The numerically stable reconstruction of a Jacobi matrix from spectral data

[摘要] We show how to construct, from certain spectral data, a discrete inner product for which the associated sequence of monic orthogonal polynomials coincides with the sequence of appropriately normalized characteristic polynomials of the left principal submatrices of the Jacobi matrix. The generation of these orthogonal polynomials via their three term recurrence relation, as popularized by Forsythe, then provides a stable means of computing the entries of the Jacobi matrix. The resulting algorithm might be of help in the approximate solution of inverse eigenvalue problems for Sturm-Liouville equations. Our construction provides, incidentally, very simple proofs of known results concerning existence and uniqueness of a Jacobi matrix satisfying given spectral data and its continuous dependence on that data.

[学科分类] 计算机科学（综合）