| 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)
桜井 鉄也 筑波大学, 電子・情報工学系, 講師 (60187086)
久野 誉人 筑波大学, 電子・情報工学系, 助教授 (00205113)
北川 高嗣 筑波大学, 電子・情報工学系, 助教授 (60153095)
稲垣 敏之 筑波大学, 電子・情報工学系, 教授 (60134219)
名取 亮 筑波大学, 電子・情報工学系, 教授 (70013745)
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| Project Period (FY) |
1994 – 1995
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| Project Status |
Completed (Fiscal Year 1995)
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| Budget Amount *help |
¥2,200,000 (Direct Cost: ¥2,200,000)
Fiscal Year 1995: ¥900,000 (Direct Cost: ¥900,000)
Fiscal Year 1994: ¥1,300,000 (Direct Cost: ¥1,300,000)
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| Keywords | Eigenvalue of Infinite Matrices / Conjugate Symmetric Tridiagonal Matrial / Special Function / Bessel Function / Mathive Function / Visualization / 可視化 |
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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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