I suspect this is a well studied problem with multiple solution, and I just don't know what terms to search for. If I knew what to search for, I suspect I could figure out how to apply those known techniques to best result for my problem.
So the question is: What is the name of this category of problem? What technique are know for working with that that I should look into?
Abstract formulation
Given:
- $U$ is an known $n\times m$ ($n>m$) matrix of rational values.
- $u$ is an $1\times m$ vector, also rational.
Solve $s\times U=u$ optimized such that $s$ is:
- sparse
- mostly integers
- small values
Practical formulation
I have a value that is tagged with physical units in terms of the 7 base SI units raised to rational powers and a collection of units (more properly; classes of units) expressed in the same form. I want to find a "reasonable" representation of that physical unit in terms of a small number of named complex units to simple powers.
For example: $\frac{m^2 kg}{s^3 mol}=\frac{J}{s\cdot mol}$