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Engenvalue Problem of Infinite Matrices and its Application.

Research Project

Project/Area Number 09640284
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field General mathematics (including Probability theory/Statistical mathematics)
Research InstitutionUniveristy of Aizu

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)

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)
Project Period (FY) 1997 – 1998
Project Status Completed (Fiscal Year 1998)
Budget Amount *help
¥3,100,000 (Direct Cost: ¥3,100,000)
Fiscal Year 1998: ¥1,000,000 (Direct Cost: ¥1,000,000)
Fiscal Year 1997: ¥2,100,000 (Direct Cost: ¥2,100,000)
KeywordsEigenvalue of Infinite Metrix / Conjugate Symmetric Tridiagonal Matrices / Special Function / Bessel Function / Mathiue Function / Visualization
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.

Report

(3 results)
  • 1998 Annual Research Report   Final Research Report Summary
  • 1997 Annual Research Report
  • Research Products

    (12 results)

All Other

All Publications (12 results)

  • [Publications] Y.Ikebe,Y.Kikuchi,N.Asai,Y.Miyazaki,D.Cai: "The Eigenvalue Problem for Infinite Matrices:New Area of Application of Numerical Linear Algebra" Proceedings of Fourth IMACS International Symposium on Scientific Computation(honoring Professor David M.Young). (1998)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] 宮崎佳典,浅井信吉,蔡東生,池辺八州彦: "Mathieu微分方程式の逆固有値問題" 応用数理. 8. 199-222 (1998)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] N.Asai,et al: "Matrix methods for the Numerical Solution of zJ'v(z)+HJv(z)=0" Electronics and Communications in Japan. Vol.80,7. 44-54 (1997)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] Y.Ikebe, Y.Kikuchi, N.Asai, Y.Miyazaki, D.Cai: "The Eigenvalue Problem for Infinite Matrices : New Area of Application of Numerical Linear Algebra" Proceedings of Fourth IMACS International Symposium on Scientific Computation (honoring Professor David M.Young). (1998)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] K.Miyazaki, N.Asai, D.Cai and Y.Ikebe: "Inverse Eigenvalue problem of Mathieu's Differential Equation" Bulletin of the Japan Society for Industrial and Applied Mathematics. No.8. 199-222 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] Asai, N., Y.Miyazaki, D.Cai, K.Hirasawa, and Y.Ikebe: "Matrix methods for the Numerical Solution of Jv'(z)+HJv(z)=0, (Selected Special Paper by Editor)" Electronics and Communications in Japan. Vol.807. 44-54 (1997)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] Y. Ikebe, Y. Kikuchi, N. Asai, Y. Miyazaki, D.Cai: "The Eigenvalue Problem for Infinite Matrices: New Area of Application of Numerical Linear Algebra." Proceedings of Fourth IMACS International Symposium on Scientific Computation (honoring Professor David M.Young). (1999)

    • Related Report
      1998 Annual Research Report
  • [Publications] 宮崎佳典,浅井信吉,蔡東生,池辺八洲彦: "Mathieu微分方程式の逆固有値問題" 応用数理. 8. 199-222 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] N.Asai.et.al: "Matrix methods for the Numerical Solution of zJ'v(z)+HJv(z)=0" Electronics and Communications in Japan. Vol.80,7. 44-54 (1997)

    • Related Report
      1998 Annual Research Report
  • [Publications] 宮崎佳典,浅井信吉,蔡東生 池辺八洲彦: "あるクラスの三項漸化式の無限行列固有値問題再定式化および二重固有値計算法" 応用数理学会論文誌. (採録決定). (1998)

    • Related Report
      1997 Annual Research Report
  • [Publications] 浅井信吉,宮崎佳典,蔡東生 平沢一紘,池辺八洲彦: "行列算法によるZJ^1_ν(Z)+HJ_2(Z)=0の数値解法" 電子情報通信学会論文誌. J79. 1256-1265 (1996)

    • Related Report
      1997 Annual Research Report
  • [Publications] Y.Ikebe,N.Asai,Y.Miyazaki,D.S.Cai: "The Eigenvalue Problem for Infinite Complex Symmetric Tridiagonal Matrices with Application" Linear Algebra and Its Applications. 241-243. 599-618 (1996)

    • Related Report
      1997 Annual Research Report

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Published: 1997-04-01   Modified: 2016-04-21  

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