The present article is mainly devoted for solving bordered k-tridiagonal linear systems of equations. Two efficient and reliable symbolic algorithms for solving such systems are constructed. The computational cost of the algorithms is obtained. Some illustrative examples are given.
References
[1]
El-Mikkawy, M.E.A. (2004) A Fast Algorithm for Evaluating nth Order Tri-Diagonal Determinants. Journal of Computational and Applied Mathematics, 166, 581-584. http://dx.doi.org/10.1016/j.cam.2003.08.044
[2]
El-Mikkawy, M.E.A. and Karawia, A. (2006) Inversion of General Tridiagonal Matrices. Applied Mathematics Letters, 19, 712-720. http://dx.doi.org/10.1016/j.aml.2005.11.012
[3]
Akin, H. (2012) On 1D Reversible Cellular Automata with Reflective Boundary over the Prime Field of Order p. International Journal of Modern Physics C, 23, 1-13. http://dx.doi.org/10.1142/s0129183111017020
[4]
Chant, T.F. and Resascot, D.C. (1986) Generalized Deflated Block-Elimination. SIAM Journal on Numerical Analysis, 23, 913-924. http://dx.doi.org/10.1137/0723059
[5]
El-Mikkawy, M.E.A. (1991) An Algorithm for Solving Tridiagonal Systems. Journal of Institute of Mathematics and Computer Sciences. Computer Sciences Series, 4, 205-210.
[6]
El-Mikkawy, M.E.A. (2003) A Note on a Three-Term Recurrence for a Tridiagonal Matrix. Applied Mathematics and Computation, 139, 503-511. http://dx.doi.org/10.1016/S0096-3003(02)00212-6
[7]
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
[8]
El-Mikkawy, M.E.A. (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
[9]
El-Mikkawy, M.E.A. 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
[10]
El-Mikkawy, M.E.A. and Rahmo, E.-D. (2009) A New Recursive Algorithm for Inverting General Periodic Pentadiagonal and Anti-Pentadiagonal Matrices. Applied Mathematics and Computation, 207, 164-170.
http://dx.doi.org/10.1016/j.amc.2008.10.010
[11]
El-Mikkawy, M.E.A. and Rahmo, E. (2010) Symbolic Algorithm for Inverting Cyclic Pentadiagonal Matrices Recursively—Derivation and Implementation. Computers & Mathematics with Applications, 59, 1386-1396.
http://dx.doi.org/10.1016/j.camwa.2009.12.020
[12]
El-Mikkawy, M.E.A. (2012) A Generalized Symbolic Thomas Algorithm. Applied Mathematics, 3, 342-345.
http://dx.doi.org/10.4236/am.2012.34052
[13]
Kavcic, A. and Moura, J.M.F. (2000) Matrices with Banded Inverses: Inversion Algorithms and Factorization of Gauss-Markov Processes. IEEE Transactions on Information Theory, 46, 1495-1509.
http://dx.doi.org/10.1109/18.954748
[14]
Li, H.B., Huang, T.Z., Liu X.P. and Li, H. (2010) On the Inverses of General Tridiagonal Matrices. Linear Algebra and Its Applications, 433, 965-983. http://dx.doi.org/10.1016/j.laa.2010.04.042
[15]
Shapiro, L.W. (1984) Positive Definite Matrices and Catalan Numbers, Revisited. Proceedings of the American Mathematical Society, 90, 488-496. http://dx.doi.org/10.1090/S0002-9939-1984-0728375-5
[16]
Sogabe, T. (2007) On a Two-Term Recurrence for the Determinant of a General Matrix. Applied Mathematics and Computation, 187, 785-788. http://dx.doi.org/10.1016/j.amc.2006.08.156
[17]
Sugimoto, T. (2012) On an Inverse Formula of a Tridiagonal Matrix. Operators and Matrices, 6, 465-480.
http://dx.doi.org/10.7153/oam-06-30
[18]
Usmani, R. (1994) Inversion of a Tridiagonal Jacobi Matrix. Linear Algebra and Its Applications, 212/213, 413-414.
http://dx.doi.org/10.1016/0024-3795(94)90414-6
[19]
Akbudak, K., Kayaaslan, E. and Aykanat, C. (2012) Analyzing and Enhancing OSKI for Sparse Matrix-Vector Multiplication. www.prace-ri.eu
[20]
Amodio, P., Gladwelly, I. and Romanazzi, G. (2006) Numerical Solution of General Bordered ABD Linear Systems by Cyclic Reduction. Journal of Numerical Analysis, Industrial and Applied Mathematics, 1, 5-12.
[21]
Doedel, E. (1991) Numerical Analysis and Control of Bifurcation Problems (I): Bifurcation in Finite Dimensions. International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, 1, 493-520.
http://dx.doi.org/10.1142/S0218127491000397
[22]
Duff, I.S. and Scott, J.A. (2004) Stabilized Bordered Block Diagonal Forms for Parallel Sparse Solvers, RAL-TR-2004-006. http://www.numerical.rl.ac.uk/reports/reports.html
[23]
da Fonseca, C.M. (2006) A Note on the Inversion of Acyclic Matrices. International Journal of Pure and Applied Mathematics, 31, 307-317.
[24]
Martin, A. and Boyd, I.D. (2010) Variant of the Thomas Algorithm for Opposite-Bordered Tridiagonal Systems of Equations. International Journal for Numerical Methods in Biomedical Engineering, 26, 752-759.
http://dx.doi.org/10.1002/cnm.1172
[25]
Pajic, S. (2007) Power System State Estimation and Contingency Constrained Optimal Power Flow a Numerically Robust Implementation. PhD Thesis, Worcester-Polytechnic Institute, Worcester.
[26]
Rashidinia, J. and Jalilian, R. (2009) Non-Polynomial Spline Solution of Fourth-Order Obstacle Boundary-Value Problems. World Academy of Science, Engineering and Technology. International Scholarly and Scientific Research & Innovation, 3, 167-173.
[27]
Udala, A., Reedera, R., Velmrea, E. and Harrisonb, P. (2006) Comparison of Methods for Solving the Schrodinger Equation for Multiquantum Well Heterostructure Applications. Proceedings of the Estonian Academy of Sciences, Engineering, 12, 246-261.
[28]
Wang, X.B. (2009) A New Algorithm with Its Scilab Implementation for Solution of Bordered Tridiagonal Linear Equations. 2009 IEEE International Workshop on Open-Source Software for Scientific Computation (OSSC), Guiyang, 18-20 September 2009, 11-14.
[29]
Golub, G. and Van Loan, C. (1996) Matrix Computations. Third Edition, The Johns Hopkins University Press, Baltimore and London.
[30]
El-Mikkawy, M.E.A. and Atlan, F. (2014) Algorithms for Solving Doubly Bordered Tridiagonal Linear Systems. British Journal of Mathematics & Computer Science, 4, 1246-1267. http://dx.doi.org/10.9734/BJMCS/2014/8835
Karawia, A.A. (2013) Symbolic Algorithm for Solving Comrade Linear Systems Based on a Modified Stair-Diagonal Approach. Applied Mathematics Letters, 26, 913-918. http://dx.doi.org/10.1016/j.aml.2012.10.019
[33]
Karawia, A.A. (2012) A New Recursive Algorithm for Inverting a General Comrade Matrix. CoRR abs/1210.4662.
[34]
Karawia, A.A. and Rizvi, Q.M. (2013) On Solving a General Bordered Tridiagonal Linear System. International Journal of Mathematics and Mathematical Sciences, 33, 1160-1163.
