Sparse dot product by merging indices

For two sparse vectors stored as increasing nonzero-coordinate streams, advance whichever current index is smaller; that coordinate has no possible partner in the other remaining stream. At equal indices, accumulate the product and advance both. The loop invariant accounts for all matched coordinates already passed and excludes any future match for skipped smaller coordinates. Termination occurs when either stream ends. The method uses $O(k_1+k_2)$ time and constant auxiliary space, with floating-point results subject to the chosen evaluation order.