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In the paper, we present a high order fast algorithm with almost optimummemory for the Caputo fractional derivative, which can be expressed as aconvolution of $u'(t)$ with the kernel $(t_n-t)^{-\alpha}$. In the fastalgorithm, the interval $[0,t_{n-1}]$ is split into nonuniform subintervals.The number of the subintervals is in the order of $\log n$ at the $n$-th timestep. The fractional kernel function is approximated by a polynomial functionof $K$-th degree with a uniform absolute error on each subinterval. We save$K+1$ integrals on each subinterval, which can be written as a convolution of$u'(t)$ with a polynomial base function. As compared with the direct method,the proposed fast algorithm reduces the storage requirement and computationalcost from $O(n)$ to $O((K+1)\log n)$ at the $n$-th time step. We prove that theconvergence rate of the fast algorithm is the same as the direct method even ahigh order direct method is considered. The convergence rate and efficiency ofthe fast algorithm are illustrated via several numerical examples.

High order fast algorithm for the Caputo fractional derivative