This work presents a new algorithm to compute the matrix exponential within a given tolerance. Combined with the scaling and squaring procedure, the algorithm incorporates Taylor, partitioned and classical Pad\'e methods shown to be superior in performance to the approximants used in state-of-the-art software. The algorithm computes matrix--matrix products and also matrix inverses, but it can be implemented to avoid the computation of inverses, making it convenient for some problems. If the matrix A belongs to a Lie algebra, then exp(A) belongs to its associated Lie group, being a property which is preserved by diagonal Pad\'e approximants, and the algorithm has another option to use only these. Numerical experiments show the superior performance with respect to state-of-the-art implementations.

翻译：本文提出了一种新算法，可在给定容差内计算矩阵指数。结合缩放与平方过程，该算法融合了泰勒、分块和经典Padé方法，其性能优于当前先进软件中使用的近似函数。该算法需计算矩阵乘积及矩阵逆，但可通过实现避免求逆运算，从而适用于特定问题。若矩阵A属于李代数，则exp(A)属于其关联李群——这一性质可由对角Padé逼近保持，算法还提供了仅使用此类逼近的选项。数值实验表明，该算法相较现有先进实现具有更优性能。