Development of high speed singular value decomposition algorithm for sparse matrices of large scale and upload of its source code

• Project/Area Number: 24360038
• Research Category: Grant-in-Aid for Scientific Research (B)
• Allocation Type: Partial Multi-year Fund
• Section: 一般
• Research Field: Engineering fundamentals
• Research Institution: Kyoto University
• Principal Investigator: Nakamura Yoshimasa 京都大学, 情報学研究科, 教授 (50172458)
• Co-Investigator(Kenkyū-buntansha): KIMURA Kinji 京都大学, 大学院情報学研究科, 特定准教授 (10447899)
• Project Period (FY): 2012-04-01 – 2016-03-31
• Project Status: Completed (Fiscal Year 2015)
• Budget Amount: ¥18,850,000 (Direct Cost: ¥14,500,000、Indirect Cost: ¥4,350,000); Fiscal Year 2014: ¥6,500,000 (Direct Cost: ¥5,000,000、Indirect Cost: ¥1,500,000); Fiscal Year 2013: ¥5,720,000 (Direct Cost: ¥4,400,000、Indirect Cost: ¥1,320,000); Fiscal Year 2012: ¥6,630,000 (Direct Cost: ¥5,100,000、Indirect Cost: ¥1,530,000)
• Keywords: 特異値分解 / 大規模スパース行列 / Golub-Kahan-Lanczos法 / 部分特異対 / 再直交化 / 部分特異対計算 / ２分法 / 逆反復法 / 並列計算 / Golub-Kahan-Lanczos前処理 / 応用数学 / アルゴリズム / 固有対計算 / 固有値分解 / 実装コード公開

Outline of Final Research Achievements

Singular value decomposition of sparse matrices of large scale is an important matrix operation which is fundamental for analyzing big data. In this research we apply the Golub-Kahan-Lanczos algorithm with reorthogonalization to sparse matrices of large scale to generate approximated tridiagonal matrices by high performance computation in parallel processing. Secondly the bisection and the inverse iteration methods are applied to them to giving a subset of eigenpairs of the originals with high reusability of data. The corresponding source codes have been uploaded one by one. When the sparse matrices are positive definite, then the resulting eigenpairs lead to a sebset of singular triplets.

Research Products

• [Journal Article] A new parallel symmetric tridiagonal eigensolver based on bisection and inverse iteration algorithms for shared-memory multi-core processors (2015)
  • Author(s): Hiroyuki Ishigami, Kinji Kimura and Yoshimasa Nakamura
  • Journal Title: Proceedings of the 10th International Conference on P2P, Parallel, Grid, Cloud and Internet Computing (3PGCIC-2015) Volume: 1 Pages: 216-223
  • Peer Reviewed / Open Access
• [Journal Article] Performance evaluation of Golub-Kahan-Lanczos algorithm with reorthogonalization by classical Gram-Schmidt algorithm and OpenMP (2015)
  • Author(s): Masami Takata, Hiroyuki Ishigami, Kinji Kimura, Yuki Fujii, Hiroki Tanaka, Yoshimasa Nakamura
  • Journal Title: Proceedings of The 2015 International Conference on Parallel and Distributed Processing Techniques and Applications (PDPTA2015) Volume: 2 Pages: 243-249
  • Peer Reviewed / Open Access
• [Journal Article] Implementation details of an extended oqds algorithm for singular values (2015)
  • Author(s): Sho Araki, Kinji Kimura, Yusaku Yamamoto and Yoshimasa Nakamura
  • Journal Title: JSIAM Letters Volume: 7 Pages: 9-12
  • NAID: 130004953465
  • Peer Reviewed / Open Access
• [Journal Article] A finite-step construction of totally nonnegative matrices with specified eigenvalues (2015)
  • Author(s): Kanae Akaiwa, Yoshimasa Nakamura, Masashi Iwasaki, Hisayoshi Tsutsumi, Koichi Kondo
  • Journal Title: Numerical Algorithms Volume: 未定 Issue: 3 Pages: 469-484
  • DOI: 10.1007/s11075-015-9957-x
  • Peer Reviewed / Open Access / Acknowledgement Compliant
