# Triangle on top of diagonal least squares

I need to solve many least squares problems with the following matrices: $$\pmatrix{ R \\ D_i }$$ where $$R$$ is upper triangular and $$D_i$$ is diagonal. $$R$$ is the same for all the problems, while $$D_i$$ changes for each problem. Is there some way to factor $$R$$ to make it efficient to solve these problems?

A Givens approach doesn't seem to work, since it will introduce many non-zero entries into the $$D_i$$ part of the matrix.

Right now I am doing a full QR factorization of each matrix, which doesn't account for the sparse structure of the problem and is inefficient.

• I am afraid that this is a variant of the classical question "can I update matrix factorizations after a diagonal update" which has been asked many times here and has answer "no". – Federico Poloni May 25 at 18:18