Solving Doubly Bordered Tridiagonal Linear Systems via Partition

Affiliation(s)

^{1}
Mathematics Department, Faculty of Science, Mansoura University, Mansoura, Egypt.

^{2}
Mathematics Department, Faculty of Science, Damietta University, Egypt.

ABSTRACT

This paper presents new numeric and symbolic algorithms for solving doubly bordered tridiagonal linear system. The proposed algorithms are derived using partition together with UL factorization. Inversion algorithm for doubly bordered tridiagonal matrix is also considered based on the Sherman-Morrison-Woodbury formula. The algorithms are implemented using the computer algebra system, MAPLE. Some illustrative examples are given.

This paper presents new numeric and symbolic algorithms for solving doubly bordered tridiagonal linear system. The proposed algorithms are derived using partition together with UL factorization. Inversion algorithm for doubly bordered tridiagonal matrix is also considered based on the Sherman-Morrison-Woodbury formula. The algorithms are implemented using the computer algebra system, MAPLE. Some illustrative examples are given.

KEYWORDS

Doubly Bordered Tridiagonal Matrices, UL Factorization, Block Matrices, Computer Algebra Systems, Sherman-Morrison-Woodbury Formula

Doubly Bordered Tridiagonal Matrices, UL Factorization, Block Matrices, Computer Algebra Systems, Sherman-Morrison-Woodbury Formula

Cite this paper

El-Mikkawy, M. , El-Shehawy, M. and Shehab, N. (2015) Solving Doubly Bordered Tridiagonal Linear Systems via Partition.*Applied Mathematics*, **6**, 967-978. doi: 10.4236/am.2015.66089.

El-Mikkawy, M. , El-Shehawy, M. and Shehab, N. (2015) Solving Doubly Bordered Tridiagonal Linear Systems via Partition.

References

[1] Udala, A., Reedera, R., Velmrea, E. and Harrisonb, P. (2006) Comparison of Methods for Solving the Schrödinger Equation for Multiquantum Well Heterostructure Applications. Proceedings of the Estonian Academy of Sciences, Engineering, 12, 246-261.

[2] Pajic, S. (2007) Power System State Estimation and Contingency Constrained Optimal Power Flow—A Numerically Robust Implementation. Polytechnic Ins. [S.l.], Worcester.

[3] Mazilu, I., Mazilu, A.D. and Williams, H.T. (2012) Applications of Tridiagonal Matrices in Non-Equilibrium Statistical Physics. Electronic Journal of Linear Algebra, 24, 7-17.

[4] Borowska, J., Lacińska, L. and Rychlewska, J. (2012) Application of Difference Equation to Certain Tridiagonal Matrices. Scientific Research of the Institute of Mathematics and Computer Science, 3, 15-20.

[5] Fischer, C.F. and Usmani, R.A. (1969) Properties of Some Tridiagonal Matrices and Their Application to Boundary Value Problems. Society for Industrial and Applied Mathematics, 6, 127-142.

http://dx.doi.org/10.1137/0706014

[6] El-Mikkawy, M. and Atlan, F. (2014) Algorithms for Solving Doubly Bordered Tridiagonal Linear Systems. British Journal of Mathematics and Computer Science, 4, 1246-1267.

http://dx.doi.org/10.9734/BJMCS/2014/8835

[7] Al-Hassan, Q. (2005) An Algorithm for Computing Inverses of Tridiagonal Matrices with Applications. Soochow Journal of Mathematics, 31, 449-466.

[8] El-Mikkawy, M.E.A. (2004) On the Inverse of a General Tridiagonal Matrix. Applied Mathematics and Computation, 150, 669-679. http://dx.doi.org/10.1016/S0096-3003(03)00298-4

[9] Ran, R.S., Huang, T.Z., Liu, X.P. and Gu, T.X. (2009) An Inversion Algorithm for General Tridiagonal Matrix. Applied Mathematics and Mechanics (English Edition), 30, 247-253.

http://dx.doi.org/10.1007/s10483-009-0212-x

[10] Hadj, A.D. and Elouafi, M. (2008) A Fast Numerical Algorithm for the Inverse of a Tridiagonal and Pentadiagonal Matrix. Applied Mathematics and Computation, 202, 441-445.

