非奇异阵特征极值和条件数的近似计算

雷光耀

非奇异阵特征极值和条件数的近似计算

雷光耀

中国科学院应用数学研究所, 北京

基金项目: 国家自然科学基金资助的课题

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出版历程
- 收稿日期: 1990-05-03
- 刊出日期: 1992-02-15

On the Approximate Computation of Extreme Eigenvalues and the Condition Number of Nonsingular Matrices

Lei Quang-yao

Institute of Applied Mathematics, Academia Sinica, Beijing

摘要

摘要: 本文从共轭梯度法的公式推导出对称正定阵A与三对角阵B的相似关系,B的元素由共轭梯度法的迭代参数确定.因此,对称正定阵的条件数计算可以化成三对角阵条件数的计算,并且可以在共轭梯度法的计算中顺带完成.它只需增加O(s)次的计算量,s为迭代次数.这与共轭梯度法的计算量相比是可以忽略的.当A为非对称正定阵时,只要A非奇异,即可用共轭梯度法计算A^TA的特征极值和条件数,从而得出A的条件数.对不同算例的计算表明,这是一种快速有效的简便方法.
- 对称正定矩阵 /
- 共轭梯度法 /
- 特征值 /
- 条件数
Abstract: From the formulas of the conjugate gradient, a similarity between a symmetric positive definite(SPD) matrix A and a tridiagonal matrix B is obtained.The elements of the matrix B are determined by the parameters of the conjugate gradient.The computation of eigenvalues of A is then reduced to the case of the tridiagonal matrix B.The approximation of extreme eigenvalues of A can be obtained as a ‘by-product' in the computation of the conjugate gradient if a computational cost of O(s) arithmetic operations is added, where s is the number of iterations This computational cost is negligible compared with the conjugate gradient.If the matrix A is not SPD, the approximation of the condition number of A can be obtained from the computation of the conjugate gradient on A^TA.Numerical results show that this is a convenient and highly efficient method for computing extreme eigenvalues and the condition number of nonsingular matrices.
- symmetric positive definite matria /
- conjugate gradient /
- eigenvalues /
- condition number

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参考文献(6)

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[5]	雷光耀,ICCG及MICCG的讨论与改进,应用数学学报(待发表).
[6]	Lei,G.Y.,Block preconditioned conjugate gradient method for large sparse systems,Technical Report 050,Institute of Applied Mathematics,Academia Sinica,Beijing(1989),1-18.