Update based on Michael's answer (thanks again!) - Can the LLL or PSLQ algorithms provide a (knowably - i.e. not just incidental) unique solution for the set of integer multiplicative factors? Are there other algorithms (perhaps with worse run-time complexities) that can?


Imagine that one has a set of 'n' finite-precision real numbers - (r\_1, r\_2, ..., r\_n), each with an associated positive integer multiplicative factor, (i\_1, i\_2, ..., i\_n). Here, one only has access to the values in the set of real numbers, as well as the sum of the real numbers multiplied by their corresponding multiplicative factors, 'S', i.e. - Sum(i\_1\*n\_1, i\_2\*n\_2, ..., i\_n\*r\_n). 


What's the most efficient way to test that the sum of a particular set of real numbers will (or, obviously, will not) always allow us to extract a unique solution for the set of integer multiplicative factors? Or to find/do this with fewest restrictions on the values of the integer multiplicative factors?


In the case that this has a very simple answer, apologies.


(Edit - Changed "each with an associated "set of" positive integer multiplicative factors" --> "each with an associated positive integer multiplicative factor". One finite-precision real number 'r\_k' has one integer multiplicative factor 'i\_k'.)



Extracting integer multiplicative factors from the sum of certain sets of (finite-precision) real numbers?
You want [integer relation algorithms](http://en.wikipedia.org/wiki/Integer_relation_algorithm), such as the [LLL algorithm](http://en.wikipedia.org/wiki/Lenstra%E2%80%93Lenstra%E2%80%93Lov%C3%A1sz_lattice_basis_reduction_algorithm). 


