A fast numerical algorithm for solving nearly penta-diagonal linear systems

Abstract

In this paper, we present a fast numerical algorithm for solving nearly penta-diagonal linear systems and show that the computational cost is less than those of three algorithms in El-Mikkawy and Rahmo, [Symbolic algorithm for inverting cyclic penta-diagonal matrices recursively–Derivation and implementation, Comput. Math. Appl. 59 (2010), pp. 1386–1396], Lv and Le [A note on solving nearly penta-diagonal linear systems, Appl. Math. Comput. 204 (2008), pp. 707–712] and Neossi Nguetchue and Abelman [A computational algorithm for solving nearly penta-diagonal linear systems, Appl. Math. Comput. 203 (2008), pp. 629–634.]. In addition, an efficient way of evaluating the determinant of a nearly penta-diagonal matrix is also discussed. The algorithm is suited for implementation using computer algebra systems (CAS) such as MATLAB, MACSYMA and MAPLE. Some numerical examples are given in order to illustrate the efficiency of our algorithm.

Keywords:

• nearly penta-diagonal matrices
• Sherman–Morrison–Woodbury formula
• linear systems
• determinants
• computer algebra systems (CAS)

2010 AMS Subject Classification :

• 15A09
• 15A23
• 65F30
• 33F10

Acknowledgements

This work was supported by the Natural Science Foundation of China (NSFC) under grant 11071192 and the International Science and Technology Cooperation Program of China under grant 2010DFA14700. The authors would like to thank the referees who substantially enhanced the quality of the paper.