On Feb 9, 2008 1:41 PM, Shunsuke Tsuchioka <[EMAIL PROTECTED]> wrote:
>
> Dear Linbox Support team,
>
> I am a graduate student in the math department of Kyoto university,
> and want to calculate a basis of the nullspace
>
> \{ x\in F^n \mid Ax=0 \}
>
> for a given m times n matrix A whose entries are in a field F.
> What is a good way to calculate a nullspace basis
> for a given matrix A by LinBox?
>
You might want to try doing this with Sage (http://sagemath.org), which
is a free open source program with a very nice interpreter interface that
is built on top of Linbox and provides the above functionality. E.g.,
dhcp46-76:hnf was$ sage
----------------------------------------------------------------------
| SAGE Version 2.10.1, Release Date: 2008-02-02 |
| Type notebook() for the GUI, and license() for information. |
----------------------------------------------------------------------
sage: a = random_matrix(QQ, 5,3); a
[ -1 -2 1]
[-1/2 0 0]
[ 1/2 -2 1]
[ 1 2 1]
[ 0 -2 -1/2]
sage: a.kernel()
Vector space of degree 5 and dimension 2 over Rational Field
Basis matrix:
[ 1 0 -8/5 9/5 12/5]
[ 0 1 -1/5 3/5 4/5]
sage: a.kernel().basis()
[
(1, 0, -8/5, 9/5, 12/5),
(0, 1, -1/5, 3/5, 4/5)
]
Note that kernel returns the set of row vectors v such that v*A = 0.
You can do quite large problems using Sage (which again, builds on
Linbox, so it's
OK that I'm posting here):
sage: a = random_matrix(QQ, 100, 102)
sage: time v = a.kernel()
CPU times: user 0.04 s, sys: 0.02 s, total: 0.06 s
sage: a = random_matrix(QQ, 400, 402)
sage: time v = a.kernel()
CPU times: user 0.59 s, sys: 0.09 s, total: 0.68 s
Working modulo 97:
sage: a = random_matrix(GF(97), 100, 102)
sage: time v = a.kernel()
CPU times: user 0.04 s, sys: 0.00 s, total: 0.04 s
Wall time: 0.06
If the specific functionality (the field etc) that you need isn't available
in Sage or is too slow, let me know, and Clement Pernet (one of the main Linbox
developers) and I can very likely add support for using that
functionality of Linbox in Sage.
> The following is what I have tried:
>
> Fortunately in my research, a nullspace is mathematically guaranteed
> to be 1-dimensional. Thus, I first thought "linbox/solutions/solve.h"
> may help. But the code like the following
>
> --------------------------------------------------------------
> typedef LinBox::Modular< double > Field;
> Field F(q);
> LinBox::DenseMatrix< Field > A(F,row,col);
> std::vector<Field::Element> x( A.coldim() ),b( A.rowdim() );
> A.setEntry( abracadabra, xyz, ... )
> for( std::vector<Field::Element>::iterator it=b.begin(); it != b.end(); ++it
> ) { *it = 0; }
> LinBox::solve( x, A, b, LinBox::Method::BlasElimination() );
> ---------------------------------------------------------------
>
> always returns the solution x = 0 (of course, it is a valid solution).
>
> I am sorry for such a basic question, but
> it would be great if you colud help me.
>
> Shunsuke, TSUCHIOKA
> Research Institute for Mathematical Sciences Kyoto University
>
> >
>
--
William Stein
Associate Professor of Mathematics
University of Washington
http://wstein.org
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