[R] Gaussian elimination - singular matrix

[Spencer Graves]

  RSiteSearch("generalized inverse", "fun") produced 194 hits for me 
just now, including references to the following:

Ginv {haplo.stats}
ginv {MASS}
invgen {far}
Ginv {haplo.score}

At least some of these should provide a solution to a singular system. 
You could further provide side constraints as Moshe suggested, but I 
suspect that 'ginv', for example, might return the minimum length 
solution automatically.

Hope this helps.
Spencer Graves

Moshe Olshansky wrote:
> All the nontrivial solutions to AX = 0 are the
> eigenvectors of A corresponding to eigenvalue 0 (try
> eigen function).
> The non-negative solution may or may not exist.  For
> example, if A is a 2x2 matrix Aij = 1 for 1 <=i,j <=2
> then the only non-trivial solution to AX = 0 is X =
> a*(1,-1), where a is any nonzero scalar.  So in this
> case there is no non-negative solution. 
> Let X1, X2,...,Xk be all the k independent
> eigenvectors corresponding to eigenvalue 0, i.e. AXi =
> 0 for i = 1,2,...,k.  Any linear combination of them,
> X = X1,...,Xk, i.e. a1*X1 + ... + ak*Xk is also a
> solution of AX = 0.  Let B be a matrix whose COLUMNS
> are the vectors X1,...,Xk (B = (X1,...,Xk).  Then
> finding a1,...,ak for which all elements of X are
> non-negative is equivalent to finding a vector a =
> (a1,...,ak) such that B*a >= 0.  Of course a =
> (0,...,0) is a solution.  The question whether there
> exists another one.  Try to solve the following Linear
> Programming problem:
> max a1
> subject to B*a >= 0
> (you can start with a = (0,...,0) which is a feasible
> solution).
> If you get a non-zero solution fine.  If not try to
> solve
> min a1
> subject to B*a >= 0
> if you still get 0 try this with max a2, then min a2,
> max a3, min a3, etc.  If all the 2k problems have only
> 0 solution then there are no nontrivial nonnegative
> solutions.  Otherwise you will find it.
> Instead of 2k LP (Linear Programming) problems you can
> look at one QP (Quadratic Programming) problem:
> max a1^2 + a2^2 + ... + ak^2 
> subject to B*a >= 0
> If there is a nonzero solution a = (a1,...,ak) then X
> = a1&X1 +...+ak*Xk is what you are looking for. 
> Otherwise there is no nontrivial nonnegative solution.
>
> --- Bruce Willy <croero at hotmail.com> wrote:
>
>   
>> I am sorry, there is just a mistake : the solution
>> cannot be unique (because it is a vectorial space)
>> (but then I might normalize it)
>>  
>> can R find one anyway ?
>>  
>> This is equivalent to finding an eigenvector in
>> fact> From: croero at hotmail.com> To:
>> r-help at stat.math.ethz.ch> Date: Wed, 27 Jun 2007
>> 22:51:41 +0000> Subject: [R] Gaussian elimination -
>> singular matrix> > > Hello,> > I hope it is not a
>> too stupid question.> > I have a singular matrix A
>> (so not invertible).> > I want to find a nontrivial
>> nonnegative solution to AX=0 (kernel of A)> > It is
>> a special matrix A (so maybe this nonnegative
>> solution is unique)> > The authors of the article
>> suggest a Gaussian elimination method> > Do you know
>> if R can do that automatically ? I have seen that
>> "solve" has an option "LAPACK" but it does not seem
>> to work with me :(> > Thank you very much>
>>
>>     
> _________________________________________________________________>
>   
>> Le blog Messenger de Michel, candidat de la Nouvelle
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>>
>>     
> _________________________________________________________________
>   
>> Le blog Messenger de Michel, candidat de la Nouvelle
>> Star : analyse, news, coulisses… A découvrir !
>>
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>>     
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