The numerically stable reconstruction of a Jacobi matrix from spectral data

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. Our construction provides, incidentally, very simple proofs of known results concerning the existence and uniqueness of a Jacobi matrix satisfying given spectral data and its continuous dependence on those data.     

On the construction of a Jacobi matrix from mixed given data

It is shown that a n×n Jacobi matrix is uniquely determined by its n eigenvalues and by the selected set of n ? 1 entries in the matrix.     

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     

Determination of an infinite non-self-adjoint Jacobi matrix from its generalized spectral function

Restoration from the generalized spectral function of the equations $$b_0 y_0 + a_0 y_1 = \lambda y_{0,} a_{n - 1} y_{n - 1} + b_n y_n + a_n y_{n + 1} = \lambda y_{n,} n = 1,2,3,...,$$ wherea _n and b_n are arbitrary complex numbers,a _n≠ 0 (n = o, i, 2,...), λ is a complex parameter, and {P _n (λ)} ₀ ^∞ is the required solution, is investigated. Necessary and sufficient conditions for solvability of the inverse problem are obtained, and the restoration procedure is described.     

On an inverse spectral problem for a quadratic Jacobi matrix pencil

Given two monic polynomials P2n and P2n−2 of degree 2n and 2n−2 (n?2) with complex coefficients and with disjoint zero sets. We give necessary and sufficient conditions on these polynomials such that there exist two n×n Jacobi matrices B and C for which

     

Construction of band matrices from spectral data

A stable numerical algorithm is presented to generate a symmetric p-band matrix from the given eigenvalues of the p greatest leading submatrices. The algorithm consists of two parts. First a matrix with the given spectral data is constructed; then this matrix is transformed into a p-band matrix leaving invariant the eigenvalues of the p greatest leading submatrices.     

A Jacobi matrix inverse eigenvalue problem with mixed data

In this paper, the inverse eigenvalue problem of reconstructing a Jacobi matrix from part of its eigenvalues and its leading principal submatrix is considered. The necessary and sufficient conditions for the existence and uniqueness of the solution are derived. Furthermore, a numerical algorithm and some numerical examples are given.     

On a spectral property of Jacobi matrices

Let be a Jacobi matrix with elements on the main diagonal and elements on the auxiliary ones. We suppose that is a compact perturbation of the free Jacobi matrix. In this case the essential spectrum of coincides with , and its discrete spectrum is a union of two sequences 2, x^-_j<-2$">, tending to . We denote sequences and by and , respectively.

The main result of the note is the following theorem.

Theorem.     Let be a Jacobi matrix described above and be its spectral measure. Then if and only if

-\infty,\qquad {ii)} \sum_j(x^\pm_j\mp2)^{7/2}<\infty. \end{displaymath}">

     

The spectral measure of a Jacobi matrix in terms of the Fourier transform of the perturbation

We study the spectral properties of Jacobi matrices. By combining Killip's technique [12] with the technique of Killip and Simon [13] we obtain a result relating the properties of the elements of Jacobi matrices and the corresponding spectral measures. This theorem is a natural extension of a recent result of Laptev-Naboko-Safronov [17]. The author thanks Sergei Naboko for useful discussions and Barry Simon for pointing out the conjecture.     

Spectrum of a Jacobi matrix with exponentially growing matrix elements

A Jacobi matrix with an exponential growth of its elements and the corresponding symmetric operator are considered. It is proved that the eigenvalue problem for some self-adjoint extension of this operator in some Hilbert space is equivalent to the eigenvalue problem of the Sturm-Liouville operator with a discrete self-similar weight. An asymptotic formula for the distribution of eigenvalues is obtained.     

On the spectral radius of the Jacobi iteration matrix for a rectangular region with two different media

