• Search Research Projects
  • Search Researchers
  • How to Use
  1. Back to previous page

A Challenge to Relative Errors by Numerical Algorithms with Positivity

Research Project

Project/Area Number 23654032
Research Category

Grant-in-Aid for Challenging Exploratory Research

Allocation TypeMulti-year Fund
Research Field General mathematics (including Probability theory/Statistical mathematics)
Research InstitutionKyoto University

Principal Investigator

NAKAMURA Yoshimasa  京都大学, 情報学研究科, 教授 (50172458)

Project Period (FY) 2011 – 2013
Project Status Completed (Fiscal Year 2013)
Budget Amount *help
¥3,510,000 (Direct Cost: ¥2,700,000、Indirect Cost: ¥810,000)
Fiscal Year 2013: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Fiscal Year 2012: ¥1,820,000 (Direct Cost: ¥1,400,000、Indirect Cost: ¥420,000)
Fiscal Year 2011: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000)
Keywords固有値計算 / 特異値計算 / 相対誤差 / 正値性 / 可積分アルゴリズム / 離散可積分系 / 漸近解析 / 行列式 / 計算数学
Research Abstract

In this project, we investigate the perturbations on singular values and the forward errors of the mdLVs variables, which occur in the mdLVs algorithm, through two kinds of error analysis in floating point arithmetic. Therefore the forward stability of the mdLVs algorithm is proved. Next we present a new similarity transformation named the dLV_\infty similarity transformation by considering the case where the discretization parameter \delta(n) goes to infinity in the integrable dLV system. It is proved that the singular values of bidiagonal matrix are computable by using the dLV_\infty transformation repeatedly. Therefore we clarify that the relative perturbations in the singular values are sufficiently small through the mixed forward-backward error analysis. We finally show that the dLV_\infty transformation is forward stable.

Report

(4 results)
  • 2013 Annual Research Report   Final Research Report ( PDF )
  • 2012 Research-status Report
  • 2011 Research-status Report
  • Research Products

    (53 results)

All 2014 2013 2012 2011 Other

All Journal Article (17 results) (of which Peer Reviewed: 10 results,  Open Access: 1 results) Presentation (33 results) (of which Invited: 4 results) Remarks (3 results)

  • [Journal Article] A tridiagonal matrix construction by the quotient difference recursion formula in the case of multiple eigenvalues2014

    • Author(s)
      Kanae Akaiwa, Masashi Iwasaki, Koichi Kondo and Yoshimasa Nakamura
    • Journal Title

      J. Math-for-Industry

      Volume: 6

    • Related Report
      2013 Annual Research Report 2013 Final Research Report
  • [Journal Article] Accurate similarity transformation derived from the discrete Lotka-Volterra system for bidiagonal singularvalues2014

    • Author(s)
      Munehiro Nagata, Masashi Iwasaki and Yoshimasa Nakamura
    • Journal Title

      Journal Calcolo

      Volume: 未定 Issue: 2 Pages: 305-317

    • DOI

      10.1007/s10092-013-0085-5

    • Related Report
      2013 Annual Research Report 2013 Final Research Report
    • Peer Reviewed / Open Access
  • [Journal Article] An application of centre manifold theory to the qd-type discrete Lotka-Volterra system2013

    • Author(s)
      You Takahashi, Masashi Iwasaki, Akiko Fukuda, Emiko Ishiwata, Yoshimasa Nakamura
    • Journal Title

      Applicable Analysis

      Volume: 92 Pages: 586-594

    • Related Report
      2013 Annual Research Report 2013 Final Research Report 2012 Research-status Report
  • [Journal Article] On a shifted LR transformation derived from the discrete hungry Toda equation2013

    • Author(s)
      Akiko Fukuda, Yusaku Yamamoto, Masashi Iwasaki, Emiko Ishiwata and Yoshimasa Nakamura
    • Journal Title

      Monatshefte fur Mathematik

      Volume: 170 Pages: 11-26

    • Related Report
      2013 Final Research Report 2012 Research-status Report
  • [Journal Article] ハングリー型の離散可積分系と非対称行列の固有値計算-可積分アルゴリズムにおける最近の発展-2013

    • Author(s)
      福田亜希子, 岩崎雅史, 山本有作, 石渡恵美子, 中村佳正
    • Journal Title

      日本応用数理学会論文誌

      Volume: 23 Pages: 109-181

    • Related Report
      2013 Annual Research Report
    • Peer Reviewed
  • [Journal Article] On a shifted LR transformation derived from the discrete hungry Toda equation2013

