On the contrary, this convergence issue in diagonalization did not appear in the test of the benzene system, and other systems with no transition metal element. Thence, the performance of the single-precision sweep method for one sweep is much better than the double precision method. The speed-up of the single-precision sweep method is slightly larger than 2, the theoretical upper bond of the single-precision optimization. This is reasonable and can be regarded as common for single-precision optimization, when it is applied on a compute-intensive task. The reason is that in the single-precision sweep method, almost all of the data and operations are transformed into single-precision. Consequently, it is highly possible to reach the theoretical upper bond. As the single-precision data form saved memory space, more data can be stored in the CPU cache and the CPU cache hit rate is increased. Therefore, the overall computational performance can be slightly higher than the theoretical upper bond. However, we have to point out that although the performance of the single-precision sweep is very high for one sweep, the speed-up of total DMRG calculation is not as good as expected. This is because of the clean-up sweeps that are required to reach the double-precision accuracy. Therefore, the clean-up sweeps must run in doubleprecision mode, whether using full double-precision or using mixed-precision diagonalization method. Consequently, these clean-up sweeps will delay the overall DMRG calculations. On the other hand, if only the mixed-precision diagonalization method is applied, all the sweeps will run with fine-grained mixed-precision method, and there will be no clean-up sweeps.