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?
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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 $A^TA$.