Based on the background value of the GM (1,1) direct model, this paper finds new grey derivative matching the background value by the method of seeking derivatives to background value, and then gets a new grey differential equation. Combined with the new information principle to optimize initial conditions, this paper establishes a new GM (1,1) direct model. The example shows the new model is suitable for the low growth of approximate non-homogeneous index series, and it also has a good simulation and forecast effect for the high growth of the approximate non-homogeneous index series .so it has certainly theoretical value and practical value.