利用计算图实现矩阵对数的多项式逼近
Polynomial approximations for the matrix logarithm with computation graphs
AI总结：
本文针对矩阵对数计算中矩阵乘法与左除法开销大的问题，提出基于计算图的高效多项式求值方案，并结合舍入误差与后向误差分析，实验表明其在时间与精度上均具竞争力。
摘要：
计算矩阵对数最常用的方法是逆缩放平方方法与 Padé 逼近相结合，有时还会辅以 Schur 分解。其主要计算开销集中在矩阵-矩阵乘法和左矩阵除法上。在本工作中，我们表明，通过使用一种高效多项式求值方案的基于图的表示，可以大幅减少此类运算的次数。我们提出了一种分析舍入误差的技术，并对后向误差分析进行了适配。我们提供了大量仿真，从计算时间和舍入误差两方面说明了该方法的竞争力。
英文摘要：
The most popular method for computing the matrix logarithm is a combination of the inverse scaling and squaring method in conjunction with a Padé approximation, sometimes accompanied by the Schur decomposition. The main computational effort lies in matrix-matrix multiplications and left matrix division. In this work we illustrate that the number of such operations can be substantially reduced, by using a graph based representation of an efficient polynomial evaluation scheme. A technique to analyze the rounding error is proposed, and backward error analysis is adapted. We provide substantial simulations illustrating competitiveness both in terms of computation time and rounding errors.
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