We extend several celebrated methods in classical analysis for summing series of complex numbers to series of complex matrices. These include the summation methods of Abel, Borel, Ces\'aro, Euler, Lambert, N\"orlund, and Mittag-Leffler, which are frequently used to sum scalar series that are divergent in the conventional sense. One feature of our matrix extensions is that they are fully noncommutative generalizations of their scalar counterparts -- not only is the scalar series replaced by a matrix series, positive weights are replaced by positive definite matrix weights, order on $\mathbb{R}$ replaced by Loewner order, exponential function replaced by matrix exponential function, etc. We will establish the regularity of our matrix summation methods, i.e., when applied to a matrix series convergent in the conventional sense, we obtain the same value for the sum. Our second goal is to provide numerical algorithms that work in conjunction with these summation methods. We discuss how the block and mixed-block summation algorithms, the Kahan compensated summation algorithm, may be applied to matrix sums with similar roundoff error bounds. These summation methods and algorithms apply not only to power or Taylor series of matrices but to any general matrix series including matrix Fourier and Dirichlet series. We will demonstrate the utility of these summation methods: establishing a Fej\'{e}r's theorem and alleviating the Gibbs phenomenon for matrix Fourier series; extending the domains of matrix functions and accurately evaluating them; enhancing the matrix Pad\'e approximation and Schur--Parlett algorithms; and more.

翻译：我们将经典分析中用于求和复数级数的几种著名方法推广到复数矩阵级数的求和。这些方法包括Abel、Borel、Cesáro、Euler、Lambert、Nörlund和Mittag-Leffler求和法，这些方法常用于求和在常规意义下发散的标量级数。我们矩阵推广的一个特点是，它们是对应标量方法的完全非交换推广——不仅标量级数被矩阵级数取代，正权重被正定矩阵权重取代，$\mathbb{R}$上的序被Loewner序取代，指数函数被矩阵指数函数取代，等等。我们将证明这些矩阵求和法的正则性，即当应用于常规意义下收敛的矩阵级数时，我们得到的和值相同。我们的第二个目标是提供与这些求和方法协同工作的数值算法。我们讨论了如何将分块与混合分块求和算法、Kahan补偿求和算法应用于矩阵求和，并得到类似的舍入误差界。这些求和方法和算法不仅适用于矩阵的幂级数或泰勒级数，也适用于任何一般的矩阵级数，包括矩阵傅里叶级数和狄利克雷级数。我们将展示这些求和方法的实用性：建立矩阵傅里叶级数的Fejér定理并缓解吉布斯现象；扩展矩阵函数的定义域并对其进行精确求值；改进矩阵Padé逼近和Schur–Parlett算法；等等。