Stable Factorizations of Symmetric Tridiagonal and Triadic Matrices

Stable Factorizations of Symmetric Tridiagonal and Triadic Matrices

We call a matrix triadic if it has no more than two nonzero off-diagonal elements in any column. A symmetric tridiagonal matrix is a special case. In this paper we consider LXLT factorizations of symmetric triadic matrices, where L is unit lower triangular and X is diagonal, block diagonal with 1×1 and 2×2 blocks, or the identity with L lower triangular. We prove that with diagonal pivoting, th...

On twisted factorizations of block tridiagonal matrices

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...

Fine Spectra of Tridiagonal Symmetric Matrices

Eigenvalues of symmetric tridiagonal interval matrices revisited

In this short note, we present a novel method for computing exact lower and upper bounds of a symmetric tridiagonal interval matrix. Compared to the known methods, our approach is fast, simple to present and to implement, and avoids any assumptions Our construction explicitly yields those matrices for which particular lower and upper bounds are attained.

Exponentials of Symmetric Matrices through Tridiagonal Reductions

A simple and efficient numerical algorithm for computing the exponential of a symmetric matrix is developed in this paper. For an n× n matrix, the required number of operations is around 10/3 n. It is based on the orthogonal reduction to a tridiagonal form and the Chebyshev uniform approximation of e−x on [0,∞).