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So why give exact rational solutions of linear systems, then?

The reason such programs are implemented are largely psychological --- people want exact solutions, even though they are impractical for large systems and give nothing more useful than the floating-point ones anyway, for the vast majority of practical cases. There is a deep reason for that whim, however --- it is one of the most powerful ideas in mathematics that the details of the context of a problem can be neglected for the purpose of modelling, and insight can be gained from exact solutions. Unfortunately, this idea can also cause more work than necessary --- insight can be gained from exact solutions of equivalent problems, also, and by allowing the computer to change the problem slightly we can make massive efficiency gains which make the difference between practical and impractical computations.

There is little pedagogical reason for such solutions either --- when learning, the student must do his or her own work, and not sit back and watch another person or computer do it.

The truth is that there are very, very few situations where one really needs the exact rational solution to linear systems of equations with rational coefficients. Such problems do exist, and should not be forgotten, but the caution in this chapter is that the reader should avoid using an expensive tool, meant for very special purposes, for everyday calculations.



Robert Corless
Wed Jan 31 11:33:59 EST 1996