Both sample entropy and approximate entropy are measurements of complexity. The two methods have received a great deal of attention in the last few years, and have been successfully verified and applied to biomedical applications and many others. However, the algorithms proposed in the literature require O(N(2)) execution time, which is not fast enough for online applications and for applications with long data sets. To accelerate computation, the authors of the present paper have developed a new algorithm that reduces the computational time to O(N(3/2))) using O(N) storage. As biomedical data are often measured with integer-type data, the computation time can be further reduced to O(N) using O(N) storage. The execution times of the experimental results with ECG, EEG, RR, and DNA signals show a significant improvement of more than 100 times when compared with the conventional O(N(2)) method for N=80,000 (N=length of the signal). Furthermore, an adaptive version of the new algorithm has been developed to speed up the computation for short data length. Experimental results show an improvement of more than 10 times when compared with the conventional method for N>4000.