• To: mathgroup at yoda.physics.unc.edu
• Subject: RE: Real vs. Integer paradox
• From: fulling at sarastro.math.tamu.edu (Stephen A. Fulling)
• Date: Fri, 5 Mar 93 17:15:33 CST

```>In computing the NullSpace of a matrix with Mathematica, there seems
>to be a significantly different algorithm used depending on whether
>the matirx has real or symbolic (e.g. integer) entries.  Here's an
>example: [condensed]

>In[3]:= NullSpace[A]

5    3
>Out[3]= {{-4, 1, 0, 1}, {-, -(-), 1, 0}}
2    2

>In[6]:= NullSpace[Ar]

>Out[6]= {{-0.943569, 0.211961, 0.0273498, 0.252986},
>    {0., -0.59588, 0.681005, 0.425628}}

>why are they so different?
>Any help or explanation would be appreciated.

The answers are consistent:  Both sets are bases for the same
two-dimensional space.  Standard row reduction (quite pleasant to do
interactively within Mathematica) reduces both to the same set, within
roundoff error:

{1., 0, -0.285714, -0.428571}, {0, 1., -1.14286, -0.714286}

{1., 0., -0.285714, -0.428571}, {0., 1., -1.14286, -0.714285}

Presumably WRI chose the integer algorithm for algebraic simplicity
(speed) and the numerical algorithm to minimize roundoff error.  It's
not surprising that these give different results to a question whose