```   Hi,

> I'd like to use ruby-gsl for some singular value decompositions, i.e.
Which library do you use? There are two extensions,
ruby-gsl and rb-gsl(Ruby/GSL).

ruby-gsl http://ruby-gsl.sourceforge.net/
rb-gsl (Ruby/GSL) http://rubyforge.org/projects/rb-gsl/

> but these are printed out and apparently also calculated only to three
> decimal digits
If you use Ruby/GSL, this is just because
matrices are displayed with the printf format
%4.3e, not because of precision. You can get
more significant figures by displaying elements
as m[0][1].

> by the command
>
>      eigval, eigvec =  Eigen::symmv(m*m.transposed).
>
> This causes the matrix u, which is constructed from the values
> eigval,
> to have quite strange values for its determinant  (I get values of
> 1.06,
> 0.96  for u*u.transpose.det, but u is unitary by definition, i.e.,
> u*u.transpose.det=1).
Is it unitary? Isn't it an orthogonal matrix?

>
> Then, of course, m isn't remotely equal to u*s*v.transposed ....

The following is rb-gsl(Ruby/GSL) outputs.

irb(main):001:0> require("gsl")
=> true
irb(main):002:0> m = GSL::Matrix[[10, 5, -10], [2, -11, 10]].trans
=> GSL::Matrix
[  1.000e+01  2.000e+00
5.000e+00 -1.100e+01
-1.000e+01  1.000e+01 ]
irb(main):003:0> u, v, s = m.SV_decomp
=> [GSL::Linalg::SV::UMatrix
[ -2.981e-01  8.944e-01
-5.963e-01 -4.472e-01
7.454e-01  5.356e-17 ], GSL::Linalg::SV::VMatrix
[ -7.071e-01  7.071e-01
7.071e-01  7.071e-01 ], GSL::Linalg::SV::SingularValues
[ 1.897e+01 9.487e+00 ]]
irb(main):004:0> u.trans*u        (U is orthogonal)
=> GSL::Matrix
[  1.000e+00  1.408e-16
1.408e-16  1.000e+00 ]
irb(main):005:0> v*v.trans        (V is also orthogonal)
=> GSL::Matrix
[  1.000e+00 -1.015e-17
-1.015e-17  1.000e+00 ]
irb(main):006:0> u*Matrix.diagonal(s)*v.trans
=> GSL::Matrix                    (Reconstruct m)
[  1.000e+01  2.000e+00
5.000e+00 -1.100e+01
-1.000e+01  1.000e+01 ]

Is this what you expect?

Yoshiki

```