Let A be a n×n positive-definite Hermitian matrix. I already have the QR decomposition of A. Is there an efficient way to utilize this knowledge to speed up the Cholesky decomposition of A?
No. In general, the QR decomposition has no relation to the Cholesky decomposition of a symmetric positive definite matrix. Moreover, the QR decomposition is substantially more expensive to compute than Cholesky 's decomposition.
However, the QR decomposition of tall matrix $A$ of full rank is closely related to the problem of computing a Cholesky factorization of the nonsingular matrix $A^T A$. Specifically, if $A = QR$, then $A^TA = R^T Q^T Q R = L L^T$ where the lower triangular matrix $L = R^T$ can now be identified as the Cholesky factor of $A^TA$.