1998 Fiscal Year Final Research Report Summary
Engenvalue Problem of Infinite Matrices and its Application.
| Project/Area Number |
09640284
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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 | Univeristy 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) |
KIKUCHI Yasushi Univ.of Aizu, School of Computer Science and Engineering Instructor, コンピュータ理工学部, 講師 (60254059)
CAI Dongsheng Univ.of Tsukuba, Inst.of Information Sciences and Electronics, Assistant Profess, 電子・情報工学系, 助教授 (70202075)
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| Project Period (FY) |
1997 – 1998
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| Keywords | Eigenvalue of Infinite Metrix / Conjugate Symmetric Tridiagonal Matrices / Special Function / Bessel Function / Mathiue 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 a maximal domai n 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 lambda of T by an eigen value lambda_n of T_n, the n-th order principal submatrix of T.Let X = [x^<(1)>, x^<(2)>, ...]^T be an eigenvector corresponding to lambda. Assuning 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> x^<(n+1)> x^n[1+omicron(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) solve Jv(z) = 0 for v, given z * O 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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