The numerically stable reconstruction of Jacobi matrices from spectral data

Abstract

Summary. We present an expos6 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.

m-Functions and inverse spectral analysis for finite and semi-infinite Jacobi matrices

“…We prove an extension of Hochstadt's theorem (who proved the result in the case n = N ) that n eigenvalues of an N × N Jacobi matrix, H, can replace the first n matrix elements in determining H uniquely. We completely solve the inverse problemThere is an enormous literature on inverse spectral problems for − [1,29,[56][57][58][59][60]64] and references therein), but considerably less for their discrete analog, the infinite and semi-infinite Jacobi matrices (see, e.g., [3, 4, 6-8, 13-22, 24, 26-28, 30, 32, 37, 38, 42-44, 50-52, 61-63, 66, 67, 69-71]) and even less for finite Jacobi matrices (where references include, e.g., [9,10,23,25,39,40,41,[45][46][47][48]). Our goal in this paper is to study the last two problems using one of the most powerful tools from the spectral theory of − d 2 dx 2 + V (x), the m-functions of Weyl.…”

Discrete Least Squares Approximation by Trigonometric Polynomials

“…Numerical examples in §5 illustrate that for many distributions of nodes the Stieltjes procedure is very sensitive to roundoff errors. This parallels the behavior of the Stieltjes procedure for the generation of polynomials orthogonal on a point set in a real interval [15,11,7].…”

“…One can therefore use other numerical methods for this inverse eigenvalue problem to generate the recurrence coefficients and, implicitly, the values of the orthogonal polynomials. In particular, an efficient method for this problem that uses elementary orthogonal similarity transformations, due to Rutishauser, is described in [11]. This method is applied to the generation of orthogonal polynomials that satisfy a three-term recurrence relation in [15], and it is observed that this method is numerically more reliable than the classical Stieltjes procedure.…”

A parallel algorithm for discrete least squares rational approximation

“…, Q m ] . From the relation 5) it follows that the property is true for the first block column of Q:…”

“…It generalizes the algorithm of Reichel [6] for the polynomial case. The latter is inspired by the Rutishauser-Gragg-Harrod algorithm [8,5,1] for the computation of Jacobi matrices. A similar way of reasoning was followed by Golub and others [3,4].…”