单向收缩QR算法在奇异值分解中的收敛特性  

Convergence Characteristic of Single Direction Shrink QR Algorithm in the Singular Value Decomposition

作　　者：赵学智[1] 叶邦彦[1] 

机构地区：[1]华南理工大学机械与汽车工程学院,广州510640

出　　处：《电子科技大学学报》2010年第5期762-767,共6页Journal of University of Electronic Science and Technology of China

基　　金：国家自然科学基金(50875086);中央高校基本科研业务费专项资金(2009ZM0287)

摘　　要：针对大型矩阵奇异值分解的数值计算问题,总结了单向收缩QR算法的特点,通过实例证明了该算法在处理由某些小幅度信号构造的大型矩阵的奇异值分解时存在不收敛的情况。从理论上分析了QR迭代过程中Givens变换矩阵的变化特点,发现算法出现不收敛现象的根本原因在于大型矩阵首行对角带元素的衰减,最终会使QR迭代时的第一个Givens右矩阵变为单位阵,从而导致后面所有Givens矩阵全部成为单位阵,引起QR算法失效。在此基础上进一步研究了首行元素的衰减对QR算法收敛速度的影响。对理论分析用实际数据进行了验证,从本质上探明了该QR算法的收敛特性。Aimed at the numerical computation of singular value decomposition（SVD） of large scale matrix,the characteristic of single direction shrink quadrature right-triangle（QR） algorithm is summarized systematically,and it is revealed by an example that this QR algorithm may not converge when it is used to process the SVD of large scale matrices created by some small amplitude signals.The variation characteristic of Givens matrices in QR iteration process is studied theoretically and it is found out that the essential reason of non-convergence of this QR algorithm lies in the attenuation of elements in diagonal belt of first row of large scale matrix.This attenuation will finally make the first right Givens matrix become an identity matrix,and then all the back Givens matrices will become identity matrices too,so the invalidation of QR algorithm is caused.On this basis,the influence of attenuation of elements of first row on convergence speed of QR algorithm is further studied.All the theoretical analysis is verified by the real data,and therefore the convergence characteristic of this QR algorithm is ascertained essentially.

关 键 词：收敛性 Givens矩阵 单位阵 QR算法 奇异值分解 

分 类 号：O241.6[理学—计算数学]