Computing Special Functions in the Complex Domain through Matrix Equation Reformulation

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。

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.