Accurate Eigensystem Computations by Jacobi Methods
Accurate Eigensystem Computations by Jacobi Methods
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DOI：
10.1137/s089547989324820x
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发表时间：
1995-07
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影响因子：
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通讯作者：
R. Mathias
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文献类型：
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作者：
R. Mathias
Demmel和Veselic证明，在有一个较小的但书的情况下，Jacobi方法计算正定矩阵的特征值和特征向量比首先将矩阵三对角化的方法更准确。我们推广了他们的分析，从而：我们去掉了他们结果中的次要条件，从而保证了雅可比方法的准确性。我们展示了如何廉价地后验地检查一个特定矩阵的三对角化是否在该矩阵的特征值中引起了较大的相对扰动。这在处理分级矩阵时很有用。我们推导出的混合Jacobi算法具有与Jacobi方法相同的精度，但速度更快，至少在串口计算机上是这样。我们证明了，如果$G$是一个$m\x n$矩阵，并且$m>；>；n$，那么Jacobi方法计算奇异值的速度几乎和标准方法一样快，但可能更准确。
Demmel and Veselic showed that, subject to a minor proviso, Jacobi's method computes the eigenvalues and eigenvectors of a positive definite matrix more accurately than methods that first tridiagonalize the matrix. We extend their analysis and thereby: \begin{remunerate} \item We remove the minor proviso in their results and thus guarantee the accuracy of Jacobi's method. \item We show how to cheaply check, a posteriori, whether tridiagonalizing a particular matrix has caused a large relative perturbation in the eigenvalues on the matrix. This can be useful when dealing with graded matrices. \item We derive hybrid Jacobi algorithms that have the same accuracy of Jacobi's method but are faster, at least on a serial computer. \item We show that if $G$ is an $m \times n$ matrix and $m >>n$ then Jacobi's method computes the singular values almost as quickly as standard methods, but potentially much more accurately.