For all hyperbolic polynomials we proved in [11] a Lipschitz estimate of Jacobi matrices built by orthogonalizing polynomials with respect to measures in the orbit of classical Perron-Frobenius-Ruelle operators associated to hyperbolic polynomial dynamics (with real Julia set). Here we prove that for all sufficiently hyperbolic polynomials this estimate becomes exponentially better when the dim...

The Jacobi matrix of the (2n+1)-point Gauss-Kronrod quadrature rule for a given measure is calculated efficiently by a five-term recurrence relation. The algorithm uses only rational operations and is therefore also useful for obtaining the Jacobi-Kronrod matrix analytically. The nodes and weights can then be computed directly by standard software for Gaussian quadrature formulas.

The asymptotic convergence behavior of cyclic versions of the nonsymmetric Jacobi algorithm for the computation of the Schur form of a general complex matrix is investigated. Similar to the symmetric case, the nonsymmetric Jacobi algorithm proceeds by applying a sequence of rotations that annihilate a pivot element in the strict lower triangular part of the matrix until convergence to the Schur...

We study spaces of reflectionless Jacobi matrices. The main theme is the following type of question: Given a reflectionless Jacobi matrix, is it possible to approximate it by other reflectionless and, typically, simpler Jacobi matrices of a special type? For example, can we approximate by periodic operators?

If a Jacobi matrix J is reflectionless on (−2, 2) and has a single an0 equal to 1, then J is the free Jacobi matrix an ≡ 1, bn ≡ 0. I’ll discuss this result and its generalization to arbitrary sets and present several applications, including the following: if a Jacobi matrix has some portion of its an’s close to 1, then one assumption in the Denisov-Rakhmanov Theorem can be dropped. 1. Statemen...

A Jacobi matrix with matrix entries is a self-adjoint block tridiagonal matrix with invertible blocks on the off-diagonals. The Weyl surface describing the dependence of Green’s matrix on the boundary conditions is interpreted as the set of maximally isotropic subspace of a quadratic from given by the Wronskian. Analysis of the possibly degenerate limit quadratic form leads to the limit point/l...

We show that the finite Toda lattice with complex-valued initial data can be integrated by the inverse spectral method. For this goal spectral data for complex Jacobi matrices are introduced and an inverse spectral problem from the spectral data is solved. The time evolution of the spectral data for the Jacobi matrix associated with the solution of the Toda lattice is computed. Using the soluti...

In this paper, we investigate the Zero-Forcing equalizer based on Fast One Sided Jacobi (ZF-FOSJ) algorithm for the Multiple Input Multiple Output Power Line Communication (MIMO PLC) systems. The proposed algorithm determines the pseudo-inverse of estimated channel matrix using only real matrix and operations free from trigonometric functions. The hardware implementation complexity is reduced c...

Introduction Non-invasive Fluorescent Reflectance Imaging (FRI) is used for accessing physiological and molecular processes in biological media. The aim of this article is to separate the overlapping emission spectra of quantum dots within tissue-equivalent phantom using SVD, Jacobi SVD, and NMF methods in the FRI mode. Materials and Methods In this article, a tissue-like phantom and an optical...