First, let us reexamine the diagonalization procedure. The result x = V • y, can be seen as the weighted summation of v i , and should be not much different from the initial v 0 and v 1 . Thus, the other v i can be seen as the correction of these initial guesses, and should be not large. Then, if we make this initial guess (i.e. v 0 and v 1 ) evaluated in double-precision, and other corrections (i.e. v 2 , v 3 , . . . ) evaluated in single precision, we can accelerate the overall calculation speed and maintain the double-precision accuracy. This mixed-precision diagonalization method is shown in Table . In this method, the orthogonalization procedure should also be calculated in double precision, as the coarse-grained level method, to guarantee the correctness.