2000 Fiscal Year Final Research Report Summary
Computing Special Functions in the Complex Domain through Matrix Equation Reformulation
| Project/Area Number |
11640130
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
General mathematics (including Probability theory/Statistical mathematics)
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| Research Institution | University of Aizu |
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Principal Investigator |
IKEBE Yasuhiko Univ. of Aizu, School of Computer Science and Engineering, Professor, コンピュータ理工学部, 教授 (10114034)
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| Co-Investigator(Kenkyū-buntansha) |
MIYAZAKI Yoshinori Shizuoka Sangyo Univ., Faculty of Communications and Informatics, Assistant Professor, 国際情報学部, 講師 (00308701)
KIKUCHI Yasushi Univ. of Aizu, School of Computer Science and Engineering, Assistant Professor, コンピュータ理工学部, 講師 (60254059)
CAI Dongsheng Univ. of Tsukuba, Inst. of Information Sciences and Electronics, Associate Professor, 電子・情報工学系, 助教授 (70202075)
ASAI Nobuyoshi Univ. of Aizu, School of Computer Science and Engineering, Assistant Professor, コンピュータ理工学部, 講師 (80325969)
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| Project Period (FY) |
1999 – 2000
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| Keywords | Eigenvalue of Infinite Matrix / Computation of Special Functions / Computing Zeros / Multiplicity of Eignevalues / Regular Coulomb Wave Funcation |
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| Research Abstract |
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.
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