大規模行列の固有値問題：演算と通信を同時に最小化する世界初の数値解法

• Project/Area Number: 26870149
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Multi-year Fund
• Research Field: High performance computing; Computational science
• Research Institution: The University of Tokyo
• Principal Investigator: 中務 佑治 東京大学, 情報理工学(系)研究科, 助教 (10723554)
• Project Period (FY): 2014-04-01 – 2016-03-31
• Project Status: Discontinued (Fiscal Year 2015)
• Budget Amount: ¥3,640,000 (Direct Cost: ¥2,800,000、Indirect Cost: ¥840,000); Fiscal Year 2016: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2015: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000); Fiscal Year 2014: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
• Keywords: 固有値アルゴリズム / 固有値の他の分野への応用 / 国際交流、アメリカ / 国際共同研究、イギリス＋ドイツ / 国際研究者交流、イギリス＋ドイツ / 固有値の他分野への応用

Outline of Annual Research Achievements

申請タイトルの固有値分解アルゴリズムについてはカリフォルニア大学デービス校のRoland Freund教授との共著論文が応用数理の分野では最も権威のある論文誌といえるSIAM Reviewの受理された．研究結果をMITでの招待講演やSIAM ALA conferenceで発表した．
またマンチェスター大学のStefan Guettel氏との共著で行列指数関数の計算アルゴリズムの論文を書き、SIAM journal on Matrix analysis and applicationsに出版された．
その他にも行列固有値を使った数理最適化問題の数値解法を進展させ、論文にして投稿中である．固有値を用いた関数近似論アルゴリズムの論文も執筆中である．
他にもJSIAM Letters,EPASA学会会議録などに出版した．
国際交流という意味ではJared Aurentz,Vladimir Kostic,Robert Luce氏を招聘してセミナー講演とともに研究議論をした．

