Tridiagonal matrix algorithm

Abstract: Non-symmetric and symmetric twisted block factorizations of block tridiagonal matrices are discussed. In contrast to non-blocked factorizations of this type, localized pivoting strategies can be integrated which improves numerical stability without causing any extra fill-in. Moreover, the application of such factorizations for approximating an eigenvector of a block tridiagonal matrix, given an approximation of the corresponding eigenvalue, is outlined. A heuristic strategy for determining a suitable starting vector for the underlying inverse iteration process is proposed.

Abstract: Binary LFSRs with tridiagonal matrices are interesting for their application to the design of very fast stream ciphers. Recently, explicit existence conditions for tridiagonal matrices with given characteristic polynomial have been reported. Here, these conditions are reviewed with the aim of giving an easily accessible and unified view of methods and proofs.

Abstract: This paper concerns the LBM T factorization of unsymmetric tridiagonal matrices, where L and M are unit lower triangular matrices and B is block diagonal with 1×1 and 2×2 blocks. In some applications, it is necessary to form this factorization without row or column interchanges while the tridiagonal matrix is formed. Bunch and Kaufman proposed a pivoting strategy without interchanges specifically for symmetric tridiagonal matrices, and more recently, Bunch and Marcia proposed pivoting strategies that are normwise backward stable for linear systems involving such matrices. In this paper, we extend these strategies to the unsymmetric tridiagonal case and demonstrate that the proposed methods both exhibit bounded growth factors and are normwise backward stable. Copyright © 2009 John Wiley & Sons, Ltd.

Abstract: An algorithm for computing the inverse of a general tridiagonal matrix is introduced. This algorithm is obtained by factoring this matrix into the product of two bidiagonal matrices using Crout’s LU factorization, one upper and one lower bidiagonal. A simple recurrence relation is used to generate a sequence of numbers, this sequence is then used to fill in the matrices L, U, L −1, U −1 and consequently the required inverse.