and Applied Mathematics, 123 (2000), 177-199.
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M. Hladík, Interval linear programming: A survey, Linear Programming: New Frontiers in Theory and Applications, Nova Science Publishers, New York, (2012), 85–120. |
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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. |
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M. Hladík and J. Horáček,
Interval linear programming techniques in constraint programming and global optimization, Constraint Programming and Decision Making, 539 (2014), 44-59.
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M. Hladík and I. Skalna,
Relation between various methods for solving linear interval and parametric equations, Linear Alg. Appl., 574 (2019), 1-21.
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R. A. Horn and C. R. Johnson, Matrix Analysis, Cambridge University Press, 1990.
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R. A. Horn and C. R. Johnson, Topics in Matrix Analysis, Cambridge University Press, 1994.
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L. Jaulin, M. Kieffer, O. Didrit and É. Walter, Applied Interval Analysis, Springer, London, 2001.
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R. Kearfott and V. Kreinovich, Applications of Interval computations, Kluwer, Dordrecht, 1996.
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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.
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G. Mayer, Interval Analysis, and Automatic Result Verification, Walter de Gruyter GmbH & Co KG, Vol(65), 2017.
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R. E. Moore, Methods and Applications of Interval Analysis, SIAM, Philadelphia, PA, 1979. |
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A. Neumaier,
New techniques for the analysis of linear interval equations, Linear Algebra and its Applications, 58 (1984), 273-325.
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A. Neumaier, Interval Methods For Systems of Equations, Cambridge University Press, 37, 1990.
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L. Qingrong and and J. Zhiying,
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J. Rohn,
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J. Rohn and S. Shary,
Interval matrices: regularity generates singularity, Linear Algebra and its Applications, 540 (2018), 149-159.
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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.
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Y. Saad, Iterative Methods for Sparse Linear Systems, SIAM, 2003.
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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.
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R. S. Varga, Matrix Iterative Analysis, Prentice-Hall, Englewood Cliffs, NJ, 1962. |
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. |
| [2] |
A. Berman and R. J. Plemmons, Nonnegative Matrices in the Mathematical Science, SIAM, Philadelphia 1979.
doi: 10.1137/1.9781611971262. |
| [3] |
L. Cvetković and H. Dragoslav,
The AOR method for solving linear interval equations, Computing, 41 (1989), 359-364.
doi: 10.1007/BF02241224. |
| [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. |
| [5] |
M. Fiedler, J. Nedoma, J. Ramík, J. Rohn and K. Zimmermann, Linear Optimization Problems with Inexact Data, Springer, New York, 2006. |
| [6] |
A. Hadjidimos,
Accelerated overrelaxation method, Mathematics of Computation, 32 (1978), 149-157.
doi: 10.2307/2006264. |
| [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. |
| [8] |
M. Hladík, Interval linear programming: A survey, Linear Programming: New Frontiers in Theory and Applications, Nova Science Publishers, New York, (2012), 85–120. |
| [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. |
| [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.
|
| [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. |
| [12] |
R. A. Horn and C. R. Johnson, Matrix Analysis, Cambridge University Press, 1990.
|
| [13] |
R. A. Horn and C. R. Johnson, Topics in Matrix Analysis, Cambridge University Press, 1994.
|
| [14] |
L. Jaulin, M. Kieffer, O. Didrit and É. Walter, Applied Interval Analysis, Springer, London, 2001.
doi: 10.1007/978-1-4471-0249-6. |
| [15] |
R. Kearfott and V. Kreinovich, Applications of Interval computations, Kluwer, Dordrecht, 1996.
doi: 10.1007/978-1-4613-3440-8_1. |
| [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. |
| [17] |
G. Mayer, Interval Analysis, and Automatic Result Verification, Walter de Gruyter GmbH & Co KG, Vol(65), 2017.
doi: 10.1515/9783110499469. |
| [18] |
R. E. Moore, Methods and Applications of Interval Analysis, SIAM, Philadelphia, PA, 1979. |
| [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. |
| [20] |
A. Neumaier, Interval Methods For Systems of Equations, Cambridge University Press, 37, 1990.
|
| [21] |
L. Qingrong and and J. Zhiying,
The SOR method for solving linear interval equations, Freiburger Intervall-Berichte, 87 (1987), 1-7.
|
| [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. |
| [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. |
| [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. |
| [25] |
Y. Saad, Iterative Methods for Sparse Linear Systems, SIAM, 2003.
doi: 10.1137/1.9780898718003. |
| [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.
|
| [27] |
R. S. Varga, Matrix Iterative Analysis, Prentice-Hall, Englewood Cliffs, NJ, 1962. |
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