一种经典的矩阵预处理算法分析
Analysis of a Classical Matrix Preconditioning Algorithm
摘要：
我们研究了一个经典的迭代算法平衡矩阵的$L_\infty$规范通过缩放变换。这种算法可以追溯到20世纪60年代的Osborne和Parlett \& Reinsch，在许多数值线性代数软件包中作为标准预条件子实现。令人惊讶的是，尽管它被广泛使用了几十年，但它的收敛速度却没有任何限制。本文证明了对于任何不可约的$n\times n$（实数或复数）输入矩阵~ $A$，算法的一个自然变体收敛于$O(n^3\log(nρ/\varepsilon))$初等平衡运算，其中$ρ$度量~ $A$的初始不平衡，而$\varepsilon$是输出矩阵的目标不平衡. (The~ $A$的不平衡是$\max_i |\log(a_i^{\text{out}}/a_i^{\text{in}})|$，其中$a_i^{\text{out}},a_i^{\text{in}}$分别是$i$第行和第列中的幅度的最大条目。该界限紧到$\log n$因子。平衡操作缩放$i$的第行和第列，使得它们的最大条目相等，并且平均需要$O(m/n)$算术运算，其中$m$是~ $A$中非零元素的数目。因此，迭代算法的运行时间是$\tilde{O}(n^2m)$。这是第一个时间界限的任何种类的任何变种的Osspell-Parlett-Reinsch算法。我们还证明了一个猜想陈的特点，这些矩阵的平衡过程的限制是独立的顺序，其中进行平衡操作。
英文摘要：
We study a classical iterative algorithm for balancing matrices in the $L_\infty$ norm via a scaling transformation. This algorithm, which goes back to Osborne and Parlett \& Reinsch in the 1960s, is implemented as a standard preconditioner in many numerical linear algebra packages. Surprisingly, despite its widespread use over several decades, no bounds were known on its rate of convergence. In this paper we prove that, for any irreducible $n\times n$ (real or complex) input matrix~$A$, a natural variant of the algorithm converges in $O(n^3\log(nρ/\varepsilon))$ elementary balancing operations, where $ρ$ measures the initial imbalance of~$A$ and $\varepsilon$ is the target imbalance of the output matrix. (The imbalance of~$A$ is $\max_i |\log(a_i^{\text{out}}/a_i^{\text{in}})|$, where $a_i^{\text{out}},a_i^{\text{in}}$ are the maximum entries in magnitude in the $i$th row and column respectively.) This bound is tight up to the $\log n$ factor. A balancing operation scales the $i$th row and column so that their maximum entries are equal, and requires $O(m/n)$ arithmetic operations on average, where $m$ is the number of non-zero elements in~$A$. Thus the running time of the iterative algorithm is $\tilde{O}(n^2m)$. This is the first time bound of any kind on any variant of the Osborne-Parlett-Reinsch algorithm. We also prove a conjecture of Chen that characterizes those matrices for which the limit of the balancing process is independent of the order in which balancing operations are performed.
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