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What is the best known algorithm for exactly solving a large sparse system of linear equations? The system I'm working on is not symmetric, not positive definite and integer. The only benefit is being sparse. I also need to point out that the matrix is not square. The dimension is m×n and it is not generally either underestimate or overestimate.

Paul
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Star
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  • possible duplicate of What guidelines should I follow when choosing a sparse linear system solver? – J. M. Jul 12 '12 at 09:54
  • In which sense do you want to solve your system? How large are your m and n? – Arnold Neumaier Jul 12 '12 at 10:41
  • I want to solve this system exactly. m and n can be very large i.e., more than $10^5$. – Star Jul 12 '12 at 11:39
  • exactly = rounding-error free? Are the matrix entries rational? You must be prepared to get answers with very big fractions, or do you have additional information that forbids this? - Also if $m>n$ there will be generally no solution while for $m<n$ there will be infinitely many. Is this really what you want? – Arnold Neumaier Jul 12 '12 at 12:29
  • By exact, I mean rounding-error free. My matrix has rational entries as well. Also, $m \leq n$. – Star Jul 12 '12 at 12:45
  • How large (order of magnitude)? – David Ketcheson Jul 12 '12 at 13:09
  • @Star, my suggestion that you post this here may not be so good given that you want an exact solution. Users here will probably only point you to floating-point solutions. – David Ketcheson Jul 12 '12 at 13:11
  • As @ArnoldNeumaier pointed out, a non-square system may have infinitely many or no solution. Are you looking for a least-square solution instead? – Paul Jul 12 '12 at 13:47
  • Star, I've closed your question because it's difficult for me to tell what's being asked. Some of the comments help to clarify what the question is asking, and these should be incorporated into the question. In particular, I don't know what you mean by "it is not generally either underestimate or overestimate". Once you make some edits to clarify what is being asked, contact one of the moderators to see if it's clear enough to be re-opened. – Geoff Oxberry Jul 13 '12 at 01:03

2 Answers2

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The exact solution of linear equations with rational coefficients belongs to the field of computer algebra. For an entry to the literature, see
http://www2.isye.gatech.edu/~dsteffy/papers/OSLifting.pdf
http://www2.isye.gatech.edu/~dsteffy/papers/rationalsolver.pdf
http://www.eecis.udel.edu/~youse/post/itersolve.pdf
http://www.lirmm.fr/~giorgi/issac06.pdf
You can do a literature search based on this and the citation facilities of http://scholar.google.com .

Arnold Neumaier
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Krylov Iterative methods are a usual choice.

If you happen to have access to Mathematica, it offers a good way to test for different method: if A is your matrix, write B=SparseArray[A]; Then use the LinearSolve function with Method->"Krylov". You can also test to see if there are advantages to retaining integer digits. Converting to real numbers may yield faster results, possibly at the cost of accuracy.

Gabriel Landi
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