# Inverse problem for symmetrie tridiagonal matrices. Calculation of the system of discrete orthogonal polynomials with arbitrary weight

Inverse problem for symmetrie tridiagonal matrices. Calculation of the system of discrete... - Numerical methods for constructing symmetric tridiagonal matrices with prescribed distinct eigenvalues are studied. The first components of orthonormal eigenvectors or Symmetry* conditions are additionally given. An analytic formula for the eigenvectors is derived. Using numerical examples we show that the Lanczos method with an additional complete 'forced' orthonormalization gives perfect results even for high-order matrices. Without the additional complete 'forced' orthonormalization, good results are obtained for low-order matrices only. An algorithm for calculating the system of discrete orthogonal polynomials with arbitrary weight is proposed. 1. INTRODUCTION This work is devoted to numerical solution of an inverse 'symmetric' problem for symmetric tridiagonal matrices. The problem is encountered when reconstructing spectra in discrete quantum mechanics [7]. The wave motion is described in discrete quantum mechanics by the finite-difference Schr dinger equation [6] ( + 1) - 2() + ( - ()() = (), When V(n) = 0, we have an eigenvalue problem for the matrix 2 - 1 0 ... 0 0 0 -1 D= -1 ... -1 0 ... -1 The eigenvalues of D are known to be with the corresponding orthonormal basis of eigenvectors E· defined by S.I.Serdyukova Note that the eigenvectors are 'symmetric': (1.1) Inverse Problem http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Russian Journal of Numerical Analysis and Mathematical Modelling de Gruyter

# Inverse problem for symmetrie tridiagonal matrices. Calculation of the system of discrete orthogonal polynomials with arbitrary weight

20 pages

/lp/de-gruyter/inverse-problem-for-symmetrie-tridiagonal-matrices-calculation-of-the-aE8QX5Oed0
Publisher
de Gruyter
ISSN
0927-6467
eISSN
1569-3988
DOI
10.1515/rnam.1993.8.3.245
Publisher site
See Article on Publisher Site

### Abstract

- Numerical methods for constructing symmetric tridiagonal matrices with prescribed distinct eigenvalues are studied. The first components of orthonormal eigenvectors or Symmetry* conditions are additionally given. An analytic formula for the eigenvectors is derived. Using numerical examples we show that the Lanczos method with an additional complete 'forced' orthonormalization gives perfect results even for high-order matrices. Without the additional complete 'forced' orthonormalization, good results are obtained for low-order matrices only. An algorithm for calculating the system of discrete orthogonal polynomials with arbitrary weight is proposed. 1. INTRODUCTION This work is devoted to numerical solution of an inverse 'symmetric' problem for symmetric tridiagonal matrices. The problem is encountered when reconstructing spectra in discrete quantum mechanics [7]. The wave motion is described in discrete quantum mechanics by the finite-difference Schr dinger equation [6] ( + 1) - 2() + ( - ()() = (), When V(n) = 0, we have an eigenvalue problem for the matrix 2 - 1 0 ... 0 0 0 -1 D= -1 ... -1 0 ... -1 The eigenvalues of D are known to be with the corresponding orthonormal basis of eigenvectors E· defined by S.I.Serdyukova Note that the eigenvectors are 'symmetric': (1.1) Inverse Problem

### Journal

Russian Journal of Numerical Analysis and Mathematical Modellingde Gruyter

Published: Jan 1, 1993

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