    • Author(s)
      Akiko Fukuda, Yusaku Yamamoto, Masashi Iwasaki, Emiko Ishiwata and Yoshimasa
    • Journal Title

      Monatshefte f¨ur Mathematik

      Volume: 170 Pages: 11-26

    • Related Report
      2013 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Accurate similarity transformation derived from the discrete Lotka-Volterra system for bidiagonal singular values2013

    • Author(s)
      Munehiro Nagata, Masashi Iwasaki and Yoshimasa Nakamura
    • Journal Title

      Journal Calcolo

      Volume: 53

    • Related Report
      2012 Research-status Report
    • Peer Reviewed
  • [Journal Article] ハングリー型の離散可積分系と非対称行列の固有値計算2013

    • Author(s)
      福田亜希子, 岩崎雅史, 山本有作, 石渡恵美子, 中村佳正
    • Journal Title

      日本応用数理学会論文誌

      Volume: 23 Pages: 109-181

    • Related Report
      2012 Research-status Report
    • Peer Reviewed
  • [Journal Article] Error analysis of the mdLVs algorithm for computing bidiagonal singular values2012

    • Author(s)
      Munehiro Nagata, Masashi Iwasaki and Yoshimasa Nakamura
    • Journal Title

      Numerical Algorithms

      Volume: 61 Issue: 2 Pages: 261-274

    • DOI

      10.1007/s11075-012-9607-5

    • Related Report
      2013 Final Research Report 2012 Research-status Report
  • [Journal Article] Subtraction-free recurrence relations for lower bounds of the minimal singular value of an upper bidiagonal matrix2012

    • Author(s)
      Takumi Yamashita, Kinji Kimura, and Yoshimasa Nakamura
    • Journal Title

      J. Math-for-Industry

      Volume: 4 Pages: 55-71

    • NAID

      40019493178

    • Related Report
      2013 Final Research Report 2011 Research-status Report
  • [Journal Article] On some properties of a discrete hungry Lotka-Volterra system of multiplicative type2012

    • Author(s)
      Yosuke Hama, Akiko Fukuda, Yusaku Yamamoto, Masashi Iwasaki, Emiko Ishiwata and Yoshimasa Nakamura
    • Journal Title

      J. Math-for- Industry

      Volume: 4 Pages: 5-15

    • Related Report
      2013 Final Research Report
  • [Journal Article] Error analysis for matrix eigenvalue algorithm based on the discrete hungry Toda equation2012

    • Author(s)
      A. Fukuda, Y. Yamamoto, M. Iwasaki, E. Ishiwata and Y. Nakamura
    • Journal Title

      Numerical Algorithms

      Volume: 61 Issue: 2 Pages: 243-260

    • DOI

      10.1007/s11075-012-9606-6

    • Related Report
      2012 Research-status Report
    • Peer Reviewed
  • [Journal Article] On some properties of a discrete hungry Lotka-Volterra system of multiplicative type2012

    • Author(s)
      Y. Hama, A. Fukuda, Y. Yamamoto, M. Iwasaki, E. Ishiwata and Y. Nakamura
    • Journal Title

      J. Math-for-Industry

      Volume: 4 Pages: 5-15

    • Related Report
      2011 Research-status Report
    • Peer Reviewed
  • [Journal Article] Positivity of dLV and mdLVs algorithms for computing singular values2011

    • Author(s)
      Masashi Iwasaki and Yoshimasa Nakamura
    • Journal Title

      Elect. Trans. Numer. Anal

      Volume: 38 Pages: 184-201

    • Related Report
      2013 Final Research Report
  • [Journal Article] Conserved quantities of the discrete finite Toda equation and lower bounds of the minimal singular value of upper bidiagonal matrices2011

    • Author(s)
      Kinji Kimura, Takumi Yamashita, Yoshimasa Nakamura
    • Journal Title

      J.Phys.A : Math.Theor.

      Volume: 44 Issue: 28 Pages: 285207-285207

    • DOI

      10.1088/1751-8113/44/28/285207

    • Related Report
      2013 Final Research Report
    • Peer Reviewed
  • [Journal Article] Positivity of dLV and mdLVs algorithms for computing singular values2011

    • Author(s)
      Masashi Iwasaki and Yoshimasa Nakamura
    • Journal Title

      Elect. Trans. Numer. Anal.

