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	Using cross-product matrices to compute the SVD
	Jia, Zhongxiao
	2010-05-06 ; 2010-05-06
关键词	cross-product matrix eigenvalue eigenvector finite precision arithmetic Rayleigh quotient singular value singular vector SVD PROJECTION METHODS EIGENPROBLEMS Mathematics, Applied
中文摘要	This paper concerns accurate computation of the singular value decomposition ( SVD) of an m x n matrix A. As is well known, cross-product matrix based SVD algorithms compute large singular values accurately but generally deliver poor small singular values. A new novel cross- product matrix based SVD method is proposed: ( a) Use a backward stable algorithm to compute the eigenpairs of A(T) A and take the square roots of the large eigenvalues of it as the large singular values of A; ( b) form the Rayleigh quotient of A(T) A with respect to the matrix consisting of the computed eigenvectors associated with the computed small eigenvalues of A(T) A; ( c) compute the eigenvalues of the Rayleigh quotient and take the square roots of them as the small singular values of A. A detailed quantitative error analysis is conducted on the method. It is proved that if small singular values are well separated from the large ones then the method can compute the small ones accurately up to the order of the unit roundoff epsilon. An algorithm is developed that is not only cheaper than the standard Golub-Reinsch and Chan SVD algorithms but also can update or downdate a new SVD by adding or deleting a row and compute certain refined Ritz vectors for large matrix eigenproblems at very low cost. Several variants of the algorithm are proposed that compute some or all parts of the SVD. Typical numerical examples confirm the high accuracy of our algorithm.
语种	英语 ; 英语
出版者	SPRINGER ; DORDRECHT ; VAN GODEWIJCKSTRAAT 30, 3311 GZ DORDRECHT, NETHERLANDS
内容类型	期刊论文
源URL	[http://hdl.handle.net/123456789/14238]
专题	清华大学
推荐引用方式 GB/T 7714	Jia, Zhongxiao. Using cross-product matrices to compute the SVD[J],2010, 2010.
APA	Jia, Zhongxiao.(2010).Using cross-product matrices to compute the SVD..
MLA	Jia, Zhongxiao."Using cross-product matrices to compute the SVD".(2010).

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