Tridiagonal matrix algorithm

Abstract: In this paper, we present a numerical algorithm for solving matrix equations $(A otimes B)X = F$ by extending the well-known Gaussian elimination for $Ax = b$. The proposed algorithm has a high computational efficiency. Two numerical examples are provided to show the effectiveness of the proposed algorithm.

Abstract: In this paper, we discuss the numerical integration with e xponential fitting factor for singularly perturbed two-point boundary value problems. It is based on t he fact that: the given SPTPBVP is replaced by an asymptotically equivalent delay differenti al equation. Then, numerical integration with exponential fitting factor is employed to obtain a tridiagonal system which is solved efficiently by Thomas algorithm. We discussed con vergence analysis of the method. Model examples are solved and the numerical results are compa red with exact solution.

Abstract: The paper contains two parts. First, by applying the results about the eigenvalue perturbation bounds for Hermitian block tridiagonal matrices in paper [1], we obtain a new efficient method to estimate the perturbation bounds for singular values of block tridiagonal matrix. Second, we consider the perturbation bounds for eigenvalues of Hermitian matrix with block tridiagonal structure when its two adjacent blocks are perturbed simultaneously. In this case, when the eigenvalues of the perturbed matrix are well-separated from the spectrum of the diagonal blocks, our eigenvalues perturbation bounds are very sharp. The numerical examples illustrate the efficiency of our methods.

Abstract: The general expression of the l-th power (l ∈ N, l ≥ 2) for one type of tridiagonal matrices of order n = 2p (p ∈ N, p ≥ 2) with zeros in the first row is derived. Expressions of eigenvectors of the matrix and of the transforming matrix and its inverse are given, too.