• Previous Article
    Discrete-time realization of fractional-order proportional integral controller for a class of fractional-order system
  • NACO Home
  • This Issue
  • Next Article
    Optimal control of a dynamical system with intermediate phase constraints and applications in cash management
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

* Corresponding author: Manideepa Saha

Received  June 2020 Revised  January 2021 Published  June 2022 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.

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
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.

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.

Figure 1.1.  Solution set of linear interval equation
Figure 4.1.  Solution set $ \sum(\textbf{A},b). $
Figure 4.2.  Solution set $ \sum(\textbf{A},b). $
[1]

Fang Chen, Ning Gao, Yao- Lin Jiang. On product-type generalized block AOR method for augmented linear systems. Numerical Algebra, Control and Optimization, 2012, 2 (4) : 797-809. doi: 10.3934/naco.2012.2.797

[2]

Nizami A. Gasilov. Solving a system of linear differential equations with interval coefficients. Discrete and Continuous Dynamical Systems - B, 2021, 26 (5) : 2739-2747. doi: 10.3934/dcdsb.2020203

[3]

Yves Bourgault, Damien Broizat, Pierre-Emmanuel Jabin. Convergence rate for the method of moments with linear closure relations. Kinetic and Related Models, 2015, 8 (1) : 1-27. doi: 10.3934/krm.2015.8.1

[4]

Yu-Ning Yang, Su Zhang. On linear convergence of projected gradient method for a class of affine rank minimization problems. Journal of Industrial and Management Optimization, 2016, 12 (4) : 1507-1519. doi: 10.3934/jimo.2016.12.1507

[5]

Leyu Hu, Xingju Cai. Convergence of a randomized Douglas-Rachford method for linear system. Numerical Algebra, Control and Optimization, 2020, 10 (4) : 463-474. doi: 10.3934/naco.2020045

[6]

Wei-Zhe Gu, Li-Yong Lu. The linear convergence of a derivative-free descent method for nonlinear complementarity problems. Journal of Industrial and Management Optimization, 2017, 13 (2) : 531-548. doi: 10.3934/jimo.2016030

[7]

Jiangxing Wang. Convergence analysis of an accurate and efficient method for nonlinear Maxwell's equations. Discrete and Continuous Dynamical Systems - B, 2021, 26 (5) : 2429-2440. doi: 10.3934/dcdsb.2020185

[8]

Anca Croitoru, Alina GavriluŢ, Alina Iosif, Anna Rita Sambucini. A note on convergence results for varying interval valued multisubmeasures. Mathematical Foundations of Computing, 2021, 4 (4) : 299-310. doi: 10.3934/mfc.2021020

[9]

Liu Liu. Uniform spectral convergence of the stochastic Galerkin method for the linear semiconductor Boltzmann equation with random inputs and diffusive scaling. Kinetic and Related Models, 2018, 11 (5) : 1139-1156. doi: 10.3934/krm.2018044

[10]

Saeed Ketabchi, Hossein Moosaei, M. Parandegan, Hamidreza Navidi. Computing minimum norm solution of linear systems of equations by the generalized Newton method. Numerical Algebra, Control and Optimization, 2017, 7 (2) : 113-119. doi: 10.3934/naco.2017008

[11]

Z. K. Eshkuvatov, M. Kammuji, Bachok M. Taib, N. M. A. Nik Long. Effective approximation method for solving linear Fredholm-Volterra integral equations. Numerical Algebra, Control and Optimization, 2017, 7 (1) : 77-88. doi: 10.3934/naco.2017004

[12]

Tahereh Salimi Siahkolaei, Davod Khojasteh Salkuyeh. A preconditioned SSOR iteration method for solving complex symmetric system of linear equations. Numerical Algebra, Control and Optimization, 2019, 9 (4) : 483-492. doi: 10.3934/naco.2019033

[13]

William Guo. The Laplace transform as an alternative general method for solving linear ordinary differential equations. STEM Education, 2021, 1 (4) : 309-329. doi: 10.3934/steme.2021020

[14]

Rajesh Kumar, Jitendra Kumar, Gerald Warnecke. Convergence analysis of a finite volume scheme for solving non-linear aggregation-breakage population balance equations. Kinetic and Related Models, 2014, 7 (4) : 713-737. doi: 10.3934/krm.2014.7.713

[15]

Ugur G. Abdulla. On the optimal control of the free boundary problems for the second order parabolic equations. II. Convergence of the method of finite differences. Inverse Problems and Imaging, 2016, 10 (4) : 869-898. doi: 10.3934/ipi.2016025

[16]

Yulan Lu, Minghui Song, Mingzhu Liu. Convergence rate and stability of the split-step theta method for stochastic differential equations with piecewise continuous arguments. Discrete and Continuous Dynamical Systems - B, 2019, 24 (2) : 695-717. doi: 10.3934/dcdsb.2018203

[17]

Cheng Wang. Convergence analysis of the numerical method for the primitive equations formulated in mean vorticity on a Cartesian grid. Discrete and Continuous Dynamical Systems - B, 2004, 4 (4) : 1143-1172. doi: 10.3934/dcdsb.2004.4.1143

[18]

Kai Qu, Qi Dong, Chanjie Li, Feiyu Zhang. Finite element method for two-dimensional linear advection equations based on spline method. Discrete and Continuous Dynamical Systems - S, 2021, 14 (7) : 2471-2485. doi: 10.3934/dcdss.2021056

[19]

Jiang-Xia Nan, Deng-Feng Li. Linear programming technique for solving interval-valued constraint matrix games. Journal of Industrial and Management Optimization, 2014, 10 (4) : 1059-1070. doi: 10.3934/jimo.2014.10.1059

[20]

Shi Jin, Yingda Li. Local sensitivity analysis and spectral convergence of the stochastic Galerkin method for discrete-velocity Boltzmann equations with multi-scales and random inputs. Kinetic and Related Models, 2019, 12 (5) : 969-993. doi: 10.3934/krm.2019037

 Impact Factor: 

Metrics

  • PDF downloads (504)
  • HTML views (601)
  • Cited by (0)

[Back to Top]