Sparse approximations of matrix functions via numerical integration of ODEs

We consider the numerical computation of matrix functions f(X) via matrix ODE integration. The solution is modeled as an asymptotic steady state of a proper differential system. The framework we propose, allows to define flows of sparse matrices leading to sparse approximations to f(X). We discuss of this approach giving stability and approximation results in a general case. We apply our method to the factorization of matrices (LU, Cholesky) as well as the computation of the square root. Numerical illustrations are presented.