http://dx.doi.org/10.1016/j.amc.2008.02.026

[11] El-Mikkwy, M. and Rahmo, E.-D. (2008) A New Recursive Algorithm for Inverting General Tridiagonal and Anti- Tridiagonal Matrices. Applied Mathematics and Computation, 204, 368-372.

http://dx.doi.org/10.1016/j.amc.2008.06.053

[12] Davis, T.A. (2006) Direct Methods for Sparse Linear Systems. University of Florida, Gainesville.

http://dx.doi.org/10.1137/1.9780898718881

[13] Golub, G.H. and Van Loan, C.F. (2013) Matrix Computations. 4rd Edition, Johns Hopkins Press, Baltimore.

[14] EL-Mikkawy, M. (2005) A New Computational Algorithm for Solving Periodic Tri-Diagonal Linear Systems. Applied Mathematics and Computation, 161, 691-696.

http://dx.doi.org/10.1016/j.amc.2003.12.114

[15] EL-Mikkawy, M. (2012) A Generalized Symbolic Thomas Algorithm. Applied Mathematics, 3, 342-345.

http://dx.doi.org/10.4236/am.2012.34052

[16] Alexandre, M. and Iain, D.B. (2010) Variant of the Thomas Algorithm for Opposite-Bordered Tridiagonal Systems of Equations. Numerical Methods in Biomedical Engineering, 26, 752-759.

[17] Strang, G. (2005) Linear Algebra and Its Applications. 4th Edition, Cengage Learning, Boston.

[18] Reddy, B.R. (2012) Recursive Method for Inversion of Lower Triangular Matrix. Journal of Experimental Sciences, 3, 2218-1768.

[19] Wang, X.-B. (2009) A New Algorithm with Its Scilab Implementation for Solution of Bordered Tridiagonal Linear Equations. IEEE International Workshop on Open-Source Software for Scientific Computation (OSSC), Guiyang, 18- 20 September 2009, 11-14.

[20] Amodio, P., Gladwell, I. and Romanazzi, G. (2007) An Algorithm for the Solution of Bordered ABD Linear Systems Arising from Boundary Value Problems. ECCOMAS Thematic Conference, Milano, 25-28 June 2007, 25-28.

[21] Amodio, P., Gladwelly, I. and Romanazzi, G. (2006) Numerical Solution of General Bordered ABD Linear Systems by Cyclic Reduction. Numerical Analysis, Industrial and Applied Mathematics, 1, 5-12.

[22] El-Mikkawy, M. (2003) A Note on a Three-Term Recurrence for a Tridiagonal Matrix. Applied Mathematics and Com- putation, 139, 503-511. http://dx.doi.org/10.1016/S0096-3003(02)00212-6

[23] Stoer, J. and Bulirsch, R (1992) Introduction to Numerical Analysis. 2nd Edition, Springer Verlag, New York.

[1] Udala, A., Reedera, R., Velmrea, E. and Harrisonb, P. (2006) Comparison of Methods for Solving the Schrödinger Equation for Multiquantum Well Heterostructure Applications. Proceedings of the Estonian Academy of Sciences, Engineering, 12, 246-261.

[2] Pajic, S. (2007) Power System State Estimation and Contingency Constrained Optimal Power Flow—A Numerically Robust Implementation. Polytechnic Ins. [S.l.], Worcester.

[3] Mazilu, I., Mazilu, A.D. and Williams, H.T. (2012) Applications of Tridiagonal Matrices in Non-Equilibrium Statistical Physics. Electronic Journal of Linear Algebra, 24, 7-17.

[4] Borowska, J., Lacińska, L. and Rychlewska, J. (2012) Application of Difference Equation to Certain Tridiagonal Matrices. Scientific Research of the Institute of Mathematics and Computer Science, 3, 15-20.

