We will offer a method to improve energy efficient consumption for processing queries on the Internet of Things. We focused on an energy efficient hierarchical clustering index tree such that we can facilitate time-correlated region queries in the I.O.T (Internet of Things). We try to improve clustering and make a change on its proposed index tree. We try to do this by optimizing the query processing. We improve clustering to increase the accuracy of the Internet of Things and prevent the network from disconnecting. In the article that we have chosen, there is a heterogeneous cluster which means there exists a large data difference in the two ends of a cluster. Also, it often happens that the same information is sent to the base station by two overlapping clusters; therefore, we save energy by eliminating duplicated data.

We will offer a method to improve energy efficient consumption for processing queries on the Internet of Things. We focused on an energy efficient hierarchical clustering index tree such that we can facilitate time-correlated region queries in the I.o.T (Internet of Things). We try to improve clustering and make a change on its proposed index tree. We try to do this by optimizing the query processing. We improve clustering to increase the accuracy of the Internet of Things and prevent the network from disconnecting. In the article that we have chosen, there is a heterogeneous cluster which means there exists a large data difference in the two ends of a cluster. Also, it often happens that the same information is sent to the base station by two overlapping clusters; therefore, we save energy by eliminating duplicated data.

Parallel programming is an effective way to increase the speed of processing applications. It is carried out simultaneously by multiple processors rather than by a single processor. We compare the number of necessary calculations for multiplying the chain matrix in normal mode with the parallel mode. Since we used the famous parallel language named CUDA in our program, we will first present a brief description of the language and secondly, we explain essential mathematical notions and compare the performance of both programs.

The double exponential formula was introduced for calculating definite integrals with singular point oscillation functions and Fourier integral. The double exponential transformation is not only useful for numerical computations but it is also used in different methods of Sinc theory. In this paper we give an upper bound estimate for the error of double exponential transformation. By improving integral estimates having singular final points, in theorem 1 we prove that the method is convergent and the rate of convergence is O(h 2 ) where h is a step size. Our main tool in the proof is DE formula in Sinc theory. The advantage of our method is that the time and space complexity is drastically reduced. Furthermore, we discovered upper bound error in DE formula independent of N truncated number, as a matter of fact we proved stability. Numerical tests are presented to verify the theoretical predictions and confirm the convergence of the numerical solution. MSC: 65D30, 65D32.

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