Subject: Re: [ublas] Fw: Re: Conjugate gradient method for large sparse matrices
From: Thomas Klimpel (Thomas.Klimpel_at_[hidden])
Date: 2009-02-17 05:39:45
Gunter Winkler wrote:
> If you have a banded matrix then you should of course use
> banded_matrix<...> class. In this case a full
> Cholesky-Decomposition can
> be made because the decomposition of a banded matrix is
> again a banded
> matrix. (AFAIR with double band width - please check this.)
The bandwidth of the Cholesky-Decomposition will be the same as the bandwidth of initial matrix, as long as no pivoting is performed. Arbitrary pivoting on the other hand could completely destroy the band structure of the matrix. However, since it is sufficient to do pivoting only for elements that are structurally nonzero in the initial matrix, the statement is more or less correct. I say more or less, because a Cholesky-Decomposition (which is only applicable to positive-definite matrices) would not "require" any pivoting, but it is clear from the context that we are really talking about a more general LU decomposition.
Sorry for lecturing...