1995 Fiscal Year Final Research Report Summary
Eigenvalue Problem for Infinite Matrices and Its Applications
| Project/Area Number |
06640290
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| Research Category |
Grant-in-Aid for General Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Research Field |
General mathematics (including Probability theory/Statistical mathematics)
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| Research Institution | University of Aizu (1995)
University of Tsukuba (1994) |
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Principal Investigator |
IKEBE Yasuhiko Univ.of Aizu Professor, コンピュータ理工学部, 教授 (10114034)
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| Co-Investigator(Kenkyū-buntansha) |
CAI Dong sheng Univ.of Tsukuba, Assist. Prof, 電子・情報工学系, 講師 (70202075)
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| Project Period (FY) |
1994 – 1995
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| Keywords | Eigenvalue of Infinite Matrices / Conjugate Symmetric Tridiagonal Matrial / Special Function / Bessel Function / Mathive Function / Visualization |
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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 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.
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[Publications] Y.Ikebe, N.Asai, Y.Miyazaki, D.Cai: "Infinite Matrices and Special Functions" Proceedings of the International Workshop on Inverse Problems with Applications to Geophysics, Industry, Medicine and Technology ( D.D.Ang, R.Gorenflo, R.S.Rutman, T.D.Van, M.Yamamoto, eds.). 100-105 (1995)
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