论文信息 - Fast block diagonalization of k-tridiagonal matrices

Fast block diagonalization of k-tridiagonal matrices

abstract In the present paper, we give a fast algorithm for block diagonalization of k-tridiagonalmatrices. The block diagonalization provides us with some useful results: e.g., another der-ivation of a very recent result on generalized k-Fibonacci numbers in [M.E.A. El-Mikkawy,T. Sogabe, A new family of k-Fibonacci numbers, Appl. Math. Comput. 215 (2010) 4456–4461]; efﬁcient (symbolic) algorithm for computing the matrix determinant. 2011 Elsevier Inc. All rights reserved. 1. Introduction and objectivesLet T ðkÞn be the k-tridiagonal matrix of order n n, i.e.T ðkÞn :¼d 1 0 0 a 1 0 00 d 2 0 ...0 a 2 ... ...... 0 ...0 ... ... ...00 ... ...d n k ... ... ...a n k b kþ1 0... ... ... ... ...00 b kþ2 ... ...0 ...0 ...... ... ...0 ...0 d n 1 00 0 b n 0 0 d n 0BBBBBBBBBBBBBBBBBBB@1CCCCCCCCCCCCCCCCCCCA: ð1ÞFor example, T ð3Þ7 is given as follows:T ð3Þ7 ¼d 1 00a 1 00 00 d 2 00a 2 0000d 3 00a 3 0b 4 00d 4 00a 4 0 b 5 00d 5 0000b 6 00d 6 0000b 7 00d 7 0BBBBBBBBBBB@1CCCCCCCCCCCA: ð2Þ 0096-3003/$ - see front matter 2011 Elsevier Inc. All rights reserved.doi:10.1016/j.amc.2011.08.014

Moawwad E. A. El-Mikkawy | Tomohiro Sogabe

[1] Aynur Yalçiner. THE LU FACTORIZATIONS AND DETERMINANTS OF THE K-TRIDIAGONAL MATRICES , 2011 .

[2] Timothy A. Davis,et al. Direct methods for sparse linear systems , 2006, Fundamentals of algorithms.

[3] Moawwad E. A. El-Mikkawy,et al. A new family of k-Fibonacci numbers , 2010, Appl. Math. Comput..

[4] Emrah Kilic. On a constant-diagonals matrix , 2008, Appl. Math. Comput..

[5] Palmer R. Schlegel,et al. The Inverse of a Tridiagonal Matrix , 1972 .

[6] Moawwad E. A. El-Mikkawy,et al. A fast algorithm for evaluating n th order tri-diagonal determinants , 2004 .

[7] Damian Slota,et al. On computing the determinants and inverses of some special type of tridiagonal and constant-diagonals matrices , 2007, Appl. Math. Comput..

[8] Ke Chen,et al. Applied Mathematics and Computation , 2022 .

[9] N. Higham. Efficient algorithms for computing the condition number of a tridagonal matrix , 1986 .

[10] Jack Dongarra,et al. Templates for the Solution of Algebraic Eigenvalue Problems , 2000, Software, environments, tools.