• [Journal Article] Accurate similarity transformation derived from the discrete Lotka-Volterra system for bidiagonal singularvalues (2014)
  • 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
  • Peer Reviewed / Open Access
• [Journal Article] A tridiagonal matrix construction by the quotient difference recursion formula in the case of multiple eigenvalues (2014)
  • Author(s): Kanae Akaiwa, Masashi Iwasaki, Koichi Kondo and Yoshimasa Nakamura
  • Journal Title: Pacic J. Math. Indust. Volume: 6 Pages: 21-29
  • Peer Reviewed / Open Access
• [Journal Article] 再直交化付きブロック逆反復法による固有ベクトルの並列計算 (2014)
  • Author(s): 石上裕之, 木村欣司, 中村佳正
  • Journal Title: 情報処理学会論文誌コンピューティングシステム Volume: 7 Pages: 1-12
  • NAID: 110009815415
  • Peer Reviewed / Open Access
• [Journal Article] Acceleration of numerical computation for positive roots of polynomials by improved bounds (2014)
  • Author(s): Kinji Kimura, Takuto Akiyama, Hiroyuki Ishigami, Masami Takata, Yoshimasa Nakamura
  • Journal Title: Proceedings of The 2014 International Conference on Parallel and Distributed Processing Techniques and Applications (PDPTA2014) Volume: II Pages: 201-207
  • Peer Reviewed / Open Access
• [Journal Article] GPU Implementation of Inverse Iteration Algorithm for Computing Eigenvectors (2014)
  • Author(s): Hiroyuki Ishigami, Kinji Kimura, Yoshimasa Nakamura
  • Journal Title: Proceedings of 2014 22nd Euromicro International Conference on Parallel, Distributed and network-based Processing Volume: I Pages: 664-671
  • Peer Reviewed
• [Journal Article] On implementation and evaluation of inverse iteration algorithm with compact WY orthogonalization (2013)
  • Author(s): Hiroyuki Ishigami, Kinji Kimura, Yoshimasa Nakamura
  • Journal Title: IPSJ Trans. Math. Modeling Appl. Volume: 6 Pages: 25-35
  • Peer Reviewed
• [Journal Article] 特異値分解アルゴリズムの性能評価のための大きな条件数を持つ行列作成 (2013)
  • Author(s): 高田雅美，木村欣司，中村佳正
  • Journal Title: 情報処理学会論文誌, 数理モデル化と応用 Volume: 6 Pages: 75-86
  • NAID: 110009656646
  • Peer Reviewed
• [Journal Article] 特異値計算アルゴリズムdqds 法およびm2dLVs 法のための新しいシフト戦略 (2013)
  • Author(s): 高田雅美, 豊川博己, 石上裕之, 木村欣司, 山下巧, 岩崎雅史, 中村佳正
  • Journal Title: 情報処理学会論文誌コンピューティングシステム Volume: 6 Pages: 94-107
  • NAID: 110009606663
  • Peer Reviewed
• [Journal Article] Implementation of the orthogonal qd algorithm for lower tridiagonal matrices (2013)
  • Author(s): Sho Araki, Hiroki Tanaka, Kinji Kimura and Yoshimasa Nakamura
  • Journal Title: Proceedings of The 2013 International Conference on Parallel and Distributed Processing Techniques and Applications Volume: II Pages: 161-167
  • Peer Reviewed
• [Journal Article] 特異値分解アルゴリズムの性能評価のための大きな条件数を持つ行列作成 (2013)
  • Author(s): 高田雅美, 木村欣司, 中村佳正
  • Journal Title: 情報処理学会,MPS-PDPTAI0-2, TOM Volume: (掲載確定)(印刷中)
  • NAID: 110009656646
  • Peer Reviewed
• [Journal Article] On implementation and evaluation of inverse iteration algorithm with compact WY orthogonalizat ion (2013)
  • Author(s): Hiroyuki Ishigami, Kinji Kimura, Yoshimasa Nakamura
  • Journal Title: 情報処理学会, MPS-PDPTA11-8, TOM Volume: (掲載確定)
  • Peer Reviewed
• [Journal Article] 特異値計算アルゴリズムdqds法およびmdLVs法のための新しいシフト戦略, (2013)
  • Author(s): 高田雅美, 石上裕之, 木村欣司, 山下巧, 岩崎雅史, 中村佳正
  • Journal Title: 情報処理学会,ハイパフォーマンスコンピューティングと計算科学シンポジウム論文集(掲載確定) Volume: (掲載確定)(印刷中)
  • NAID: 170000073607
  • Peer Reviewed
• [Journal Article] Accelerating the reorthogona1i za t i on on singular vectors with a multi-coreprocessor (2013)
