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MathGroup Archive 2006

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nullspaces

  • To: mathgroup at smc.vnet.net
  • Subject: [mg68948] nullspaces
  • From: hespeler <hespeler at gmx.de>
  • Date: Fri, 25 Aug 2006 05:34:58 -0400 (EDT)
  • Sender: owner-wri-mathgroup at wolfram.com

Hello

I have a general problem with the computation of nullspaces for large dense matrices, e.g. dim(2,108), explicit example is added at the end of the mail. Using the internal command B=NullSpace[A] (A is an inexact, potentially complex matrix) and trying to reconfirm the result by 

Chop[A.ConjugateTranspose[NullSpace[B]]]==N[ZeroMatrix[Dimensions[A][[1]],Dimensions[B][[1]]]]

the expression is evaluated as False. Looking closer at the problem reveals that the Chop[] command needs to be changed to Chop[ ,10^3] in order to get the result True. But this is an inacceptable precision error. 

I have already written and tried a package which computes the nullspace by using the corresponding last columns of the singular value decomposition of A, but the results remain inaccurate, even if the numerical error changes slightly.

Does anyone have any hints for me?

Thanks

In[162]:=
A=Normal[transversante].ConjugateTranspose[hhante].qqi

Out[162]=
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\[ImaginaryI], \(-0.0072107521385074095`\) + 0.0020962990190027935`\ \
\[ImaginaryI], \(\(0.2322579284275751`\)\(\[InvisibleSpace]\)\) - \
0.07187483163639043`\ \[ImaginaryI], \(-0.017968395434927953`\) - \
0.05213032583412297`\ \[ImaginaryI], \(\(0.04101492549617794`\)\(\
\[InvisibleSpace]\)\) - 0.05781905312972336`\ \[ImaginaryI], \
\(-2.241823563051902`\) - 0.6638806247785184`\ \[ImaginaryI], \
\(\(3.7231807156966448`\)\(\[InvisibleSpace]\)\) + 4.141064900928756`\ \
\[ImaginaryI], \(-3.2321784288246187`\) - 3.4833139232559858`\ \[ImaginaryI], \
\(-4.609629297686979`\) + 3.1846450167124956`\ \[ImaginaryI], \
\(-1.447339100854874`\) - 0.49968582628752684`\ \[ImaginaryI], \
\(\(0.2767361533870346`\)\(\[InvisibleSpace]\)\) - 0.13468033122430223`\ \
\[ImaginaryI], \(-1.8069351604027177`\) - 0.006387330181249744`\ \
\[ImaginaryI], \(\(0.09660287518422392`\)\(\[InvisibleSpace]\)\) + \
0.187845256630468`\ \[ImaginaryI], \(\(0.029581592108621362`\)\(\
\[InvisibleSpace]\)\) + 0.052873219749844165`\ \[ImaginaryI], \
\(\(0.46986808856631423`\)\(\[InvisibleSpace]\)\) - 0.12498829627797803`\ \
\[ImaginaryI], \(\(0.1494140057593214`\)\(\[InvisibleSpace]\)\) - \
0.049761274482300964`\ \[ImaginaryI], \(-0.07272327749503527`\) - 
        0.13531555329842745`\ \[ImaginaryI], \(-0.021857341056074342`\) + 
0.003440466953750463`\ \[ImaginaryI], \(-0.02273089849346567`\) - \
0.1303139194377979`\ \[ImaginaryI], \(\(0.23266712313539428`\)\(\
\[InvisibleSpace]\)\) + 0.011347412247304683`\ \[ImaginaryI], \
\(-0.25810605978716356`\) + 0.0022581817127046125`\ \[ImaginaryI]}}\)

In[163]:=
B=ConjugateTranspose[NullSpace[A]];

In[168]:=
Chop[A.B]\[Equal]N[ZeroMatrix[Dimensions[A][[1]],Dimensions[B][[2]]]]

Out[168]=
False

In[169]:=
Chop[A.B,10^(-1)]\[Equal]N[ZeroMatrix[Dimensions[A][[1]],Dimensions[B][[2]]]]

Out[169]=
False

In[172]:=
Chop[A.B,10^(3)]\[Equal]N[ZeroMatrix[Dimensions[A][[1]],Dimensions[B][[2]]]]

Out[172]=
True


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