Algorithms for the Reconstruction of Special Jacobi Matrices from Their Eigenvalues

Abstract

Two algorithms for the reconstruction of symmetric tridiagonal (not necessarily persymmetric ) matrices J with subdiagonal entries equal to one from their eigenvalues are established. The first algorithm is an iteration method using orthogonal similarity transformations in the sense of an inverted Jacobi algorithm and is shown to be locally convergent. Since reconstruction problems are often rather ill-conditioned, the algorithm may be slow, but it gives good approximations $J'$ to J. $J'$ may be used as a starting value for the second algorithm, a Newton method iterating the characteristic polynomial of $J'$. Numerical examples demonstrate the convergence behavior, also for nonpersymmetric matrices J.