I'm not sure if this is the right place to ask, or what the right terminology to use is. The problem I have is this: I have a vector for example: v = [1,4,5,6,3,1,4,5,6,7,...7]
and I have a set of other vectors for example: a1 = [0,0,0,0,0,1,2,3,0,0,...0] a2 = [0,0,0,0,1,2,5,0,0,0,...0] a3 = [0,0,0,2,2,3,0,0,0,0,...0] a4 = [0,0,1,1,1,0,0,0,0,0,...0] ... an = [0,0,1,1,1,0,0,0,0,0,...0]
I want to find out what linear combination of the vectors is closest to v. I define the distance of two vectors to be the sum of the absolute differences of each of its elements. For example the distance between [1,2,3] and [2,2,2] is abs(1-2) + abs(2-2) + abs(3-2) = 2.
Does such an algorithm exist? If it doesn't exist, are there any algorithms that could get me close to a solution?
Ideally I would like an integer multiple of each vector, does such a solution exist with integer multiples?