Let $A_1=L_1U_1$ and $A_2=L_2U_2$ be two matrices with their respective LU-factorizations ($L_i$ is lower triangular and $U_i$ upper triangular).
Is it possible to obtain the LU decomposition of the matrix product $A_1A_2$, exploiting the known factorizations of $A_1,A_2$?
Or is the best one can do just to compute the factorization of $A_1A_2$ from scratch?