In PDEs, we typically use something called a Krylov subspace iterative solver.<p>These solvers construct a solution by adding up terms of the form c_n * A^n * y and truncating the series when the residuals are small enough.  This is especially efficient in PDEs, where problem matrices are sparse so matrix-vector products are efficient (i.e. O(n)), and in practice only a few terms of the series are necessary.