Research Products

• [Int'l Joint Research] MIT(米国)
• [Int'l Joint Research] Univ. Novi Sad(セルビア)
• [Int'l Joint Research] EPFL(スイス)
• [Int'l Joint Research] Oxford(英国)
• [Journal Article] Computing Fundamental Matrix Decompositions Accurately via the Matrix Sign Function in Two Iterations: The Power of Zolotarev's Functions (2016)
  • Author(s): Yuji Nakatsukasa, Roland W. Freund
  • Journal Title: SIAM Review Volume: TBD
  • Peer Reviewed / Int'l Joint Research / Acknowledgement Compliant
• [Journal Article] Scaled and squared subdiagonal Pade approximation for the matrix exponential (2016)
  • Author(s): Stefan Guettel, Yuji Nakatsukasa
  • Journal Title: SIAM Journal on Matrix Analysis and Applications Volume: 37 Issue: 1 Pages: 145-170
  • DOI: 10.1137/15m1027553
  • Peer Reviewed / Int'l Joint Research / Acknowledgement Compliant
• [Journal Article] Roundoff error analysis of the CholeskyQR2 algorithm in an oblique inner product (2016)
  • Author(s): Yusaku Yamamoto, Yuji Nakatsukasa, Yuka Yanagisawa and Takeshi Fukaya
  • Journal Title: JSIAM Letters Volume: 8 Issue: 0 Pages: 5-8
  • DOI: 10.14495/jsiaml.8.5
  • NAID: 130005125235
  • ISSN: 1883-0609, 1883-0617
  • Peer Reviewed / Open Access / Acknowledgement Compliant
• [Journal Article] On the stability of computing polynomial roots via confederate linearizations (2015)
  • Author(s): Yuji Nakatsukasa and Vanni Noferini
  • Journal Title: Mathematics of Computation Volume: 未定
  • Peer Reviewed / Acknowledgement Compliant
• [Journal Article] Computing the common zeros of two bivariate functions via Bezout resultants (2015)
  • Author(s): Yuji Nakatsukasa, Vanni Noferini and Alex Townsend
  • Journal Title: Numerische Mathematik Volume: 129 Issue: 1 Pages: 181-209
  • DOI: 10.1007/s00211-014-0635-z
  • Peer Reviewed
• [Journal Article] Roundoff Error Analysis of the CholeskyQR2 Algorithm (2015)
  • Author(s): Yusaku Yamamoto, Yuji Nakatsukasa, Yuka Yanagisawa and Takeshi Fukaya
  • Journal Title: Electronic Transactions on Numerical Analysis. Volume: 未定
  • Peer Reviewed / Acknowledgement Compliant
• [Journal Article] Using Zolotarev's Rational Approximation for Computing the Polar, Symmetric Eigenvalue, and Singular Value Decompositions (2015)
  • Author(s): Yuji Nakatsukasa and Roland W. Freund
  • Journal Title: 未定 Volume: 未定
• [Journal Article] Vector spaces of linearizations for matrix polynomials: A bivariate polynomial approach (2015)
  • Author(s): Yuji Nakatsukasa, Vanni Noferini and Alex Townsend
  • Journal Title: 未定 Volume: 未定
• [Journal Article] Finding a low-rank basis in a matrix subspace (2015)
  • Author(s): Yuji Nakatsukasa, Tasuku Soma and Andre Uschmajew
  • Journal Title: 未定 Volume: 未定
• [Journal Article] CholeskyQR2: A Simple and communication-avoiding algorithm for computing a tall-skinny QR factorization on a large-scale parallel system (2014)
  • Author(s): Takeshi Fukaya, Yuji Nakatsukasa, Yuka Yanagisawa and Yusaku Yamamoto
  • Journal Title: ScalA 2014 Volume: なし Pages: 31-38
  • Peer Reviewed / Open Access / Acknowledgement Compliant
• [Presentation] Beest L1 polynomial approximation (2016)
  • Author(s): Yuji Nakatsukasa
  • Organizer: Oxford University Numerical Analysis seminar
  • Place of Presentation: Oxford, UK
  • Year and Date: 2016-04-19
  • Int'l Joint Research
• [Presentation] Matrix Eigenvalue Problems: Theory and Applications (2016)
  • Author(s): Yuji Nakatsukasa
  • Organizer: Stanford University Interview
  • Place of Presentation: Stanford, USA
  • Year and Date: 2016-04-13
  • Int'l Joint Research / Invited
• [Presentation] “Stable polefinding and rational least-squares fitting via eigenvalues”, (2016)
  • Author(s): Yuji Nakatsukasa
  • Organizer: Workshop on Numerical Algebra, Algorithms, and Analysis
  • Place of Presentation: NII, 東京都・千代田区
  • Year and Date: 2016-01-11
  • Int'l Joint Research / Invited
• [Presentation] Computing Fundamental Matrix Decompositions Accurately via the Matrix Sign Function in Two Iterations: The Power of Zolotarev’s Functions (2015)
  • Author(s): Yuji Nakatsukasa
  • Organizer: MIT applied mathematics seminar
  • Place of Presentation: MIT, USA
  • Year and Date: 2015-10-30
  • Int'l Joint Research / Invited
• [Presentation] “Finding a low-rank basis in a matrix subspace” (2015)
  • Author(s): Yuji Nakatsukasa
  • Organizer: Low-rank Optimization and Applications workshop,
  • Place of Presentation: Bonn, Germany
  • Year and Date: 2015-06-08
  • Int'l Joint Research / Invited
• [Presentation] 副対角 Pade 近似による行列指数関数の効率的アルゴリズム (2015)
  • Author(s): 中務佑治
  • Organizer: 応用数理学会 研究部会連合発表会
  • Place of Presentation: 中野、東京
  • Year and Date: 2015-03-06 – 2015-03-07
• [Presentation] コンパニオン型行列固有値による有理関数の根計算 (2014)
  • Author(s): 中務佑治
  • Organizer: 応用数理学会年会
  • Place of Presentation: 政策研究大学院大学, 東京
  • Year and Date: 2014-09-03 – 2014-09-05
• [Presentation] Using Zolotarev’s rational approximation for computing the polar, symmetric eigenvalue, and singular value decompositions (2014)
  • Author(s): Yuji Nakatsukasa
  • Organizer: Householder Symposium XIX
  • Place of Presentation: Spa, Belgium
  • Year and Date: 2014-06-08 – 2014-06-13
  • Invited
• [Presentation] Algorithms and Perturbation Theory for Matrix Eigenvalue Problems and the SVD (2014)
  • Author(s): Yuji Nakatsukasa
  • Organizer: Householder Symposium XIX
  • Place of Presentation: Spa, Belgium
  • Year and Date: 2014-06-08 – 2014-06-13
  • Invited
• [Presentation] Eigenvector error bounds and perturbation for nonlinear eigenvalue problems (2014)
  • Author(s): Yuji Nakatsukasa
  • Organizer: Workshop on Nonlinear Eigenvalue Problems
  • Place of Presentation: Manchester, UK
  • Year and Date: 2014-04-23 – 2014-04-25
  • Invited