The self-consistent field (SCF) iteration is a commonly used method for solving eigenvector-dependent nonlinear eigenvalue problems (NEPv). Despite its simplicity, SCF can suffer from slow convergence or even non-convergence. Our focus in this talk is on analyzing the local convergence of SCF. First, we will present a sharp estimation of the local convergence factor of SCF for the unitarily-invariant NEPv. This estimation justifies the use of a level-shift scheme to address the potential divergence issue of SCF. Next, we will explore a class of NEPv that lack unitary invariance and show how to reformulate them into equivalent NEPv with unitary invariance. This will help us understand the local convergence of a popular SCF-type iteration for solving the NEPv.

10.11.pdf