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In this paper we consider efficient algorithms for solving the algebraicequation ${\mathcal A}^\alpha {\bf u}={\bf f}$, $0< \alpha <1$, where${\mathcal A}$ is a symmetric and positive definite matrix obtained form finitedifference or finite element approximations of second order elliptic problemsin ${\mathbb R}^d$, $d=1,2,3$. The method is based on the best uniform rationalapproximation of the function $t^{\beta-\alpha}$ for $0 < t \le 1$ and natural$\beta$, and the assumption that one has at hand an efficient method (e.g.multigrid, multilevel, or other fast algorithm) for solving equations like$({\mathcal A} +c {\mathcal I}){\bf u}= {\bf f}$, $c \ge 0$. The providednumerical experiments on model problems with ${\mathcal A}$ obtained by finiteelement approximation of elliptic equations in one and three spacial dimensionsconfirm the efficiency of the proposed algorithms.

Optimal Solvers for Linear Systems with Fractional Powers of Sparse SPD Matrices