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Jacobi operator
About: Jacobi operator is a research topic. Over the lifetime, 1378 publications have been published within this topic receiving 21789 citations.
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TL;DR: This note gives the required Jacobi angles in close form for simultaneous diagonalization of several matrices.
Abstract: Simultaneous diagonalization of several matrices can be implemented by a Jacobi-like technique. This note gives the required Jacobi angles in close form.
903 citations
Book•
01 Jan 1999TL;DR: In this paper, the Toda system and the Kac-van Moerbeke system are studied. But the initial value problem is not considered in this paper, as it is in the case of Jacobi operators with periodic coefficients.
Abstract: Jacobi operators: Jacobi operators Foundations of spectral theory for Jacobi operators Qualitative theory of spectra Oscillation theory Random Jacobi operators Trace formulas Jacobi operators with periodic coefficients Reflectionless Jacobi operators Quasi-periodic Jacobi operators and Riemann theta functions Scattering theory Spectral deformations-Commutation methods Completely integrable nonlinear lattices: The Toda system The initial value problem for the Toda system The Kac-van Moerbeke system Notes on literature Compact Riemann surfaces-A review Hergoltz functions Jacobi difference equations with MathematicaR Bibliography Glossary of notations Index.
782 citations
Book•
27 Aug 2011
TL;DR: In this article, it was shown that Jacobi's method is optimally accurate in the sense that small relative errors in the entries of a matrix cause small errors in its eigenvalues.
Abstract: It is shown that Jacobi’s method (with a proper stopping criterion) computes small eigenvalues of symmetric positive definite matrices with a uniformly better relative accuracy bound than QR, divide and conquer, traditional bisection, or any algorithm which first involves tridiagonalizing the matrix. Modulo an assumption based on extensive numerical tests, Jacobi’s method is optimally accurate in the following sense: if the matrix is such that small relative errors in its entries cause small relative errors in its eigenvalues, Jacobi will compute them with nearly this accuracy. In other words, as long as the initial matrix has small relative errors in each component, even using infinite precision will not improve on Jacobi (modulo factors of dimensionality). It is also shown that the eigenvectors are computed more accurately by Jacobi than previously thought possible. Similar results are proved for using one-sided Jacobi for the singular value decomposition of a general matrix.
318 citations
TL;DR: In this article, the Jacobi elliptic function method with symbolic computation is extended to special-type nonlinear equations for constructing their doubly periodic wave solutions, such as the coupled Schrodinger-KdV equation.
308 citations