On preconditioning of penalized matrices (Q2760344)

Given a symmetric positive definite matrix \(A\in \mathbb R^{n\times n}\) and a vector \(b\in \mathbb R^n\), a quadratic functional born by \(A\) and \(b\) is to be minimized on a subspace \(V\) of \(\mathbb R^n\). To get rid of constraints, a system \(Bx=b\), \(B\equiv A+\rho C^TC,\) where \(\rho >0\) and \(C\in \mathbb R^{m\times n}\), is solved on \(\mathbb R^n\). A full matrix \(C\) defines \(V\) through \(V=\operatorname {Ker}(C)\). NEWLINENEWLINENEWLINEProving the existence of a gap in the spectrum of \(B\) if \(\rho \) is sufficiently large, the author suggests to use an incomplete factorization of \(A\) as a preconditioner for \(B\), and then to apply the conjugate gradient method. NEWLINENEWLINENEWLINEBounds for the convergence rate of the method independent of \(\rho\) and rank \(C\) are derived. The efficiency of the algorithm is illustrated by numerical experiments. NEWLINENEWLINENEWLINEThe paper is short and refers to other works. The reader keen to know more about the background and more details will probably need to consult some of them as the book of \textit{O.~Axelsson} [Iterative solution methods, Cambridge University Press (1994; Zbl 0795.65014)] for example. NEWLINENEWLINENEWLINEThe interpretation of equality (3.1) seems to be somewhat inaccurate. The parameter \(\varepsilon \) in (3.1) should be read as the upper bound of the relative error.