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linear algebra - Find NullSpace of a very sparse symbolic matrix


Below is a matrix diagram, produced in Mathematica. In this case it's a 956×950 rectangular matrix. The white parts are all zero.



sa = Import["https://pastebin.com/raw/fiErKrhU", "Package"];
MatrixPlot[sa]

MatrixPlot


I'm wondering if there is a way to efficiently compute the null space of this matrix. From a different calculation entirely (using a Molien series), I know in advance there should be 6 linearly independent vectors in this null space, and I already know one of them.


The NullSpace routine takes too long to be feasible. I am hoping that there is a better way. I know that using NullSpace[N[m]] will return the answer rather quickly, but I am hoping to be able to do this symbolically.


Any help would be appreciated.


Update


Added SparseArray data for this matrix. It was too large for this message so I put it on pastebin.


https://pastebin.com/raw/fiErKrhU




Answer



Turns out this can be done with exact methods and a good option setting. And a dose of patience. I won't copy the matrix itself. In my session I named it mat as below.


AbsoluteTiming[
 nullspace = NullSpace[mat, Method -> "OneStepRowReduction"];]

(* Out[18]= {1839.141297, Null} *)

Check size and correctness:


Length[nullspace]


(* Out[19]= 6 *)

LeafCount[nullspace]

(* Out[20]= 50803 *)

Max[Abs[mat.Transpose[N[nullspace]]]]

(* Out[21]= 1.00182351685*10^-10 *)


I also have tried the method here but with no luck thus far. Oh well (I blame the author...)


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