Why does Mathematica give a different answer for FindIntegerNullVector based on the order of the arguments? [closed]

Question

For instance,

qq = FindIntegerNullVector[{-9169.\
4529910449317529836685049212402940180640301948607925084807104675274205\
5034435184341092392543267646373701515968690891229074851663678405807145\
17622686786232514867447817140964`142., 1, EulerGamma, Pi^2, Log[2], 
Log[3]}]

gives

{-15044151968404805667153532191301833940133467621984038322223374816777\
2631007211100415129341007449731149059410554213978135216603240705107570\
58969, 0, 0, \
-139769172763599387675364620092226606423643480124197140593288493451413\
4179078124750307338440232001550774823705698830319337392850837300787936\
4436992, 0, 0}

which is the "wrong" answer.

On the other hand,

qq = FindIntegerNullVector[{-9169.\
4529910449317529836685049212402940180640301948607925084807104675274205\
5034435184341092392543267646373701515968690891229074851663678405807145\
17622686786232514867447817140964`142., 1, Pi^2, Log[2], Log[3], 
EulerGamma}]

gives

{-633004359680000, -5919486889313637261, 43840739895598800, \
-134027404823687760, -134027404823687760, -134027404823687760}

which is the answer I'm looking for.

Is there any way to force Mathematica to give me the second answer, regardless of the order of the input? I tried the extra "norm" argument, but it didn't do anything (just told me Mathematica couldn't find an answer for the first ordering).

Answer 1

Using SortBy on the argument works in this case.

list = {-9169.\
452991044931752983668504921240294018064030194860792508480710467527420550344351\
843410923925432676463737015159686908912290748516636784058071451762268678623251\
4867447817140964`142., 1, EulerGamma, Pi^2, Log[2], Log[3]};

list2 = list // SortBy[#, N] &

(*  {-9169.452991044931752983668504921240294018064030194860792508480710467\
5274205503443518434109239254326764637370151596869089122907485166367840\
58071, EulerGamma, Log[2], 1, Log[3], π^2}  *)

qq = FindIntegerNullVector[list2]

(*  {-633004359680000, -134027404823687760, -134027404823687760, \
-5919486889313637261, -134027404823687760, 43840739895598800}  *)

list2.qq

(*  0.*10^-124  *)

EDIT: Testing the performance of FindIntegerNullVector with random data:

validQ[list_?VectorQ, eps_: 10^-15] :=
  Module[{qq = FindIntegerNullVector[list]},
   Abs[list.qq] < eps];

SeedRandom[0]

data = RandomReal[{-10^4, 10^4}, {10000, 4},
   WorkingPrecision -> 50];

Select[data, Not@validQ@# &] // Length

(*  0  *)

FindIntegerNullVector did not fail for any of the 10,000 random vectors even without sorting the input. You will need to test your data yourself and determine whether sorting the input helps with whatever data is causing problems.