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DSF chain for biorthogonal polynomials and its application to matrix eigenvalue problems

JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS 2019年 347卷 273-295页

作者： Mukaihira, Atsushi Tsuboi, Yoshimasa Doshisha Univ Fac Sci & Engn Kyotanabe Kyoto Japan

A discrete-time integrable system, called the DSF chain, is derived from the compatibility condition of spectral equations for biorthogonal polynomials associated with the discrete Schur flow. The DSF algorithm for computing eigenvalues of a type of matrix eigenvalue problem is designed. A discrete analogue of the Miura transformation between the discrete Schur flow and the qd-algorithm is constructed. Two special solutions to the DSF chain related to q-special functions are presented. (C) 2018 Elsevier B.V. All rights reserved.

关键词： Schur flow Qd-algorithm matrix eigenvalue problems Numerical algorithms Integrable systems Biorthogonal polynomials

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eigenvalues of a class of fourth-order boundary value problems with transmission conditions using matrix theory

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LINEAR & MULTILINEAR ALGEBRA 2021年第9期69卷 1610-1624页

作者： Ao, Ji-jun Sun, Jiong Inner Mongolia Univ Technol Coll Sci Hohhot 010051 Peoples R China Inner Mongolia Univ Sch Math Sci Hohhot Peoples R China

A class of fourth-order boundary value problems with general self-adjoint boundary and transmission conditions is investigated. It is shown that such boundary value problems with a finite number of eigenvalues are equivalent to certain types of matrix eigenvalue problems. Moreover, the results are given under the general self-adjoint transmission conditions and the self-adjoint canonical forms of the fourth-order problems which are classified as separated, mixed and coupled, respectively.

关键词： Fourth-order boundary value problems matrix eigenvalue problems transmission conditions self-adjoint boundary conditions canonical forms

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The study of Jacobi and cyclic Jacobi matrix eigenvalue problems using Sturm-Liouville theory

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LINEAR ALGEBRA AND ITS APPLICATIONS 2011年第7期434卷 1648-1655页

作者： Kong, Qingkai Zettl, Anton No Illinois Univ Dept Math De Kalb IL 60115 USA

We study the eigenvalues of matrix problems involving Jacobi and cyclic Jacobi matrices as functions of certain entries. Of particular interest are the limits of the eigenvalues as these entries approach infinity. Our approach is to use the recently discovered equivalence between these problems and a class of Sturm-Liouville problems and then to apply the Sturm-Liouville theory. (C) 2010 Elsevier Inc. All rights reserved.

关键词： matrix eigenvalue problems Jacobi and cyclic Jacobi matrices Sturm-Liouville theory

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The study of Jacobi and cyclic Jacobi matrix eigenvalue problems using Sturm-Liouville theory

The study of Jacobi and cyclic Jacobi matrix eigenvalue prob...

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Conference on Linear and Numerical Linear Algebra - Theory, Methods and Applications

作者： Kong, Qingkai Zettl, Anton No Illinois Univ Dept Math De Kalb IL 60115 USA

关键词： matrix eigenvalue problems Jacobi and cyclic Jacobi matrices Sturm-Liouville theory

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matrix Representations of Fourth-Order Boundary Value problems with Transmission Conditions

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MEDITERRANEAN JOURNAL OF MATHEMATICS 2016年第1期13卷 205-215页

作者： Kablan, Abdullah Manafov, Manaf Dzh. Gaziantep Univ Fac Arts & Sci Dept Math TR-27310 Gaziantep Turkey Adiyaman Univ Fac Arts & Sci Dept Math TR-02040 Adiyaman Turkey

In this study, we construct a certain class of matrix eigenvalue problems correspond to a class of regular fourth-order boundary value problems with transmission conditions of Atkinson type. The relation between bound... 详细信息

关键词： Boundary value problems matrix eigenvalue problems Transmission conditions

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matrix representations of fourth-order boundary value problems with coupled or mixed boundary conditions

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LINEAR & MULTILINEAR ALGEBRA 2015年第8期63卷 1590-1598页

作者： Ao, Ji-jun Sun, Jiong Inner Mongolia Univ Technol Coll Sci Hohhot Peoples R China Inner Mongolia Univ Sch Math Sci Hohhot Peoples R China

An equivalence between a class of regular self-adjoint fourth-order boundary value problems with coupled or mixed boundary conditions and a certain class of matrix problems is investigated. Such an equivalence was pre... 详细信息

