Reconstruction of tridiagonal matrices from spectral data

Abstract

Jacobi matrices are parametrized by their eigenvalues and norming constants (first coordinates of normalized eigenvectors): this coordinate system breaks down at reducible tridiagonal matrices. The set of real symmetric tridiagonal matrices with prescribed simple spectrum is a compact manifold, admitting an open covering by open dense sets ${\cal U}^π_Λ$ centered at diagonal matrices $Λ^π$, where $π$ spans the permutations. {\it Bidiagonal coordinates} are a variant of norming constants which parametrize each open set ${\cal U}^π_Λ$ by the Euclidean space. The reconstruction of a Jacobi matrix from inverse data is usually performed by an algorithm introduced by de Boor and Golub. In this paper we present a reconstruction procedure from bidiagonal coordinates and show how to employ it as an alternative to the de Boor-Golub algorithm. The inverse bidiagonal algorithm rates well in terms of speed and accuracy.