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Title: QR與LR算則之位移策略
On the shift strategies for the QR and LR algorithms
Authors: 黃義哲
Contributors: 王太林
Keywords: 位移策略
QR algorithm, LR algorithm, modified Cholesky algorithm.
Date: 1992
Issue Date: 2016-05-02 17:07:19 (UTC+8)
Abstract: 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。
Reference: 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
(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.
Description: 碩士
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Data Type: thesis
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