On the construction of real non-selfadjoint tridiagonal matrices with prescribed three spectra

Abstract

Non-selfadjoint tridiagonal matrices play a role in the discretization and truncation of the Schrödinger equation in some extensions of quantum mechanics, a research field particularly active in the last two decades. In this article, we consider an inverse eigenvalue problem that consists of the reconstruction of such a real nonselfadjoint matrix from its prescribed eigenvalues and those of two complementary principal submatrices. Necessary and sufficient conditions under which the problem has a solution are presented, and uniqueness is discussed. The reconstruction is performed by using a modified unsymmetric Lanczos algorithm, designed to solve the proposed inverse eigenvalue problem. Some illustrative numerical examples are given to test the efficiency and feasibility of our reconstruction algorithm.

A Modified Inverse Iteration Method for Computing the Symmetric Tridiagonal Eigenvectors

“…Let us take a typical example of clustered eigenvalues to illustrate. Let α 0 be a 200 × 1 vector and α 0 (i) = i(i ∈ [1,200]) and then construct α ← [ f lip(α 0 ); 0; α 0 ]. Then, repeat α ← [α; α 0 ] by eight times totally.…”

“…Note these extreme points may not be exactly the same as the envelope vector. We show |γ k |(k ∈ [1,2001]) of Φ in Figure 1. A logarithmic scale on the y-axis has been used to emphasize the small entries.…”

“…Computing the symmetric tridiagonal (ST) eigenvector is an important task in many research fields, such as the computational quantum physics [1], mathematics [2,3], dynamics [4], computational quantum chemistry [5], etc. The ST eigenvector problem also arises while solving any symmetric eigenproblem because it is a common practice to reduce the generalized symmetric eigenproblems to an ST one.…”

A Novel Divisional Bisection Method for the Symmetric Tridiagonal Eigenvalue Problem

“…The symmetric tridiagonal matrices often arise as primary data in many computational quantum physical [1,2], mathematical [3][4][5], dynamical [6,7], computational quantum chemical [8,9], signal processing [10], or even medical [11] problems and hence are important. The current software reduces the generalized and the standard symmetric eigenproblems to a symmetric tridiagonal eigenproblem as a common practice [10,12,13].…”