Reconstruction of tridiagonal matrices from spectral data

Details Reconstruction of tridiagonal matrices from spectral data Reconstruction of tridiagonal matrices from spectral data

2005-08-04

arxiv.org/abs/math/0508099v1

Mathematics

Numerical Analysis

10 pages, 1 figure

Scientific paper

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 ${\cal U}^\pi_\Lambda$ centered at diagonal matrices $\Lambda^\pi$, where $\pi$ spans the permutations. {\it Bidiagonal coordinates} are a variant of norming constants which parametrize each open set ${\cal U}^\pi_\Lambda$ 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.

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