      Volume: 38 Pages: 184-201

    • Related Report
      2011 Research-status Report
    • Peer Reviewed
  • [Journal Article] Conserved quantities of the discrete finite Toda equation and lower bounds of the minimal singularvalue of upper bidiagonal matrices2011

    • Author(s)
      Kinji Kimura, Takumi Yamashita and Yoshimasa Nakamura,
    • Journal Title

      J. Phys. A: Math. Theor.

      Volume: 44

    • Related Report
      2011 Research-status Report
    • Peer Reviewed
  • [Presentation] Why are integrable algorithms accurate?2013

    • Author(s)
      Yoshimasa Nakamura
    • Place of Presentation
      Leeds University
    • Year and Date
      2013-03-14
    • Related Report
      2013 Final Research Report
  • [Presentation] 可積分アルゴリズムはなぜ高精度か?2013

    • Author(s)
      中村佳正
    • Organizer
      第14回情報学シンポジウム
    • Place of Presentation
      京都大学
    • Year and Date
      2013-02-19
    • Related Report
      2013 Final Research Report
  • [Presentation] 指定された固有値をもつHessenberg型帯行列の作成方法について2013

    • Author(s)
      赤岩香苗, 近藤弘一, 岩崎雅史, 中村佳正
    • Organizer
      日本応用数理学会年会
    • Place of Presentation
      アクロス福岡
    • Related Report
      2013 Final Research Report
  • [Presentation] カソラチ行列式の漸近展開定理2013

    • Author(s)
      新庄雅斗, 岩崎雅史, 中村佳正
    • Organizer
      日本応用数理学会年会
    • Place of Presentation
      アクロス福岡
    • Related Report
      2013 Annual Research Report 2013 Final Research Report
  • [Presentation] Prospect of integrable numerical algorithms2013

    • Author(s)
      Yoshimasa Nakamura
    • Organizer
      Forum on Scientific and Engineering Computing 2013
    • Place of Presentation
      Si Yuan Buildings Conference Hall, Chinese Academy of Sciences, Beijing
    • Related Report
      2013 Final Research Report
    • Invited
  • [Presentation] Aspects of integrable numerical algorithms, Integrable Algorithms2013

    • Author(s)
      Yoshimasa Nakamura
    • Organizer
      New Frontiers in Numerical Analysis
    • Place of Presentation
      Leeds University
    • Related Report
      2013 Final Research Report
  • [Presentation] 重複固有値をもつ3 重対角行列が定めるqd変数の漸近挙動について2013

    • Author(s)
      赤岩香苗, 近藤弘一, 岩崎雅史, 中村佳正
    • Organizer
      応用数理学会研究部会連合発表会
    • Place of Presentation
      東洋大学
    • Related Report
      2013 Final Research Report
  • [Presentation] mdLVsアルゴリズムに関する数値安定性と計算量の改良2013

    • Author(s)
      永田宗寛, 岩崎雅史, 木村欣司, 中村佳正
    • Organizer
      応用数理学会研究部会連合発表会
    • Place of Presentation
      東洋大学
    • Related Report
      2013 Final Research Report 2012 Research-status Report
  • [Presentation] Prospect of integrable numerical algorithms2013

    • Author(s)
      Yoshimasa Nakamura
    • Organizer
      Forum on Scientific and Engineering Computing 2013
    • Place of Presentation
      Chinese Academy of Sciences, Beijing
    • Related Report
      2013 Annual Research Report
    • Invited
  • [Presentation] 指定された固有値をもつHessenberg 型帯行列の作成方法について2013

    • Author(s)
      赤岩香苗, 近藤弘一, 岩崎雅史, 中村佳正
    • Organizer
      日本応用数理学会年会
    • Place of Presentation
      アクロス福岡
    • Related Report
      2013 Annual Research Report
  • [Presentation] 玉に番号が付いた箱玉系の保存量について2013

    • Author(s)
      柿崎苑美, 福田亜希子, 石渡恵美子, 山本有作, 岩崎雅史, 中村佳正
    • Organizer
      日本応用数理学会年会
    • Place of Presentation
      アクロス福岡
    • Related Report
      2013 Annual Research Report
  • [Presentation] 並列計算における再直交化付きブロック逆反復法の性能評価2013

