How to convert floating point input to integers and preserve maximum precision?

Question

I have to use an algorithm which expects a matrix of integers as input. The input I have is real valued, therefore I want to convert the input it to integer before passing it to the algorithm.

I though of scaling the input by a large constant and then rounding it to integers. This looks like a good solution but how does one decide a good constant to be used, specially since the range of float input could vary from case to case? Any other ideas are also welcome?

Answer 1

Probably the best general answer to this question is to find out what is the maximum integer value that your algorithm can accept as an element in the matrix without causing overflow in the algorithm itself. Once you have this maximum value, find the maximum floating point value in your input data, then scale your inputs by the ratio of these two maximum values and round to the nearest integer (avoid truncation).

In practice you probably cannot do this because you probably cannot determine what is the maximum integer value that the algorithm can accept without overflowing. Perhaps you don't know the details of the algorithm, or it depends in a complicated way on all of the input values. If this is the case, you'll just have to pick an arbitrary maximum input value that seems to work well enough.

Answer 2

First normalize your input to [0,1) range, then use a common way to scale them:

f(x) = range_max_exclusive * x + range_min_inclusive

After that, cast f(x) (or round if you wish) to integer. In that way you can handle situations such as real values are in range [0,1) or [0,n) where n>1.

In general, your favourite library contains matrix operations, which you can implement this technique easily and with better performance than your possible implementation.

EDIT: Scaling-down then Scaling-up is sure to get lost some precision. I favor it because a normalization operation is generally comes with the library. Also you can do that without downscaling by:

f(x) = range_max_exlusive / max_element * x + range_min_inclusive