The most popular method for computing the matrix logarithm is a combination of the inverse scaling and squaring method in conjunction with a Pad\'e 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.

翻译：计算矩阵对数最常用的方法是将逆缩放平方方法与Padé近似相结合，有时辅以Schur分解。其核心计算量主要集中于矩阵乘法与左矩阵除法。本研究通过构建高效多项式求值方案的计算图表示，证明上述运算次数可大幅减少。我们提出了舍入误差分析方法，并改进了向后误差分析框架。大量仿真实验表明，该方法在计算时间与舍入误差控制方面均具有显著竞争力。