Accuracy and effectiveness of the Lanczos algorithm for the symmetric eigenproblem
Accuracy and effectiveness of the Lanczos algorithm for the symmetric eigenproblem
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DOI：
10.1016/0024-3795(80)90167-6
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发表时间：
1980-12
影响因子：
1.1
通讯作者：
C. Paige
中科院分区：
文献类型：
--
作者：
C. Paige
利用Lanczos算法可以精确地、快速地求出大型稀疏对称矩阵A的特征值和特征向量，而无需重新正交化。该算法给出了基本上正确的信息特征系统的A，虽然它不一定给正确的多重性，甚至单一的，特征值。确定该算法给出的每个特征值的准确度的有用界限是很简单的。该算法的初始行为是令人惊讶的好：它产生的向量跨越Krylov子空间的矩阵非常接近A，直到这个子空间包含一个确切的特征向量的矩阵非常接近A，到这一点的有效行为的算法的特征问题是非常类似的Lanczos算法使用完全再正交化。这有助于解释基本Lanczos算法的显著行为。
Eigenvalues and eigenvectors of a large sparse symmetric matrixAcan be found accurately and often very quickly using the Lanczos algorithm without reorthogonalization. The algorithm gives essentially correct information on the eigensystem ofA, although it does not necessarily give the correct multiplicity of multiple, or even single, eigenvalues. It is straightforward to determine a useful bound on the accuracy of every eigenvalue given by the algorithm. The initial behavior of the algorithm is surprisingly good: it produces vectors spanning the Krylov subspace of a matrix very close toAuntil this subspace contains an exact eigenvector of a matrix very close toA, and up to this point the effective behavior of the algorithm for the eigenproblem is very like that of the Lanczos algorithm using full reorthogonalization. This helps to explain the remarkable behavior of the basic Lanczos algorithm.