[5] Fischer, C.F. and Usmani, R.A. (1969) Properties of Some Tridiagonal Matrices and Their Application to Boundary Value Problems. Society for Industrial and Applied Mathematics, 6, 127-142.

http://dx.doi.org/10.1137/0706014

[6] El-Mikkawy, M. and Atlan, F. (2014) Algorithms for Solving Doubly Bordered Tridiagonal Linear Systems. British Journal of Mathematics and Computer Science, 4, 1246-1267.

http://dx.doi.org/10.9734/BJMCS/2014/8835

[7] Al-Hassan, Q. (2005) An Algorithm for Computing Inverses of Tridiagonal Matrices with Applications. Soochow Journal of Mathematics, 31, 449-466.

[8] El-Mikkawy, M.E.A. (2004) On the Inverse of a General Tridiagonal Matrix. Applied Mathematics and Computation, 150, 669-679. http://dx.doi.org/10.1016/S0096-3003(03)00298-4

[9] Ran, R.S., Huang, T.Z., Liu, X.P. and Gu, T.X. (2009) An Inversion Algorithm for General Tridiagonal Matrix. Applied Mathematics and Mechanics (English Edition), 30, 247-253.

http://dx.doi.org/10.1007/s10483-009-0212-x

[10] Hadj, A.D. and Elouafi, M. (2008) A Fast Numerical Algorithm for the Inverse of a Tridiagonal and Pentadiagonal Matrix. Applied Mathematics and Computation, 202, 441-445.

http://dx.doi.org/10.1016/j.amc.2008.02.026

[11] El-Mikkwy, M. and Rahmo, E.-D. (2008) A New Recursive Algorithm for Inverting General Tridiagonal and Anti- Tridiagonal Matrices. Applied Mathematics and Computation, 204, 368-372.

http://dx.doi.org/10.1016/j.amc.2008.06.053

[12] Davis, T.A. (2006) Direct Methods for Sparse Linear Systems. University of Florida, Gainesville.

http://dx.doi.org/10.1137/1.9780898718881

[13] Golub, G.H. and Van Loan, C.F. (2013) Matrix Computations. 4rd Edition, Johns Hopkins Press, Baltimore.

[14] EL-Mikkawy, M. (2005) A New Computational Algorithm for Solving Periodic Tri-Diagonal Linear Systems. Applied Mathematics and Computation, 161, 691-696.

http://dx.doi.org/10.1016/j.amc.2003.12.114

[15] EL-Mikkawy, M. (2012) A Generalized Symbolic Thomas Algorithm. Applied Mathematics, 3, 342-345.

http://dx.doi.org/10.4236/am.2012.34052

[16] Alexandre, M. and Iain, D.B. (2010) Variant of the Thomas Algorithm for Opposite-Bordered Tridiagonal Systems of Equations. Numerical Methods in Biomedical Engineering, 26, 752-759.

[17] Strang, G. (2005) Linear Algebra and Its Applications. 4th Edition, Cengage Learning, Boston.

[18] Reddy, B.R. (2012) Recursive Method for Inversion of Lower Triangular Matrix. Journal of Experimental Sciences, 3, 2218-1768.

[19] Wang, X.-B. (2009) A New Algorithm with Its Scilab Implementation for Solution of Bordered Tridiagonal Linear Equations. IEEE International Workshop on Open-Source Software for Scientific Computation (OSSC), Guiyang, 18- 20 September 2009, 11-14.

[20] Amodio, P., Gladwell, I. and Romanazzi, G. (2007) An Algorithm for the Solution of Bordered ABD Linear Systems Arising from Boundary Value Problems. ECCOMAS Thematic Conference, Milano, 25-28 June 2007, 25-28.

[21] Amodio, P., Gladwelly, I. and Romanazzi, G. (2006) Numerical Solution of General Bordered ABD Linear Systems by Cyclic Reduction. Numerical Analysis, Industrial and Applied Mathematics, 1, 5-12.

[22] El-Mikkawy, M. (2003) A Note on a Three-Term Recurrence for a Tridiagonal Matrix. Applied Mathematics and Com- putation, 139, 503-511. http://dx.doi.org/10.1016/S0096-3003(02)00212-6

[23] Stoer, J. and Bulirsch, R (1992) Introduction to Numerical Analysis. 2nd Edition, Springer Verlag, New York.