论文标题
一种改善三角矩阵功能计算的技术
A Technique for Improving the Computation of Functions of Triangular Matrices
论文作者
论文摘要
我们提出了一种简单的技术，如果与用于计算三角矩阵函数的算法相结合，则可以使其更有效。基本上，这种技术包括特定的缩放相似性变换，从而降低了三角矩阵的正态性，从而降低了其标准并通常其功能条件数。只要它与涉及先前的Schur分解的算法结合使用，它可以很容易地扩展到非三角形矩阵。应详细讨论应使用或不使用该技术的情况。特别注意特定的算法，例如逆缩放和平方对矩阵对数以及缩放和平方的矩阵指数。理论结果支持我们的提案的优势，并通过数值实验进行了说明。
We propose a simple technique that, if combined with algorithms for computing functions of triangular matrices, can make them more efficient. Basically, such a technique consists in a specific scaling similarity transformation that reduces the departure from normality of a triangular matrix, thus decreasing its norm and in general its function condition number. It can easily be extended to non-triangular matrices, provided that it is combined with algorithms involving a prior Schur decomposition. Situations where the technique should be used or not will be discussed in detail. Special attention is devoted to particular algorithms like the inverse scaling and squaring to the matrix logarithm and the scaling and squaring to the matrix exponential. The advantages of our proposal are supported by theoretical results and illustrated with numerical experiments.
