Computing Special Functions in the Complex Domain through Matrix Equation Reformulation

通过矩阵方程重构计算复域中的特殊函数

• 批准号: 11640130
• 负责人: IKEBE Yasuhiko
• 金额: $ 2.18万
• 依托单位: University 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: 1999
• 资助国家: 日本
• 起止时间: 1999 至 2000
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-11640130/
• 关键词: Eigenvalue of Infinite Matrix; Computation of Special Functions; Computing Zeros; Multiplicity of Eignevalues; Regular Coulomb Wave Funcation; 複素対称三重対角行列; 特殊関数; ベッセル関数; マシュー関数; 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 a 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 eigenvalue λ of T by an eigenvalue λ_n of T_n, the n-th order principal sub-matrix of T.Let X=[x^<(1)>, x^<(2)>, _…]^T be an eigenvector corresponding to λ. Assuming X^T X≠O and f_<n+1> x^<(n+1)>/x^n→0 as n→∽, we will show that there exists a sequence {λ_n} of T_n such that λ-λ_n=f_<n+1> x^<(n+1)>x^n[1+o (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) Consider the multiplicity of zeros of Bessel Functions and eigenvalues of Mathieu's Equation, (b) Compute zeros of Regular Coulomb Wave Function and Consider the multiplicity of the zeros, and (c) Theoretically consider about Spheroidal Wave Function, Lame Function, and Ellipsoidal Function.

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，其对角元素发散到λ in模，其非对角元素有界。我们把T看作是一个线性算子，它将希尔伯特空间l^2中的极大域映射到l^2中。在T^存在的<-1>条件下，考虑T的n阶主子矩阵T_n的特征值λ_n逼近T的一个给定的简单特征值λ的问题。设X=[x^&lt;（1）&gt;，x^&lt;（2）&gt;，...]^T是λ对应的特征向量。设X^T X≠ 0，f_&lt;n+1&gt; x^&lt;（n+1）&gt;/x^n→0为n→ n，我们证明了存在T_n的序列{λ_n}使得λ-λ_n=f_&lt;n+1&gt; x^&lt;（n+1）&gt;x^n[1+o（1）]/（X^T X）→0，其中f_&lt;n+1&gt;表示T的（n，n+1）元。（a）考虑Bessel函数的零点重数和Mathieu方程的本征值;（B）计算正则库仑波函数的零点并考虑零点重数;（c）理论上考虑球面波函数、Lame函数和椭球函数。

项目成果

Y.Miyazaki,Y.Kikuchi,D.S.Cai,and Y.Ikebe : "The Computation of Double Eigenvalues for Infinite Matrices of a Certain Class with Newton's Method"Abstracts of Plenary and Invited Lectures Delivered at the Second Congress ISAAC 1999. 148-149 (1999)

Y.Miyazaki、Y.Kikuchi、D.S.Cai 和 Y.Ikebe：“用牛顿法计算某类无限矩阵的双特征值”ISAAC 1999 年第二次大会全体会议和特邀报告摘要。148-149

蔡東生 ほか: "アジソンウェスレイートッパン"マルチメディアフラクタル画像圧縮. 500 (1995)

蔡东升等人：“Addison-Wesley Toppan”多媒体分形图像压缩 500 (1995)。

Y.Miyazaki, Y.Kikuchi, D.S.Cai, and Y.Ikebe: "Error Analysis for the Computation of Zeros of Regular Coulomb Wave Function and Its First Derivative"Math.Comp.. (to appear).

Y.Miyazaki、Y.Kikuchi、D.S.Cai 和 Y.Ikebe：“正则库仑波函数及其一阶导数零点计算的误差分析”Math.Comp..（待发表）。

Y.Ikebe,Y.Kikuchi,Y.Miyazaki,and D.Cai: "Infinite Matrices and Special Function Computations"the Fourth IMACS International Symposium on Iterative Methods in Scientific Computation (Austin, Texas)), also appearing in a book entitled "Iterative Methods in

Y.Ikebe、Y.Kikuchi、Y.Miyazaki 和 D.Cai：“无限矩阵和特殊函数计算”第四届 IMACS 国际科学计算迭代方法研讨会（德克萨斯州奥斯汀）），也出现在题为“

Y.Miyazaki,Y.Kikuchi,D.S.Cai,and Y.Ikebe: "Error Analysis for the Computation of Zeros of Regular Coulomb Wave Function and Its First Derivative"Math.Comp.. (to appear).

Y.Miyazaki、Y.Kikuchi、D.S.Cai 和 Y.Ikebe：“正则库仑波函数及其一阶导数零点计算的误差分析”Math.Comp..（待出版）。