Cholesky of matrix plus identity

I have a positive definite matrix $A$ ($n \times n$ dimension) for which I have the Cholesky decomposition $A=LL^{'}$. I want to use this to compute

a) The Cholesky decomposition of $A+c^2\times I $ where $c$ is a constant and $I$ is the identity matrix

b) The Cholesky decomposition of $A+BB^{'}$ where $B$ is a $n \times n$ sparse matrix with each row having at most $k$ elements for some fixed $k << n$.

Is there any analytical/ computational method/ R-package that can use the already available Cholesky decomposition of $A$ and perform (a) and (b) in a computationally scalable way i.e. ($O(n)$ complexity). Note that (a) is a special case of (b) with $B=cI$. Any references will be appreciated.

Thanks

Solution 1:

I don't see what you can expect.

The direct way is to solve the system in the unknown triangular matrix $X$: $XX^T=A+BB^T,x_{i,i}>0$.

If you want to use your decomposition $A=LL^T$, then $A+BB^T=(L+Y)(L+Y)^T$; the system to solve is now $YY^T+YL^T+LY^T-BB^T=0,y_{i,i}>0$; clearly, this system is more complicated than the above one!

Note that, if $B$ is a small matrix, then we obtain a good approximation by solving the linear equation $YL^T+LY^T-BB^T=0$. This system is stable when the $l_{i,i}$ are not too small.