Engenvalue Problem of Infinite Matrices and its Application.

无限矩阵的Engenvalue问题及其应用。

• 批准号: 09640284
• 负责人: IKEBE Yasuhiko
• 金额: $ 1.98万
• 依托单位: Univeristy of Aizu
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 1997
• 资助国家: 日本
• 起止时间: 1997 至 1998
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-09640284/
• 关键词: Eigenvalue of Infinite Metrix; Conjugate Symmetric Tridiagonal Matrices; Special Function; Bessel Function; Mathiue Function; Visualization

We consider an infinite complex symmetric (not necessarily Hermitian)tridiagonal matrix T whose diagonal elements diverge to * in modulus and whose off-diagonal elements are bounded. We regard T as a linear operator mapping a maximal domai n in the Hilbert space l^2 into l^2. Assuming the existence of T^<-1> we consider the problem of approximating a given simple eigenvalue lambda of T by an eigen value lambda_n of T_n, the n-th order principal submatrix of T.Let X = [x^<(1)>, x^<(2)>, ...]^T be an eigenvector corresponding to lambda. Assuning X^T X * *0 and f_<n+1> x^<(n+1)>/x^n * 0 as n * *, we will show that there exists a sequence [lambda_n] of T_n such that lambda-lambda _n=f_<n+1> x^<(n+1)> x^n[1+omicron(1)]/(X^T X) * 0, where f_<n+1> represents the (n, n+1) element of T.Application to the following problems is included : (a) solve Jv(z) = 0 for v, given z * O and (b)compute the eigenvalues of the Mathieu equation. Fortunately, the existence of T^<-1> need not be verified for these examples since we may show that T + alphaI with alpha taken appropriately has an inverse.

我们考虑一个无限复对称（不一定是厄米）三对角矩阵T，它的对角元素在模中发散到*，其非对角元素是有界的。我们把T看作一个线性算子将希尔伯特空间中的极大定义域n映射到l^2。假设T^<-1>存在，我们考虑用T的n阶主矩阵T_n的特征值lambda_n逼近给定的简单特征值λ (T)的问题。设X = [X ^<(1)>, X ^<(2)>，…]^T是对应于的特征向量。假设X^T X * *0和f_<n+1> X^ <(n+1)>/ X^ n *0为n * *，我们将证明T_n存在一个序列[lambda_n]，使得λ - lambda_n =f_<n+1> X^ <(n+1)> X^ n[1+omicron(1)]/(X^T X) *0，其中f_<n+1>表示T的（n, n+1）元素，包括以下问题的应用：(a)在给定z * O的情况下求解v的Jv(z) = 0; (b)计算Mathieu方程的特征值。幸运的是，对于这些例子，T^<-1>的存在性不需要验证，因为我们可以证明T + α α α与α α适当地取有一个逆。

项目成果

宮崎佳典,浅井信吉,蔡東生,池辺八州彦: "Mathieu微分方程式の逆固有値問題" 応用数理. 8. 199-222 (1998)

Yoshinori Miyazaki、Shinkichi Asai、Tongsei Cai、Yashuhiko Ikebe：“Mathieu 微分方程的反特征值问题”应用数学。 8. 199-222 (1998)

Y. Ikebe, Y. Kikuchi, N. Asai, Y. Miyazaki, D.Cai: "The Eigenvalue Problem for Infinite Matrices: New Area of Application of Numerical Linear Algebra." Proceedings of Fourth IMACS International Symposium on Scientific Computation (honoring Professor David

Y. Ikebe、Y. Kikuchi、N. Asai、Y. Miyazaki、D.Cai：“无限矩阵的特征值问题：数值线性代数应用的新领域。”

Asai, N., Y.Miyazaki, D.Cai, K.Hirasawa, and Y.Ikebe: "Matrix methods for the Numerical Solution of Jv'(z)+HJv(z)=0, (Selected Special Paper by Editor)" Electronics and Communications in Japan. Vol.807. 44-54 (1997)

Asai, N.、Y.Miyazaki、D.Cai、K.Hirasawa 和 Y.Ikebe：“Jv(z) HJv(z)=0 数值解的矩阵方法，（编辑精选专题论文）”

浅井信吉,宮崎佳典,蔡東生 平沢一紘,池辺八洲彦: "行列算法によるZJ^1_ν(Z)+HJ_2(Z)=0の数値解法" 電子情報通信学会論文誌. J79. 1256-1265 (1996)

Shinkichi Asai、Yoshinori Miyazaki、Dongsei Cai、Kazuhiro Hirasawa、Yasuhiko Ikebe：“使用矩阵算术的 ZJ^1_ν(Z)+HJ_2(Z)=0 的数值解”，电子信息与通信工程师学会汇刊 J79。 1256-1265（1996）

K.Miyazaki, N.Asai, D.Cai and Y.Ikebe: "Inverse Eigenvalue problem of Mathieu's Differential Equation" Bulletin of the Japan Society for Industrial and Applied Mathematics. No.8. 199-222 (1998)

K.Miyazaki、N.Asai、D.Cai 和 Y.Ikebe：“Mathieu 微分方程的反特征值问题”日本工业与应用数学学会会刊。