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Two efficient algorithms are proposed to analyze a large-scale periodic structure with finite size using the subentire-domain (SED) basis functions and the conjugate-gradient fast Fourier transform (CG-FFT). The SED basis function is defined on the support of each single element of the periodic structure. In a simplified SED (SSED)-CG-FFT algorithm, all elements of the periodic structure share the same SED basis function. As a consequence, SSED-CG-FFT can be performed in the whole periodic structure. However, SSED-CG-FFT becomes less accurate if the gap between two unit elements is very small, where the single SED basis function cannot capture the strong mutual coupling. In order to consider the mutual coupling, an accurate SED (ASED)-CG-FFT algorithm is proposed. In this algorithm, nine types of SED basis functions are employed to distinguish interior cells, edge cells, and corner cells. As a consequence, ASED-CG-FFT can be performed in all interior cells of the periodic structure. Comparing with the conventional method of moments with subdomain basis functions, the proposed algorithms are more efficient in both the computational complexity and the memory requirement. Numerical results are given to test the validity and efficiency of the proposed methods.