I have two vectors 'A1' and 'A2' of size M and N. I have an algorithmic problem related to the 'additive outer product' (rather than the usually 'multiplicative') matrix of the two vectors
$$A'(i,j) = A1(i) + A2(j)$$
Firstly is there better mathematical terminology to describe this linear algebraic operation?
The overall problem I'm trying to solve is as follows:
I have vectors A1 and B1 of size M and A2 and B2 of size N.
A1,A2,B1 and B2 are all sorted and strictly monotonically increasing in value ( A(i+1) > A(i) for all valid $i$).
If I calculate two of these 'additive outer product' matrices that I need a better name for
$A'(i,j) = A1(i)+ A2(j)$
$B'(i,j) = B1(i)+ B2(j)$
Is there an efficient way of extracting the members of this matrix (i,j) so that A' and B' are sorted such that:
$A'(i,j) > A'(k,l) \implies B'(i,j) > B'(k,l)$ or $A'(i,j) < A'(k,l) \implies B'(i,j) < B'(k,l)$
for all $i \ne k$ and $j \ne l$.
without brute force calculation and sorting of every member of the two matrices?
Again, surely there is better terminology to describe this problem than I've used above.
Thanks very much for any suggestions or pointers for further reading and investigation.