A note on best conditioned preconditioners
In this short note, the author gives a characterization of best conditioned preconditioners \(M\) for the conjugate gradient method applied to Hermitian positive definite linear systems \(Ax= b\). Denoting by \({\mathcal M}_ Q\) and \({\mathcal M}^ +_ Q\) the sets of matrices being similar to a complex diagonal matrix and a positive definite diagonal matrix with respect to a unitary transformation \(Q\), the best conditioned preconditioner \(M_ b\) minimizes the spectral condition number \(\kappa(M^{-1/2} AM^{-1/2})\) over \({\mathcal M}^ +_ Q\). It is shown that \(M_ b\) also minimizes the Frobenius norm \(\| M- A\|_ F\) over \({\mathcal M}_ Q\) provided the matrix \(QAQ^*\) has Young's property A.
scientific article; zbMATH DE number 644799