Computational aspects of the geometric mean of two matrices: a survey

Algorithms for the computation of the (weighted) geometric mean G of two positive definite matrices are described and discussed. For large and sparse matrices the problem of computing the product \(y=Gb\) $y=Gb$ , and of solving the linear system \(Gx=b\) $Gx=b$ , without forming G, is addressed. An analysis of the conditioning is provided. Substantial numerical experimentation is carried out to test and compare the performances of these algorithms in terms of CPU time, numerical stability, and number of iterative steps.