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題名 QR與LR算則之位移策略
On the shift strategies for the QR and LR algorithms作者 黃義哲
HUANG, YI-ZHE貢獻者 王太林
WANG, TAI-LIN
黃義哲
HUANG, YI-ZHE關鍵詞 位移策略
特徵向量
特徵值
QR algorithm, LR algorithm, modified Cholesky algorithm.日期 1992
1991上傳時間 2-May-2016 17:07:19 (UTC+8) 摘要 用QR與LR迭代法求矩陣特徵值與特徵向量之過程中，前人曾提出位移策略以加速其收斂速度，其中最有效的是Wilkinson 移位值。在此我們希望尋求能使收斂更快速的位移值。
Abstract參考文獻 References: [l] Dekker, T . J. and Traub, J. F., 1971. "The Shifted QR Algorithm for Hermitian Matrices." Linear Algebra and Its Applications, 4:137-154 [2] Dubrulle, A., 1970. "A Short Note on the Implicit QL Algorithm for Symmetric Tridiagonal Matrix." Numer. Math. , 15 :450. [3] Golub, G. H. and Van Loan, C. F. , 1989. Matrix Computations. 2nd edition, Baltimore, MD: The Johns Hopkins University Press. [4] Jiang, E. and Zheng, Z., 1985. "A New Shift of the QL Algorithm for Irreducible Symmetric Tridiagonal Matrices." Linear Algebra and Its Applications,65:261-272. [5] Ortega, J. M. and Kaiser, H. F., 1963. "The LLT and QR Methods for Symmetric Tridiagonal matrices." Computer Journal, 99-101. [6] Parlett, B. N. , 1964. "The Development and Use of Methods of LR Type." SIAM Review, 6:275-295 . [7] Parlett, B. N., 1966. "Singular and Invariant Matrices Under the QR Transformation. " Math. Comp., 611-615. [8] Parlett, B. N., 1980. The Semmetric Eigenvalue Problem. Prentice-Hall Inc. , Englewood Cliffs 1980. [9] Rutishauser, H. and Schwarz, H. R., 1963. "The LR Transformation Method for Symmetric Matrices." Numer. Math. 5:273-289. [10] Saad, Y. , 1974, "Shift of Origin for the QR Algorithm." Toronto: Proceedings IFIP Congress. [11] Smith, B. T. and Boyle, J. M., 1974. Matrix Eigensystem Routines - EISPACK Guide, Springer Verlag. [12] `Ward, R. C. and Gray, L. J ., 1978. "Eigensystem Computation for Skew-Symmetric Matrices and a Class of Symmetric Matrices." A CM Trans. on Math. Software , 4:278-285 . [13] Wilkinson, J. H. and Reisch, C., 1961. Handbook for A`l?tomatric Computation. Volum. II. Linear Algebra, Springer Verlag. [14] Wilkinson, J. H. , 1968. "Global Convergence of Tridiagonal QR Algorithm with Origin Shifts." Linear Algebra and Its Applications, 1:409-420. Notation Convention: (1) CHOLESKY: This subroutine is the implementation of the modified LLT algorithm. (2)imTQLl: This subroutine from the EISPACK computes the eigenvalues. by the implicit QL algorithm. (3) imTQL2: This subroutine from the EISPACK computes the eigenvalues and eigenvectors at the same tims by the implicit QL method. (4) imTQL2s4l: This routine first computes eigenvalues by CHOLESKY and then uses these eigenvalues as shifts in imTQL2. (5) imTQL2s42: This subroutine makes the use of imTQL1 to compute the eigenvalues and then uses these computed values as shifts in imTQL2 . . (6) TQL1: This subroutine from the EISPACK computes eigenvalues by the QL method. (7) TQL1s31, TQL1s32, TQL1s33 : These subroutines are the test of the use of 83 , described in section 3. (8) TQL2: This subroutine from the EISPACK computes eigenvalues and eigenvectors simultaneously by the QL method. (9) TQL2s41: This subroutine calculate eigenvalues by CHOLESKY at first and then uses these eigenvalues as shifts in TQL2. (1 0) TQL2s42: This subroutine uses eigenvalues computed by TQL1 as shifts in TQL2. 描述 碩士
國立政治大學
