Let a matrix $\mathbf{X}$ with a known QR and cholesky decomposition for $\mathbf{X^TX}$. How to find the QR and cholesky decomposition for a new matrix $\mathbf{Y^TY}$ s.t. $\mathbf{Y=[X\;a]}$?
What is the reduction in computational complexity when compared to naive QR or cholesky decomposition of $\mathbf{Y}$ directly?