    • Author(s)
      石上裕之, 木村欣司, 中村佳正
    • Organizer
      日本応用数理学会年会
    • Place of Presentation
      アクロス福岡
    • Related Report
      2013 Annual Research Report
  • [Presentation] Aspects of integrable numerical algorithms2013

    • Author(s)
      Yoshimasa Nakamura
    • Organizer
      Integrable Algorithms: New Frontiers in Numarical Analysis
    • Place of Presentation
      Leeds University, the United Kingdom
    • Related Report
      2012 Research-status Report
    • Invited
  • [Presentation] 重複固有値をもつ3重対角行列が定めるqd変数の漸近挙動について2013

    • Author(s)
      赤岩香苗,近藤弘一,岩﨑雅史,中村佳正
    • Organizer
      応用数理学会研究部会連合発表会
    • Place of Presentation
      東洋大学
    • Related Report
      2012 Research-status Report
  • [Presentation] totally nonnegative行列の最小固有値の下界について2012

    • Author(s)
      山下巧, 木村欣司, 中村佳正
    • Organizer
      日本応用数理学会年会
    • Place of Presentation
      稚内全日空ホテル
    • Related Report
      2013 Final Research Report
  • [Presentation] A tridiagonalization for nonsymmetric complex matrix in terms of the finite discrete Toda equation2012

    • Author(s)
      Kanae Akaiwa, Koichi Kondo, Masashi Iwasaki, Yoshimasa Nakamura
    • Organizer
      International Congress on Computational and Applied Mathematics (ICCAM2012)
    • Place of Presentation
      University, Belgium
    • Related Report
      2013 Final Research Report
  • [Presentation] Application of integrable discrete hungry systems to nonsymmetric matrix eigenvalue algorithms2012

    • Author(s)
      Yusaku Yamamoto and Yoshimasa Nakamura
    • Organizer
      The 8th East Asia SIAM Conference (EASIAM 2012)
    • Place of Presentation
      National Taiwan University, Taipei, Taiwan
    • Related Report
      2013 Final Research Report
  • [Presentation] Integrable discrete hungry systems and their related eigenvalue algorithms2012

    • Author(s)
      Yoshimasa Nakamura, Akiko Fukuda, Yusaku Yamamoto, Masashi Iwasaki and Emiko Ishiwata
    • Organizer
      Symmetries and Integrability of Difference Equations (SIDE 10)
    • Place of Presentation
      Yin Feng Holiday Resort, Xikou, Ningbo, China
    • Related Report
      2013 Final Research Report 2012 Research-status Report
  • [Presentation] Application of integrable discrete hungry systems to nonsymmetric matrix eigenvalue algorithms2012

    • Author(s)
      Yoshimasa Nakamura, Akiko Fukuda, Emiko Ishiwata, Yusaku Yamamoto and Masashi Iwasaki
    • Organizer
      Workshop on Methods and Applications of Industrial and Applied Mathematics
    • Place of Presentation
      RIMS Kyoto Univ
    • Related Report
      2013 Final Research Report
  • [Presentation] 最小特異値の下界を構成するための逆行列のトレースを計算する新しい減算なし公式について2012

    • Author(s)
      木村欣司, 山本有作, 山下巧, 高田雅美, 中村佳正
    • Organizer
      応用数理学会研究部会連合発表会
    • Place of Presentation
      九州大学
    • Related Report
      2013 Final Research Report
  • [Presentation] 有限戸田格子方程式に関連する3重対角行列の固有値について2012

    • Author(s)
      赤岩香苗, 近藤弘一, 岩崎雅史, 中村佳正
    • Organizer
      応用数理学会研究部会連合発表会
    • Place of Presentation
      九州大学
    • Related Report
      2013 Final Research Report
  • [Presentation] 離散可積分系に基づく高精度Darboux変換について2012

    • Author(s)
      永田宗寛, 岩崎雅史, 中村佳正
    • Organizer
      応用数理学会研究部会連合発表会
    • Place of Presentation
      九州大学
    • Related Report
      2013 Final Research Report 2011 Research-status Report
  • [Presentation] totally nonnegative 行列の 最小固有値の下界について2012

    • Author(s)
      山下巧, 木村欣司, 中村佳正
    • Organizer
      日本応用数理学会年会
    • Place of Presentation
      稚内全日空ホテル
    • Related Report
      2012 Research-status Report
  • [Presentation] A tridiagonalization for nonsymmetric complex matrix in terms of the finite discrete Toda equation2012

