Suppose that $a,b,c,d$ are real number and $k$ is a positive integer number. Define the following function:
$f(k) = \frac{a + ck}{b + dk}.$
So, $f(0) = \frac{a}{b}$, $f(1) = \frac{a+c}{b+d}$, and so on. Is there a way to incrementally compute $f(1), f(2),\ldots, f(n)$ avoiding any division and possibly multiplications?
PS. This is related to this (Fast patch extraction using homography). I want to avoid homography-vector multiplications in a real-time image processing code. I think i got a way to do that, but for every pixel coordinates that i compute using the homography, i need to divide by the third component (please, look at that question to understand better what i am saying). I was wondering if i could avoid the 2 extra division-per-pixel, since i need it fast...