Let 
be an 
matrix and 
be a sequence of 
vectors. We consider the problem to find 
vectors 
such that:
We assume that the vectors 
are not known simultaneously. In particular, it is quite a common situation that the 
th problem has to be solved before 
becomes available, for example in the context of the simplified Newton iteration, see [24].
factorization is a way to organize the classical Gauss elimination method in such a way that the computation is done in two steps:
- A factorization step of the matrix
to get matrices in triangular form - A relatively cheap backward and forward elimination step that works on the instances of
and benefits from the more time-consuming factorization step
The method also uses the fact that if 
is a permutation matrix such that 
is the original matrix with its rows permuted, the two systems 
and 
have the same...