Eigenvalue Problem for Infinite Matrices and Its Applications

无限矩阵的特征值问题及其应用

• 批准号: 06640290
• 负责人: IKEBE Yasuhiko
• 金额: $ 1.41万
• 依托单位: University of Aizu (1995)University of Tsukuba (1994)
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for General Scientific Research (C)
• 财政年份: 1994
• 资助国家: 日本
• 起止时间: 1994 至 1995
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-06640290/
• 关键词: Eigenvalue of Infinite Matrices; Conjugate Symmetric Tridiagonal Matrial; Special Function; Bessel Function; Mathive Function; Visualization; 可視化

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 maximal domain in the Hilbert space l^2 into l^2. Assuming the existence of T^<-1> we consider the problem of approximating a given simple eigenvalu lambda of T by an eigenvalue lambda_n of T_nthe n-th order principal submatrix of T.Let T=[x^<(1)>, x^<(2)>, ...]^T be an eigenvector corresponding to lambda. Assuming 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>^<(n)>x^<(n+1)>[1+o(1)]/(X^TX)*0, where f_<n+1> represents the (n, n+1) element of T.Application to the following problems is included : (a) solve J_<nu>(z)=0 for nu, given z * 0 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（不一定是埃尔米特）三对角矩阵T，其对角元素的模发散到 *，其非对角元素有界。我们把T看作是Hilbert空间l^2中极大域到l^2的线性映射。在T^存在<-1>的前提下，考虑用T_n的n阶主子阵的特征值λ_n逼近T的给定单特征值λ的问题。设T=[x^&lt;（1）&gt;，x^&lt;（2）&gt;，.]^ T是对应于λ的特征向量。设X^T X * 0和f_&lt;n+1&gt;x^&lt;（n+1）&gt;/x^&lt;（n）&gt;*0为n**，我们证明存在T_n的序列{lambda_n}使得λ_n=f_&lt;n+1&gt;^&lt;（n）&gt;x^&lt;（n+1）&gt;[1+o（1）]/（X^TX）*0，其中f_&lt;n+1&gt;表示T的（n，n+1）元。（a）求出J_<nu>（z）=0的nu，给定z * 0;（B）计算Mathieu方程的特征值。幸运的是，对于这些例子，T^的存在性<-1>不需要验证，因为我们可以证明，适当地取α的T+ α I具有逆。

项目成果

稲垣敏之: "Adaptive Choice of a Safety Management Scheme upon an Alarm under Supervisory Controre of a Large-Complex System" Reliability Enginnering and System Safety. 31. 81-87 (1993)

Toshiyuki Inagaki：“大型复杂系统监控下报警时安全管理方案的自适应选择”可靠性工程和系统安全。31. 81-87 (1993)。

池辺八洲彦 他: "行列算法によるzJ′ν(Z)+HJν(z)=0の数値解法" 電子情報通信学会. (採録決定).

Yasuhiko Ikebe 等人：“使用矩阵算术的 zJ′ν(Z)+HJν(z)=0 的数值解”IEICE（已接受）。

DongSheng Cai: "A data-parallel particle-in-cell code using a scan-vector model" Supercomputing‘95. (to appear).

DongSheng Cai：“使用扫描矢量模型的数据并行粒子内代码”超级计算95（即将出现）。

Y.Ikebe, N.Asai, Y.Miyazaki, and D.Cai: "The Eigenvalue Problem for Infinite Complex Symmetric Tridiagonal Matrices with Application" to appear in Linear Algebra and Its Applications.

Y.Ikebe、N.Asai、Y.Miyazaki 和 D.Cai：“无限复对称三对角矩阵的特征值问题及其应用”出现在《线性代数及其应用》中。

池辺八洲彦 他3名: "Infinite matrices and special functions" Proc.Int.Workshop on Inverse Problems. (1995)

Yasuhiko Ikebe 和其他 3 人：“无限矩阵和特殊函数”Proc.Int.Workshop on Inverse Problems (1995)。