The numerically stable reconstruction of Jacobi matrices from spectral data

Summary We present an exposé of the elementary theory of Jacobi matrices and, in particular, their reconstruction from the Gaussian weights and abscissas. Many recent works propose use of the diagonal Hermitian Lanczos process for this purpose. We show that this process is numerically unstable. We recall Rutishauser's elegant and stable algorithm of 1963, based on plane rotations, implement it efficiently, and discuss our numerical experience. We also apply Rutishauser's algorithm to reconstruct a persymmetric Jacobi matrix from its spectrum in an efficient and stable manner.Dedicated to Professor F.L. Bauer on the occasion of his 60th birthdayThis work was supported by the National Science Foundation under grant MCS-81-02344, and by the Mathematics Research Institute of the Swiss Federal Institute of Technology, Zürich