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(This is on MMA 7.0.1.0 on OS X)

I've just found a large matrix m for which NullSpace[m] and NullSpace[m, Method->"OneStepRowReduction"] give different answers (the first one is the correct answer).

I put the matrix up at pastebin as won't fit here!

What's going on? How might I guess ahead of time which arguments are going to break Method->"OneStepRowReduction"?

(Update; it seems the bug has got worse in 8, rather than better.)

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On a side note, Mathematica 8 yields an empty list in both cases. I'm going to assume the bug has been fixed. –  David Feb 2 '12 at 0:57
    
Could have been either of two bugs in "OneStepRowReduction" when algebraics are present. Both were fixed prior to version 8 release. –  Daniel Lichtblau Feb 2 '12 at 14:47
    
No, I think this is a real problem --- and your answers indicate it's got worse in Mathematica 8, not better! The NullSpace should not be empty. Here's the result of RootReduce[NullSpace[m]]: {{Root[3 - 80 #1^2 + 9 #1^4 &, 1], Root[3 - 80 #1^2 + 9 #1^4 &, 4], Root[-3 + 4 #1^2 + 3 #1^4 &, 4], Root[3 - 32 #1^2 + 81 #1^4 &, 3], Root[1 - 38 #1 + 116 #1^2 - 90 #1^3 + 27 #1^4 &, 4], Root[1 + 38 #1 + 116 #1^2 + 90 #1^3 + 27 #1^4 &, 4], Root[-1 + 23 #1^2 + 27 #1^4 &, 3], 1}}, which you can verify really is in the kernel. –  Scott Morrison Feb 2 '12 at 19:08
    
@DanielLichtblau, is there any reference for these bugs? I'd like to be able to ascertain which code running in 7 I can or cannot 'trust' (to the extend that trusting Mathematica is ever possible). –  Scott Morrison Feb 2 '12 at 20:04
    
@Scott Morrison (1) No reference I'm aware of. (2) I verified that matrix.your_vector has a 97th component of around -10.8. So it's not a serious contender for a null vector. –  Daniel Lichtblau Feb 2 '12 at 22:02
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