The most important step is the subtraction of two numbers that are extremely close to each other, hence resulting in a great loss of significant digits. Consequently, the Gram-Schmidt orthogonalization should be conducted under double-precision. In fact, using a higher precision in the orthogonalization process to improve convergence has already been applied by Alvermann, et al. to achieve a robust orthogonalization in Jacobi-Davidson method. Even though they were applying a two-double projection in a double-precision diagonalization. This method was also been applied by Carson, et al. in the mixed-precision s-step Lanczos method for the same reason and in a similar way to Alvermann. It was also found that if the evaluation of H k remain in double-precision, the accuracy of the result will be improved. In fact, in the diagonalization procedure, the Ĥv calculation consumes more than 90% of the computational cost. Thus, other operations whether in double-precision or in single-precision, have little influence on the overall performance. Moreover, the additional conversion operations will have little effect of the entire computational cost.