# The fastest way to invert a big exact linear system

I have a big linear system of ~ $5000$ unknowns.

It is a symbolic problem in the sense that I need to find exact solutions, but I don't have variables, everything is a number (If I represent the matrix of the system it is a matrix full of symbolic numbers, not variables).

(By the way if there is a name for "symbolic but numbers", I would appreciate to know it).

It seems to be too long for my computer, so I am asking :

• Do you think such a problem is feasible on a standard computer ?
• What are the techniques I could use to increase the speed of computation at the maximum (can I parallelize, can I do other things that I don't know to increase at the maximum the speed ?)

Note : The matrix of the problem is dense.

• If the entries are integers, the best approach perhaps is the symbolic-numeric method by Zhendong Wan. It can be extended to handle rationals and Gaussians. – Daniel Lichtblau Jul 2 '17 at 16:32
• Could you make available a sample system so we can play with it? – Rolf Mertig Jul 2 '17 at 19:38
• "symbolic numbers, not variables" Does this mean exact rationals only? Or do you have forms like Pi, E or Sqrt[2[. – m_goldberg Jul 2 '17 at 21:01
• @m_goldberg I have rationals numbers only. I will make an example to work with. – StarBucK Jul 2 '17 at 22:50

## 1 Answer

I expect your exact answer will involve many rational numbers with many digits in the numerator and denominator. That will require a great deal of computation with numbers that are not machine numbers. Hence it will require a long time and a huge amount of memory, and may be impractical without using a super computer. However, you may be able settle for an answer if sufficiently high precision is used. Your best bet may be to solve something like the following instead.

N[equatiion,200];