I have been using Krylov subspace methods for a long time in my research.
I know from general theory of Krylov subspace methods that for solving a problem:
$$\bf Ax = b$$
if $\bf A$ is square and positive definite for example symmetric, then we can get away with $\dim({\bf x})$ iterations of Conjugate Gradient (C-G).
For a particular problem I am solving recently, because of lack of mentioned properties for $\bf A$ I build the following Normal-equations:
$${\bf A}^T{\bf Ax = A}^T {\bf b}$$
Where we will be sure that $({\bf A}^T{\bf A})$ will be positive semi definite. Maybe for stability we can add Tykhonov regularizing term : $+\lambda \bf I$:
$$({\bf A}^T{\bf A}+\lambda{\bf I}){\bf x} = {\bf A}^T {\bf b}$$
Then it seems I get good solution only for iteration count of $\geq 2\dim({\bf x})$.
My questions are two-fold:
- Can we prove this, and
- Does there exist any methods to avoid this doubling in number of iterations?
My own work is mostly restricted to speculation about some clever choosing of $\lambda$ for regularization or perhaps if we can find some suitable preconditioner?