  • Author(s): Hiroki Toyokawa, Hiroyuki Ishigami, Kinji Kimura, Masami Takata and Yoshimasa Nakamura
  • Journal Title: Lecture Notes in Computer Science (LNCS), Springer - Verlag, Volume: (掲載確定)(印刷中)
  • Peer Reviewed
• [Journal Article] An application of the Kato-Temple inequality on matrix eigenvalues to the dqds algorithm for singular values (2013)
  • Author(s): Takumi Yamashita, Kinji Kimura, and Yoshimasa Nakamura
  • Journal Title: JSIAM Letters Volume: 4 Pages: 21-24
  • NAID: 130003371115
  • Peer Reviewed
• [Presentation] dqds 法および mdLVs 法の新しい実装法について (2016)
  • Author(s): 木村欣司, 中村佳正
  • Organizer: 日本応用数理学会研究部会連合発表会
  • Place of Presentation: 神戸学院大学（兵庫県・神戸市）
  • Year and Date: 2016-03-04
• [Presentation] クラスタ行列に対する直交QD法の特異値分解精度を改善する停止条件について (2016)
  • Author(s): 荒木翔, 木村欣司, 中村佳正
  • Organizer: 日本応用数理学会研究部会連合発表会
  • Place of Presentation: 神戸学院大学（兵庫県・神戸市）
  • Year and Date: 2016-03-04
• [Presentation] Thick-Restart GKL法の新しいリスタートの方法について (2016)
  • Author(s): 石田遊也，木村欣司，中村佳正
  • Organizer: 日本応用数理学会研究部会連合発表会
  • Place of Presentation: 神戸学院大学（兵庫県・神戸市）
  • Year and Date: 2016-03-04
• [Presentation] Bisection and inverse iteration algorithms for computing eigenpair subsets of symmetric band matrices (2015)
  • Author(s): Hiroyuki Ishigami, Kinji Kimura, Yoshimasa Nakamura
  • Organizer: International Workshop on Eigenvalue Problems: Algorithms; Software and Applications, in Petascale Computing (EPASA2015)
  • Place of Presentation: Tsukuba International Congress Center(茨城県・つくば市）
  • Year and Date: 2015-09-14
  • Int'l Joint Research / Invited
• [Presentation] 直交 QD 法 (random-time Toda molecule equation) の実装の詳細につ いて (2015)
  • Author(s): 木村欣司, 石上裕之, 中村佳正
  • Organizer: 日本応用数理学会年会
  • Place of Presentation: 金沢大学（石川県・金沢市）
  • Year and Date: 2015-09-09
• [Presentation] Xeon Phiにおける並列2分法による固有値計算の性能評価 (2015)
  • Author(s): 木村欣司, 石上裕之, 中村佳正
  • Organizer: 情報処理学会 2015年並列／分散／協調処理に関するサマー・ワークショップ (SWoPP2015)
  • Place of Presentation: 別府ビーコンプラザ（大分県・別府市）
  • Year and Date: 2015-08-04
• [Presentation] Performance evaluation of Golub-Kahan-Lanczos algorithm with reorthogonalization by classical Gram-Schmidt algorithm and OpenMP (2015)
  • Author(s): Masami Takata, Hiroyuki Ishigami, Kinji Kimura, Yuki Fujii, Hiroki Tanaka, Yoshimasa Nakamura
  • Organizer: The 2015 International Conference on Parallel and Distributed Processing Techniques and Applications(PDPTA2015)
  • Place of Presentation: Las Vegas, Monte Carlo Resort（Las Vegas, USA)
  • Year and Date: 2015-07-27
  • Int'l Joint Research
• [Presentation] Xeon Phi における部分固有対計算の高速化に向けて (2015)
  • Author(s): 石上裕之, 木村欣司, 中村佳正
  • Organizer: 日本応用数理学会研究部会連合発表会
  • Place of Presentation: 明治大学
  • Year and Date: 2015-03-06 – 2015-03-07
• [Presentation] 直交QD アルゴリズムによる高精度特異値分解について (2015)
  • Author(s): 田中博基, 木村欣司, 石上裕之, 中村佳正
  • Organizer: 日本応用数理学会研究部会連合発表会
  • Place of Presentation: 明治大学
  • Year and Date: 2015-03-06
• [Presentation] 部分固有対計算ソフトウェアARPACK の改良について (2014)
  • Author(s): 藤井祐貴, 石上裕之, 木村欣司, 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18 回研究会
  • Place of Presentation: 東京大学工学部
  • Year and Date: 2014-12-25
• [Presentation] 高精度特異値分解のための直交QD アルゴリズムの改良 (2014)
  • Author(s): 田中博基, 石上裕之, 木村欣司, 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18 回研究会
  • Place of Presentation: 東京大学工学部
  • Year and Date: 2014-12-25
• [Presentation] Accelerating the numerical computation of positive roots of polynomials using improved bounds (2014)
