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The fast multipole method (FMM) is developed to analyse electromagnetic scattering using higher-order hierarchical Legendre basis functions, which have been proposed recently and make the impedance matrix, from method of moments, a low condition number. In this method, the point-to-point based-on-sample quadrature is used because the basis functions are defined on large surface patches of scatterers. The numerical implementations of matrix-vector multiply by FMM are particularly presented. These numerical results show that the proposed second- or third-order FMM (high-order FMM) is accurate and efficient.