Fast Computation of the Fractional Power of a Matrix

Abstract. The matrix power of fractional order is often required in the simulation of fractional partial differential equations, which is a core bottleneck in nonlocal or fractional models. This paper develops a uniformly fast method for computing the matrix power of the form [Formula: see text], where [Formula: see text] and [Formula: see text] is a positive-definite matrix. The new method overcomes the disadvantages of the existing methods that do not work well when [Formula: see text] approaches zero or one. We then extend our method to calculate [Formula: see text], where [Formula: see text] is an identity matrix and [Formula: see text]. Sufficient numerical simulations are performed to show the effectiveness of the new methods, and comparisons with the state-of-the-art methods are also made to show the advantages of the new methods.