The spectral radius of the Jacobi iteration matrix plays an important role to estimate the optimum relaxation factor, when the successive overrelaxation (SOR) method is used for solving a linear system. The specific systems are finite difference forms of the Laplace equation satisfied on a rectanglar region with two different media. Though the potential function for the inhomogeneous closed region is continuous, the first order derivative is not continuous. So this requires internal boundary conditions or interface conditions. In this paper, the spectral radius of the Jacobi iteration matrix for the inhomogeneous rectangular region is formulated and the approximation for the explicit formula, suitable for the computation of the spectral radius, is deduced. It is also found by the proposed formula that the spectral radius and the optimum relaxation factor rigorously depend on the inhomogeneity or the internal boundary conditions in the closed region, and especially vary with the position of the internal boundary. These findings are also confirmed by the numerical results of the power method.The stationary iterative method using the proposed formula for calculating estimates of the spectral radius of the Jacobi iteration matrix is compared with Carré's method, Kulstrud's method and the stationary iterative method using Frankel's theoretical formula, all for the case of some numerical models with two different media. According to the results our stationary iterative method gives the best results ffor the estimate of the spectral radius of the Jacobi iteration matrix, for the required number of iterations to calculate solutions, and for the accuracy of the solutions.As a numerical example the microstrip transmission line is taken, the propating mode of which can be approximated by a TEM mode. The cross section includes inhomogeneous media and a strip conductor. Upper and lower bounds of the spectral radius of the Jacobi iteration matrix are estimated. Our method using these estimates is also compared with the other methods. The upper bound of the spectral radius of the Jacobi iteration matrix for more general closed regions with two different media might be given by the proposed formula.     

Nonnegative matrix factorization for spectral data analysis

Data analysis is pervasive throughout business, engineering and science. Very often the data to be analyzed is nonnegative, and it is often preferable to take this constraint into account in the analysis process. Here we are concerned with the application of analyzing data obtained using astronomical spectrometers, which provide spectral data, which is inherently nonnegative. The identification and classification of space objects that cannot be imaged in the normal way with telescopes is an important but difficult problem for tracking thousands of objects, including satellites, rocket bodies, debris, and asteroids, in orbit around the earth. In this paper, we develop an effective nonnegative matrix factorization algorithm with novel smoothness constraints for unmixing spectral reflectance data for space object identification and classification purposes. Promising numerical results are presented using laboratory and simulated datasets.     

A spectral equivalence for Jacobi matrices

We use the classical results of Baxter and Golinskii–Ibragimov to prove a new spectral equivalence for Jacobi matrices on . In particular, we consider the class of Jacobi matrices with conditionally summable parameter sequences and find necessary and sufficient conditions on the spectral measure such that and lie in or for s1.     

Inverse spectral theory of finite Jacobi matrices

We solve the following physically motivated problem: to determine all finite Jacobi matrices and corresponding indices such that the Green's function

is proportional to an arbitrary prescribed function . Our approach is via probability distributions and orthogonal polynomials.

We introduce what we call the auxiliary polynomial of a solution in order to factor the map

(where square brackets denote the equivalence class consisting of scalar multiples). This enables us to construct the solution set as a fibration over a connected, semi-algebraic coordinate base. The end result is a wealth of explicit constructions for Jacobi matrices. These reveal precise geometric information about the solution set, and provide the basis for new existence theorems.

     

Construction of Jacobi forms

Partially supported KOSEF Research Grant 91-08-00-07 and KOSEF 921-0100-018-2     

Determination of a differential pencil from interior spectral data

In this paper, inverse spectra problems for a differential pencil are studied. By using the approach similar to those in Hochstadt and Lieberman (1978) [14] and Ramm (2000) [26], we prove that (1) if p(x) (or q(x)) is full given on the interval [0,π], then a set of values of eigenfunctions at the mid-point of the interval [0,π] and one spectrum suffice to determine q(x) (or p(x)) on the interval [0,π] and all parameters in the boundary conditions; (2) if p(x) (or q(x)) is full given on the interval [0,π], then some information on eigenfunctions at some internal point and parts of two spectra suffice to determine q(x) (or p(x)) on the interval [0,π] and all parameters in the boundary conditions.     

Characterization of the absence of a spectral gapfor a class of periodic Jacobi operators

Let q ∈ {2, 3} and let 0 = s₀ < s₁ < … < s_q = T be integers. For m, n ∈ Z, we put ¯m,n = {j ∈ Z| m? j ? n}. We set l_j = s_j − s_j−1 for j ∈ 1, q. Given (p₁,, p_q) ∈ R^q, let b: Z → R be a periodic function of period T such that b(·) = p_j on s_j−1 + 1, s_j for each j ∈ 1, q. We study the spectral gaps of the Jacobi operator (Ju)(n) = u(n + 1) + u(n − 1) + b(n)u(n) acting on l²(Z). By [λ_2j , λ_2j−1] we denote the jth band of the spectrum of J counted from above for j ∈ 1, T. Suppose that p_m ≠ p_n for m ≠ n. We prove that the statements (i) and (ii) below are equivalent for λ ∈ R and i ∈ 1, T − 1.