Recursive Least Square (RLS) filter has been applied to real-time lossless compression of hyperspectral imagery and been proved a high performance onboard algorithm. Recent research has revealed that the RLS filter with Adaptive-Length-Prediction (ALP) can significantly improve the compression performance. However, the prediction procedure with numerous bands slows down the run-time and is nearly impossible to be applied onboard. In this letter, we proposed a fast RLS algorithm which can accelerate the ALP stage by exploiting the feature of the projection matrix of the RLS algorithm. The experiment results illustrated that with the same compression ratio, the proposed algorithm is 100 times faster than the traditional RLS algorithm with ALP.