We propose a fast and efficient approach for solving the Bogoliubov-de Gennes (BdG) equations in superconductivity, with a numerical matrix-size reduction procedure proposed by Sakurai and Sugiura [J. Comput. Appl. Math. 159 (2003) 119]. The resultant small-size Hamiltonian contains the information of the original BdG Hamiltonian in a given low-energy domain. Thus, the present approach leads to a numerical construction of a low-energy effective theory in superconductivity. The combination with the polynomial expansion method allows a self-consistent calculation of the BdG equations. Through numerical calculations of quasi-particle excitations in a vortex lattice, thermal conductivity, and nuclear magnetic relaxation rate, we show that our approach is suitable for evaluating physical quantities in a large-size superconductor and a nano-scale superconducting device, with the mean-field superconducting theory.