Summing divergent matrix series

Abstract

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.