Sharpness in rates of convergence for the symmetric Lanczos method
Sharpness in rates of convergence for the symmetric Lanczos method
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DOI：
10.1090/s0025-5718-09-02258-3
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发表时间：
2010
期刊：
影响因子：
--
通讯作者：
Ren-Cang Li
中科院分区：
文献类型：
--
作者：
Ren-Cang Li
Lanczos方法常用于求解大型稀疏对称矩阵特征值问题。有一个完善的收敛理论，产生的界限，预测收敛速度好的几个极端的特征对。这些界限建议至少线性收敛的Lanczos步骤的数量，假设有个别特征值之间的差距。在实践中，经常观察到超线性收敛。问题是“现有的界限是否能告诉正确的收敛速度？".一个肯定的答案是这里给出的两个极端的特征值的例子，其Lanczos近似的误差可比的误差界的所有Lanczos步骤。
The Lanczos method is often used to solve a large and sparse symmetric matrix eigenvalue problem. There is a well-established convergence theory that produces bounds to predict the rates of convergence good for a few extreme eigenpairs. These bounds suggest at least linear convergence in terms of the number of Lanczos steps, assuming there are gaps between individual eigenvalues. In practice, often superlinear convergence is observed. The question is "do the existing bounds tell the correct convergence rate in general?". An affirmative answer is given here for the two extreme eigenvalues by examples whose Lanczos approximations have errors comparable to the error bounds for all Lanczos steps.