  CholeskyQR is a simple and fast QR decomposition via Cholesky decomposition,
while it has been considered highly sensitive to the condition number. In this
paper, we provide a randomized preconditioner framework for CholeskyQR
algorithm. Under this framework, two methods (randomized LU-CholeskyQR and
randomized QR-CholeskyQR) are proposed and discussed. We prove the proposed
preconditioners can effectively reduce the condition number, which is also
demonstrated by numerical tests. Abundant numerical tests indicate our methods
are more stable and faster than all the existing algorithms and have good
scalability.
