[petsc-users] Best way to compute the null space of a sparse maximum-rank rectangular matrix

[Luca Argenti]

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


-- 
Luca Argenti
Departamento de Química
Universidad Autónoma de Madrid
28049 Madrid, Spain
Module 13
Office 308
e-mail: luca.argenti at uam.es
tel : +34 914973360
fax: +34 914975238
group homepage: http://www.xchem.uam.es

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