From: "M. Edward (Ed) Borasky" <znmeb@...>
Date: 2006-07-31T08:09:41+09:00
Subject: Re: Setting precision in Ruby-GSL calculations ?

M. Edward (Ed) Borasky wrote:
> Nuralanur@aol.com wrote:
>> Dear Ed,
>>  
>> thank you for responding.
>> Well, I tried transposing as you said too, but then I lost
>> confidence in the algorithm implemented there, because
>>  
>> for m=[[10,5,-10],[2,-11,10]].to_gsl_matrix, I then get
>> a matrix u of size (3,2), (it should be 3,3), and u*u.transposed
>> is  
>> [  8.889e-001 -2.222e-001 -2.222e-001 -2.222e-001   5.556e-001 
>> -4.444e-001 -2.222e-001 -4.444e-001  5.556e-001  ],
>>  
>> so nowhere near a unit matrix, either ...
>>  
>> With respect to the sizes, I am expecting to be able to keep
>> (m1,m2) somewhere in the range of 50 or below each.
>>  
>> Best regards,
>>  
>> Axel   
>
> Here's what I get from R:
>
> > m=matrix(c(10, 5 ,-10 ,2 ,-11 ,10), nrow=3, ncol=2)
> > m
>     [,1] [,2]
> [1,]   10    2
> [2,]    5  -11
> [3,]  -10   10
> > svd(m)
> $d
> [1] 18.973666  9.486833
>
> $u
>           [,1]          [,2]
> [1,] -0.2981424  8.944272e-01
> [2,] -0.5962848 -4.472136e-01
> [3,]  0.7453560 -1.489694e-16
>
> $v
>           [,1]      [,2]
> [1,] -0.7071068 0.7071068
> [2,]  0.7071068 0.7071068
>
>
>
> > m=t(m)
> > m
>     [,1] [,2] [,3]
> [1,]   10    5  -10
> [2,]    2  -11   10
> > svd(m)
> $d
> [1] 18.973666  9.486833
>
> $u
>           [,1]      [,2]
> [1,] -0.7071068 0.7071068
> [2,]  0.7071068 0.7071068
>
> $v
>           [,1]          [,2]
> [1,] -0.2981424  8.944272e-01
> [2,] -0.5962848 -4.472136e-01
> [3,]  0.7453560 -1.859678e-16
Oh ... I see what's up now. m is 3x2. Thus its rank must be 2 or less. 
So we do the svd (the first one above) and get a 3x2 u, which is 
correct. Now multiply u.transpose by u. In R, that's

 > m
     [,1] [,2]
[1,]   10    2
[2,]    5  -11
[3,]  -10   10
 > svd=svd(m)
 > svd
$d
[1] 18.973666  9.486833

$u
           [,1]          [,2]
[1,] -0.2981424  8.944272e-01
[2,] -0.5962848 -4.472136e-01
[3,]  0.7453560 -1.489694e-16

$v
           [,1]      [,2]
[1,] -0.7071068 0.7071068
[2,]  0.7071068 0.7071068

 > t(svd$u) %*% svd$u
              [,1]          [,2]
[1,]  1.000000e+00 -1.275693e-16
[2,] -1.275693e-16  1.000000e+00
 >


See if you get the same things when you run it in Ruby/GSL.