Re: Socis 16 - Problem with pcg

[Marco Caliari]

> Moreover my mentors told me that, since it is difficult to determine if A is
> positive definite or not, the simplest way to check it is to run pcg or the
> Cholesky factorization (another function that works with positive definite
> matrices). If pcg / Cholesky doesn't converge then we are sure that A is not
> positive definite.

Dear Cristiano,

the above statement is a little bit unprecise. In exact arithmetic

A is SPD iff it admits a Choleksy factorization
if A is SPD, pcg converges but there are non-SPD matrices for which it 
converges

And then there are floating point arithmetic and Hilbert matrices... 
Anyway, for your non-SPD matrix A=[2,2+1i,...] and b=A*ones(3,1), pcg 
gives the correct result (due to complex comparison between alpha and 0). 
What to do in this case? I'm in favor of stopping pcg (in this case it is 
easy since the second alpha has a large negative real part). If I'm using 
pcg, it is because I think A is SPD. If it not the case, I did something 
wrong, or the matrix is very ill-conditioned and I would like to be 
informed. I don't like to have the right solution only by chance.

Marco