Optimal scaling for p‐norms and componentwise distance to singularity
Optimal scaling for p‐norms and componentwise distance to singularity
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DOI：
10.1093/imanum/23.1.1
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发表时间：
2003
影响因子：
2.1
通讯作者：
S. Rump
中科院分区：
文献类型：
--
作者：
S. Rump
在这篇注记中，我们给出了双边对角标度可达到的最优p-范数的上下界。对某些矩阵的不可约性没有任何假设。对于2-范数，这些界被证明是最优的。对于1-范数和inf-范数，证明了最优条件数的(已知)精确值。给出了极小对角线矩阵的计算方法。此外，还给出了到最近奇异矩阵的分支距离的一类新的下界。事实证明，它们比现有的要好。
In this note we give lower and upper bounds for the optimal p-norm condition number achievable by two-sided diagonal scalings. There are no assumptions on the irreducibility of certain matrices. The bounds are shown to be optimal for the 2-norm. For the 1-norm and inf-norm the (known) exact value of the optimal condition number is conflrmed. We give means how to calculate the minimizing diagonal matrices. Furthermore, a class of new lower bounds for the componentwise distance to the nearest singular matrix is given. They are shown to be superior to existing ones.