Computing Matrix-Matrix Addition using QR and/or SVD

Apologies if this sounds like an uninformed question but I was wondering if there are theoretical results that talk about the following problem:

Suppose we have two matrices and with and they can be written as the following via QR decomposition:

and

Is there a way we can get the QR decomposition of the matrix without explicitly adding and together and by only using the individual QR decomposition of both and . Specifically, I want to know if there are theoretical results that either talk about the feasibility of this algorithmically or if not? provide a justification of why it cannot be done. Also, can the same be said about the SVD of ?

[Jan]

This question has no relation to Matlab.

[Tarek Hajj Shehadi]

I could provide a possible computational motivation since I am motivated to consider cases where addition of two matrices will lead to absorption since we are solving finite precision. Let us say that elements of A are of scale e<<<1 and elements of B are of scale 1/e then can an approach involving QR or SVD help in finding A+B? while trying to avoid as much numerical loss as possible?

[Matt J]

I don't think I see how that would help you. Let's take the simple case where,

A=e*eye(2);
B=1/e*eye(2);

The QR decomposition of A+B is

Q=eye(2);
R=(e+1/e)*eye(2);

How do you use this to deal with the the case where (e+1/e) absorbs to 1/e in double float precision?

[Tarek Hajj Shehadi]

Thank you very much Matt, I will accept your counter example.