Matrix arithmetic-geometric mean and the computation of the algorithm

Abstract

We investigate the stability of the matrix arithmetic-geometric mean (AGM) iteration. We show that the classical formulation of this iteration may be not stable (a necessary and su cient condition for its stability is given) and investigate the numerical properties of alternative formulations. It turns out that the so-called Legendre form is the right choice for matrices. Due to its fast convergence and good numerical properties, our AGM formulation has the potential to play an important role in the computation of matrix functions. In fact, we developed an algorithm, whose main block is an optimized AGM scheme, for the computation of the logarithm of a matrix, which is shown to be competitive, in terms of accuracy, with the state-of-the-art methods. Methods that do not require an initial reduction to the Schur form are potentially more e cient on parallel computers. For this reason, our current implementation does not include such reduction and operates with full matrices till the end. As compared to the state-of-the-art reduction free algorithm, our method relies more heavily on matrix multiplications, which are highly suited to modern architectures, and requires a smaller number of multiple right-hand-side linear systems, making it competitive also in terms of computational e ciency. Our claims are supported with analysis and also with numerical results produced with a MATLAB code.