    • Author(s)
      Kanae Akaiwa, Koichi Kondo, Masashi Iwasaki, Yoshimasa Nakamura
    • Organizer
      International Congress on Computational and Applied Mathematics (ICCAM2012)
    • Place of Presentation
      Ghent University, Belgium
    • Related Report
      2012 Research-status Report
  • [Presentation] 対称行列の逆行列の固有値に対するKato-Temple不等式の導出と応用2011

    • Author(s)
      山下巧, 木村欣司, 高田雅美, 中村佳正
    • Organizer
      日本応用数理学会「行列・固有値問題の解法とその応用」 研究部会第12回研究会
    • Place of Presentation
      国立情報学研究所
    • Year and Date
      2011-11-21
    • Related Report
      2013 Final Research Report
  • [Presentation] Some aspects of the integrable discrete Lotka-Volterra system2011

    • Author(s)
      Masashi Iwasaki and Yoshimasa Nakamura
    • Organizer
      International Conference on Scientific Computing (SC2011)
    • Place of Presentation
      Cagliari, Italy
    • Year and Date
      2011-10-13
    • Related Report
      2013 Final Research Report
    • Invited
  • [Presentation] Some Fibonacci sequences appear in the dLV algorithm2011

    • Author(s)
      Kanae Akaiwa, Masashi Iwasaki and Yoshimasa Nakamura
    • Organizer
      International Workshop on Application of Iterative Methods to Engineering and Its Mathematical Element
    • Place of Presentation
      Doshisha Univ
    • Related Report
      2013 Final Research Report
  • [Presentation] Numerical stability of dLV transformation2011

    • Author(s)
      Munehiro Nagata, Masashi Iwasaki and Yoshimasa Nakamura
    • Organizer
      International Workshop on Application of Iterative Methods to Engineering and Its Mathematical Element
    • Place of Presentation
      Doshisha Univ
    • Related Report
      2013 Final Research Report
  • [Presentation] 負の差分間隔をもつ離散ロトカ・ボルテラ系に現れる数列の再帰性について2011

    • Author(s)
      赤岩香苗, 岩崎雅史, 中村佳正
    • Organizer
      日本応用数理学会年会ポスターセッション
    • Place of Presentation
      同志社大学
    • Related Report
      2013 Final Research Report
  • [Presentation] 中心多様体理論を用いた離散ハングリー系の局所解析2011

    • Author(s)
      高橋悠, 岩崎雅史, 福田亜希子, 石渡恵美子, 中村佳正
    • Organizer
      日本応用数理学会年会
    • Place of Presentation
      同志社大学
    • Related Report
      2013 Final Research Report 2011 Research-status Report
  • [Presentation] 特異値計算アルゴリズムmdLVs法におけるシフト戦略2011

    • Author(s)
      木村欣司, 山下巧, 高田雅美, 中村佳正
    • Organizer
      第40回数値解析シンポジウム
    • Place of Presentation
      鳥羽シーサイドホテル
    • Related Report
      2013 Final Research Report 2011 Research-status Report
  • [Presentation] Numerical stability of dLV transformation2011

    • Author(s)
      Munehiro Nagata, Masashi Iwasaki and Yoshimasa Nakamura
    • Organizer
      International Workshop on Application of Iterative Methods to Engineering and Its Mathematical Element
    • Place of Presentation
      同志社大学
    • Related Report
      2011 Research-status Report
  • [Presentation] シフトを考慮したLR変換に関連する離散ハングリー系について2011

    • Author(s)
      濱洋輔,福田亜希子,石渡恵美子,岩崎雅史,山本有作,中村佳正
    • Organizer
      日本応用数理学会年会
    • Place of Presentation
      同志社大学
    • Related Report
      2011 Research-status Report
  • [Remarks]

    • URL

      http://www.amp.i.kyoto-u.ac.jp/

    • Related Report
      2013 Final Research Report
  • [Remarks] 倍精度2 重対角行列シフト付き用特異値計算ルーチンDLVS の公開

    • URL

      http://www-is.amp.i.kyoto-u.ac.jp/svd/index.html

    • Related Report
      2013 Annual Research Report
  • [Remarks]

    • URL

      http://www-is.amp.i.kyoto-u.ac.jp/svd/index.html

    • Related Report
      2011 Research-status Report

URL: 

Published: 2011-08-05   Modified: 2019-07-29  

Information User Guide FAQ News Terms of Use Attribution of KAKENHI

Powered by NII kakenhi