El 08/03/2014, a las 20:42, Luca Argenti escribió:
> Dear all,
>
> I need to evaluate the null space Span{v_i} of a sparse rectangular
> matrix A \in C^{ N x (N+k)},
>
> A v_i = 0, (1)
>
> where
>
> i) N is typically very big ( N ~ 500’000 ) and k, in comparison, is very
> small ( k ~ 500 ).
> ii) The sub-matrix A_ij i,j<= N is hermitian and non-singular. Equation (1),
> therefore, has exactly k solutions.
> iii)The matrix is sparse, with a fill typically <= 3%, and its columns/rows
> can be reordered in such a way that
> a very large block, A_ij with i,j > n, & i,j<=N, n << N, is
> band-diagonal.
> iv) A has a dominant diagonal.
> v) For large values of i,j, the number of non-zero diagonals in the central
> band drop by about an order of magnitude.
> vi) Finally, this problem must be solved for several (thousands)
> closely-spaced values of an external parameter
> Q on which A depends continuously, A = A(Q). Most of the time,
> therefore, the null space at Q_{i+1} is arguably
> very close to the null-space at Q_i .
>
> My feeling is that this problem is very well defined, and that a parallel
> sparse iterative method should be
> able to solve it with no issues or unnecessary operations. Yet, probably
> because I am not an expert
> of either PETSc or SLEPc, the two libraries I have considered so far, all the
> possible solutions that I found
> seem to provide much more information than needed (thus, consuming much more
> resources than
> warranted). For example: is it really necessary to make a sparse LU
> factorization for the *whole* matrix?
> In practice, one is looking for the null eigenspace of A^h A. However, SLEPc
> suggests that this operation is
> much more expensive than for a sparse A matrix alone (is it so? Shouldn’t
> Lanczos be implementable at just
> twice the cost?), or maybe I misinterpreted the user guide.
>
> Your suggestions will be greatly appreciated. Thank you so much for your help!
>
> Cheers,
>
> Luca
>
The nullspace of A^h A is equal to the right singular space of A corresponding
to the zero singular value. It should be possible to compute this with SLEPc's
SVD. Computing a large number of zeros may be problematic, so I cannot say in
advance if the method will succeed. If you can generate a small matrix with
these properties, send it to my personal address (not the list) and I will give
it a try.
Jose