关键词： 34B40 Primary 34B05 Secondary 15A60 coupled boundary conditions mixed boundary conditions matrix eigenvalue problems fourth-order boundary value problems

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matrix representations of Sturm-Liouville problems with coupled eigenparameter-dependent boundary conditions

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APPLIED MATHEMATICS AND COMPUTATION 2014年第0期244卷 142-148页

作者： Ao, Ji-jun Sun, Jiong Inner Mongolia Univ Technol Coll Sci Hohhot 010051 Peoples R China Inner Mongolia Univ Sch Math Sci Hohhot 010021 Peoples R China

We study the matrix representations of Sturm-Liouville problems with coupled eigenparameter-dependent boundary conditions with a finite spectrum. We prove for any positive integer n, the considered problems have at most n + 3 eigenvalues, and show that this kind of Sturm-Liouville problems with coupled eigenparameter-dependent boundary conditions is equivalent to a class of matrix eigenvalue problems in the sense that they have exactly the same eigenvalues. (C) 2014 Elsevier Inc. All rights reserved.

关键词： Sturm-Liouville problems matrix eigenvalue problems Finite spectrum Coupled eigenparameter-dependent boundary conditions

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matrix representations of Sturm-Liouville problems with transmission conditions

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COMPUTERS & MATHEMATICS WITH APPLICATIONS 2012年第8期63卷 1335-1348页

作者： Ao, Ji-jun Sun, Jiong Zhang, Mao-zhu Inner Mongolia Univ Sch Math Sci Hohhot 010021 Peoples R China Inner Mongolia Univ Technol Coll Sci Hohhot 010051 Peoples R China Taishan Univ Sch Math & Syst Sci Tai An 271021 Shandong Peoples R China

We identify a class of Sturm-Liouville equations with transmission conditions such that any Sturm-Liouville problem consisting of such an equation with transmission condition and an arbitrary separated or real coupled self-adjoint boundary condition has a representation as an equivalent finite dimensional matrix eigenvalue problem. Conversely, given any matrix eigenvalue problem of certain type and an arbitrary separated or real coupled self-adjoint boundary condition and transmission condition, we construct a class of Sturm-Liouville problems with this specified boundary condition and transmission condition, each of which is equivalent to the given matrix eigenvalue problem. (C) 2012 Elsevier Ltd. All rights reserved.

关键词： Sturm-Liouville problems Transmission condition matrix eigenvalue problems Finite spectrum

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matrix Representations of Sturm-Liouville problems with Finite Spectrum

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RESULTS IN MATHEMATICS 2009年第1-2期54卷 103-116页

作者： Kong, Qingkai Volkmer, Hans Zettl, Anton No Illinois Univ Dept Math Sci De Kalb IL 60115 USA Univ Wisconsin Dept Math Sci Milwaukee WI 53201 USA

We identify a class of Sturm-Liouville equations such that any Sturm-Liouville (SL) problem consisting of such an equation and an arbitrary separated or coupled real self-adjoint boundary condition has a representation as an equivalent finite dimensional matrix eigenvalue problem. Conversely, given any matrix eigenvalue problem of certain type and an arbitrary separated or coupled real self-adjoint (SL) boundary condition, we construct a class of Sturm-Liouville problems with the specified boundary condition, each of which is equivalent to the given matrix eigenvalue problem.

关键词： Sturm-Liouville problems matrix eigenvalue problems finite spectrum

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Arnoldi versus nonsymmetric Lanczos algorithms for solving matrix eigenvalue problems

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BIT NUMERICAL MATHEMATICS 1996年第3期36卷 470-493页

作者： Cullum, J IBM CORP THOMAS J WATSON RES CTR DIV RES DEPT MATH SCI YORKTOWN HTS NY 10598 USA

We present theoretical and numerical comparisons between Arnoldi and nonsymmetric Lanczos procedures for computing eigenvalues of nonsymmetric matrices. In exact arithmetic we prove that any type of eigenvalue convergence behavior obtained using a nonsymmetric Lanczos procedure may also be obtained using an Arnoldi procedure but on a different matrix and with a different starting vector. In exact arithmetic we derive relationships between these types of procedures and normal matrices which suggest some interesting questions regarding the roles of nonnormality and of the choice of starting vectors in any characterizations of the convergence behavior of these procedures. Then, through a set of numerical experiments on a complex Arnoldi and on a complex nonsymmetric Lanczos procedure, we consider the more practical question of the behavior of these procedures when they are applied to the same matrices.

关键词： matrix eigenvalue problems Arnoldi methods Lanczos methods

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