The exact diagonalization is the most accurate approach for solving the Hubbard model. The approach calculates the ground state of the Hamiltonian derived exactly from the model. Since the Hamiltonian is a large sparse symmetric matrix, we usually utilize an iteration method. It has been reported that the LOBPCG method is one of the most effectual solvers for this problem. Since most operations of the method are linear operations, the method can be executed on CUDA GPU, which is one of the mainstream processors, by using cuBLAS and cuSPARSE libraries straightforwardly. However, since the routines are executed one after the other, cached data can not be reused among other routines. In this research, we tune the routines by fusing some of their loop operations in order to reuse cached data. Moreover, we propose the tuning strategies for the Hamiltonian-vector multiplication with shared memory system in consideration of the character of the Hamiltonian. The numerical test on NVIDIA Tesla P100 shows that the tuned LOBPCG code is about 1.5 times faster than the code with cuBLAS and cuSPARSE routines.