應用數學系資料來源 http://thesis.lib.nccu.edu.tw/record/#B2002004735 資料類型 thesis dc.contributor.advisor 王太林 zh_TW dc.contributor.advisor WANG, TAI-LIN en_US dc.contributor.author (Authors) 黃義哲 zh_TW dc.contributor.author (Authors) HUANG, YI-ZHE en_US dc.creator (作者) 黃義哲 zh_TW dc.creator (作者) HUANG, YI-ZHE en_US dc.date (日期) 1992 en_US dc.date (日期) 1991 en_US dc.date.accessioned 2-May-2016 17:07:19 (UTC+8) - dc.date.available 2-May-2016 17:07:19 (UTC+8) - dc.date.issued (上傳時間) 2-May-2016 17:07:19 (UTC+8) - dc.identifier (Other Identifiers) B2002004735 en_US dc.identifier.uri (URI) http://nccur.lib.nccu.edu.tw/handle/140.119/89759 - dc.description (描述) 碩士 zh_TW dc.description (描述) 國立政治大學 zh_TW dc.description (描述) 應用數學系 zh_TW dc.description.abstract (摘要) 用QR與LR迭代法求矩陣特徵值與特徵向量之過程中，前人曾提出位移策略以加速其收斂速度，其中最有效的是Wilkinson 移位值。在此我們希望尋求能使收斂更快速的位移值。 zh_TW dc.description.abstract (摘要) Abstract en_US dc.description.tableofcontents Contents 1 Introductions..........2 2 The modified LLT algorithm..........3 3 Shift strategy ..........8 4 Numerical experiments ..........9 5 Conclusion ..........13 Reference ..........14 Notation convection ..........15 zh_TW dc.source.uri (資料來源) http://thesis.lib.nccu.edu.tw/record/#B2002004735 en_US dc.subject (關鍵詞) 位移策略 zh_TW dc.subject (關鍵詞) 特徵向量 zh_TW dc.subject (關鍵詞) 特徵值 zh_TW dc.subject (關鍵詞) QR algorithm, LR algorithm, modified Cholesky algorithm. en_US dc.title (題名) QR與LR算則之位移策略 zh_TW dc.title (題名) On the shift strategies for the QR and LR algorithms en_US dc.type (資料類型) thesis en_US dc.relation.reference (參考文獻) References: [l] Dekker, T . J. and Traub, J. F., 1971. "The Shifted QR Algorithm for Hermitian Matrices." Linear Algebra and Its Applications, 4:137-154 [2] Dubrulle, A., 1970. "A Short Note on the Implicit QL Algorithm for Symmetric Tridiagonal Matrix." Numer. Math. , 15 :450. [3] Golub, G. H. and Van Loan, C. F. , 1989. Matrix Computations. 2nd edition, Baltimore, MD: The Johns Hopkins University Press. [4] Jiang, E. and Zheng, Z., 1985. "A New Shift of the QL Algorithm for Irreducible Symmetric Tridiagonal Matrices." Linear Algebra and Its Applications,65:261-272. [5] Ortega, J. M. and Kaiser, H. F., 1963. "The LLT and QR Methods for Symmetric Tridiagonal matrices." Computer Journal, 99-101. [6] Parlett, B. N. , 1964. "The Development and Use of Methods of LR Type." SIAM Review, 6:275-295 . [7] Parlett, B. N., 1966. "Singular and Invariant Matrices Under the QR Transformation. " Math. Comp., 611-615. [8] Parlett, B. N., 1980. The Semmetric Eigenvalue Problem. Prentice-Hall Inc. , Englewood Cliffs 1980. [9] Rutishauser, H. and Schwarz, H. R., 1963. "The LR Transformation Method for Symmetric Matrices." Numer. Math. 5:273-289. [10] Saad, Y. , 1974, "Shift of Origin for the QR Algorithm." Toronto: Proceedings IFIP Congress. [11] Smith, B. T. and Boyle, J. M., 1974. Matrix Eigensystem Routines - EISPACK Guide, Springer Verlag. [12] `Ward, R. C. and Gray, L. J ., 1978. "Eigensystem Computation for Skew-Symmetric Matrices and a Class of Symmetric Matrices." A CM Trans. on Math. Software , 4:278-285 . [13] Wilkinson, J. H. and Reisch, C., 1961. Handbook for A`l?tomatric Computation. Volum. II. Linear Algebra, Springer Verlag. [14] Wilkinson, J. H. , 1968. "Global Convergence of Tridiagonal QR Algorithm with Origin Shifts." Linear Algebra and Its Applications, 1:409-420. Notation Convention: (1) CHOLESKY: This subroutine is the implementation of the modified LLT algorithm. (2)imTQLl: This subroutine from the EISPACK computes the eigenvalues. by the implicit QL algorithm. (3) imTQL2: This subroutine from the EISPACK computes the eigenvalues and eigenvectors at the same tims by the implicit QL method. (4) imTQL2s4l: This routine first computes eigenvalues by CHOLESKY and then uses these eigenvalues as shifts in imTQL2. (5) imTQL2s42: This subroutine makes the use of imTQL1 to compute the eigenvalues and then uses these computed values as shifts in imTQL2 . . (6) TQL1: This subroutine from the EISPACK computes eigenvalues by the QL method. (7) TQL1s31, TQL1s32, TQL1s33 : These subroutines are the test of the use of 83 , described in section 3. (8) TQL2: This subroutine from the EISPACK computes eigenvalues and eigenvectors simultaneously by the QL method. (9) TQL2s41: This subroutine calculate eigenvalues by CHOLESKY at first and then uses these eigenvalues as shifts in TQL2. (1 0) TQL2s42: This subroutine uses eigenvalues computed by TQL1 as shifts in TQL2. zh_TW
