From: "M. Edward (Ed) Borasky" <znmeb@...>
Date: 2006-02-02T00:30:51+09:00
Subject: Re: Is this a floating point precision problem?

I don't know what Ruby is doing here, but the R language gives a much 
different result for the matrix.

Your input matrix:

 > m1
            [,1]        [,2]       [,3]
[1,] -0.01082270 -0.22493468 -0.9743137
[2,]  0.04682478  0.97318791 -0.2251949
[3,] -0.99884449  0.04805924  0.0000000

R's inverse, computed using the routine "ginv"

 > library(MASS)
 > m3<-ginv(m1)
 > m3
            [,1]        [,2]          [,3]
[1,] -0.01082270  0.04682478 -9.988445e-01
[2,] -0.22493468  0.97318791  4.805924e-02
[3,] -0.97431374 -0.22519489  6.223320e-17

Multiplication check:
 > m1%*%m3
              [,1]          [,2]          [,3]
[1,]  1.000000e+00 -1.509074e-16 -8.675039e-17
[2,] -1.865234e-16  1.000000e-00 -1.023683e-17
[3,]  6.958291e-17 -1.558541e-18  1.000000e+00

So ... either the Ruby matrix inverter is broken or you matrix is 
ill-conditioned enough that the Ruby matrix inverter can't handle it. I 
don't have a condition number checker handy, but I can do a singular 
value decomposition and look at the numerical rank.

 > svd(m1)
$d
[1] 1 1 1

$u
            [,1]        [,2]          [,3]
[1,] -0.01082270 -0.22493468 -9.743137e-01
[2,]  0.04682478  0.97318791 -2.251949e-01
[3,] -0.99884449  0.04805924  6.223320e-17

$v
     [,1] [,2] [,3]
[1,]    1    0    0
[2,]    0    1    0
[3,]    0    0    1

Hmmm ... no zero or small singular values ("$d") so ... looks off the 
top of my head like a nice rank 3 well conditioned matrix. So ... best 
have a look at the Ruby matrix inverter.


Todd S. wrote:
> I'm using the Matrix module (matrix.rb)  to help place a vertex into 
> local coordinates and then back into world coordinates.  Under certain 
> situations the transformation to and from local space corrupts the 
> data...
>
> This matrix...
>     -0.0108226964530337  -0.224934679233174 -0.974313737622412
>       0.0468247818757306    0.973187905351297 -0.225194894880513
>     -0.998844486916645      0.0480592442327619 0.0
>
> will invert to this...
>   64.0                               0.0 
> -0.998844486916645
>     0.0                               1.0 
> 0.0480592442327619
>   -0.974313737622412 -0.225194894880513 -2.40312135813741e-18
>
>
> starting with vertex: 0.015525, -0.28474399, 0.53221899
> multiplyng by the inverse matrix and then back again by the matrix 
> itself will not yield the same number as it should but instead 
> returns... 0.00555723611485535, -0.24161810434515, -0.4739174731596
>
> So am I seeing strange behavior due to the extreamly small (e-18) number 
> in the matrix?
>
> If so, how can I set the precision so that I don't see this error?
>
>   

-- 
M. Edward (Ed) Borasky

http://linuxcapacityplanning.com