A Novel Approach for the Efficient Computation of 1-D and 2-D Summations

A novel computational method is proposed to evaluate 1-D and 2-D summations and integrals which are relatively difficult to compute numerically. The method is based on applying a subspace algorithm to the samples of partial sums and approximating them in terms of complex exponentials. For a convergent summation, the residue of the exponential term with zero complex pole of this approximation corresponds to the result of the summation. Since the procedure requires the evaluation of relatively small number of terms, the computation time for the evaluation of the summation is reduced significantly. In addition, by using the proposed method, very accurate and convergent results are obtained for the summations which are not even absolutely convergent. The efficiency and accuracy of the method are verified by evaluating some challenging 1-D and 2-D summations and integrals.