  • Author(s): Kinji Kimura, Takuto Akiyama, Hiroyuki Ishigami, Masami Takata, Yoshimasa Nakamura
  • Organizer: The 2014 International Conference on Parallel and Distributed Processing Techniques and Applications(PDPTA2014),
  • Place of Presentation: Las Vegas, USA
  • Year and Date: 2014-07-21 – 2014-07-24
• [Presentation] Implementation of the orthogonal QD-algorithm for performing singular value decomposition with high accuracy (2014)
  • Author(s): Hiroki Tanaka, Hiroyuki Ishigami, Kinji Kimura, Yoshimasa Nakamura
  • Organizer: 8th International Workshop on Parallel Matrix Algorithms and Applications (PMAA2014)
  • Place of Presentation: Lugano, Switzerland
  • Year and Date: 2014-07-02 – 2014-07-04
• [Presentation] Speed-up of reorthogonalized block Golub-Kahan-Lanczos algorithm for preprocessing (2014)
  • Author(s): Yuki Fujii, Hiroyuki Ishigami, Kinji Kimura, Yoshimasa Nakamura
  • Organizer: 8th International Workshop on Parallel Matrix Algorithms and Applications (PMAA2014)
  • Place of Presentation: Lugano, Switzerland
  • Year and Date: 2014-07-02 – 2014-07-04
• [Presentation] 高精度特異値分解を表現するoqds 法のシフト戦略について (2014)
  • Author(s): 田中博基, 木村欣司, 中村佳正
  • Organizer: 日本応用数理学会研究部会連合発表会
  • Place of Presentation: 京都大学
• [Presentation] 再直交化付きブロック逆反復法による固有ベクトルの並列計算 (2014)
  • Author(s): 石上裕之，木村欣司，中村佳正
  • Organizer: 情報処理学会2014 年ハイパフォーマンスコンピューティングと計算科学シンポジウム
  • Place of Presentation: 一橋講堂
• [Presentation] 下3 重対角行列向けoqds 法について (2013)
  • Author(s): 荒木翔，田中博基，木村欣司，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
  • Place of Presentation: 東京大学
• [Presentation] Reorthogonalized-Golub-Kahan-Lanczos 法と拡大行列に対するReorthogonalized-Lanczos 法の比較について (2013)
  • Author(s): 藤井祐貴, 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
• [Presentation] 新シフト戦略によるoqds 法の反復回数の削減と高精度特異値分解への応用 (2013)
  • Author(s): 田中博基, 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: アクロス福岡
• [Presentation] 逆反復に基づく並列固有ベクトル計算のブロックCGS2アルゴリズムの適用による高速化 (2013)
  • Author(s): 石上裕之, 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
• [Presentation] Accelerating the Reorthogonali zat i on on Singular Vectors with a Multi-Core Processor (2012)
  • Author(s): Hiroki Toyokawa, Hiroyuki Ishigami, Kinji Kimura, Masami Takata, and Yoshimasa Nakamura
  • Organizer: The Seventh International Workshop on Automatic Performance Tuning (iWAPT2012)
  • Place of Presentation: RIKEN Advanced Institute for Computational Science, Kobe
  • Year and Date: 2012-07-17
• [Presentation] Performance Evaluation of Some Inverse Iteration Algorithms on PowerXCellTM 8i Processor (2012)
  • Author(s): Masarai Takata, Hiroyuki Ishigami, Kinji Kimura, and Yoshimasa Nakamura
  • Organizer: The 2012 International Conference on Parallel and Distributed Processing Techniques and Applications (PDPTA2012)
  • Place of Presentation: Monte Carlo Resort, Las Vegas
• [Presentation] compact WY直交化法の計算量を削減する実装について (2012)
  • 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: 稚内全日空ホテル
• [Presentation] GPUによる逆反復法の高速化について (2012)
  • Author(s): 秋山拓斗, 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: 稚内全日空ホテル
• [Remarks] Singular value decomposition code DBDSLV
  • URL: http://www-is.amp.i.kyoto-u.ac.jp/lab/en/isvd/
• [Remarks] 倍精度2 重対角行列特異値分解コードDBDSLVを含む密行列の特異値分解コードの公開
  • URL: http://www-is.amp.i.kyoto-u.ac.jp/svd/index.html
• [Remarks]
  • URL: http://www.amp.i.kyoto-u.ac.jp/