IMPROVED INVERSE SCALING AND SQUARING ALGORITHMS FOR THE MATRIX LOGARITHM
IMPROVED INVERSE SCALING AND SQUARING ALGORITHMS FOR THE MATRIX LOGARITHM
复制标题
DOI：
10.1137/110852553
复制
发表时间：
2012-01-01
影响因子：
3.1
通讯作者：
Higham, Nicholas J.
中科院分区：
文献类型：
--
作者：
Al-Mohy, Awad H.;Higham, Nicholas J.
计算矩阵对数的一种流行方法是逆缩放和平方法，其基本上以逆序执行矩阵指数的缩放和平方法的步骤。在这里，我们对该方法进行了一些改进，将其发展与我们最近的版本[SIAM J. Matrix Anal.应用程序、31（2009），pp. 970-989]的缩放和平方法的指数。特别地，我们引入后向误差分析来代替以前的前向误差分析，得到了后向误差界||A（p）||（1/p），对于几个小整数p，而不是||一||并使用特殊的技术来更准确地计算Pade近似的参数。我们推导出一个算法，采用舒尔分解，从而与三角矩阵，另一个只需要矩阵乘法和多个右侧线性系统的解决方案。数值实验表明，新的算法一般是更快，更准确地比现有的同行，并建议Schur为基础的方法是计算矩阵对数的方法的选择。
A popular method for computing the matrix logarithm is the inverse scaling and squaring method, which essentially carries out the steps of the scaling and squaring method for the matrix exponential in reverse order. Here we make several improvements to the method, putting its development on a par with our recent version [SIAM J. Matrix Anal. Appl., 31 (2009), pp. 970-989] of the scaling and squaring method for the exponential. In particular, we introduce backward error analysis to replace the previous forward error analysis; obtain backward error bounds in terms of the quantities ||A(p)||(1/p), for several small integer p, instead of ||A||; and use special techniques to compute the argument of the Pade approximant more accurately. We derive one algorithm that employs a Schur decomposition, and thereby works with triangular matrices, and another that requires only matrix multiplications and the solution of multiple right-hand side linear systems. Numerical experiments show the new algorithms to be generally faster and more accurate than their existing counterparts and suggest that the Schur-based method is the method of choice for computing the matrix logarithm.