  We present a comprehensive convergence analysis for Self-Consistent Field
(SCF) iteration to solve a class of nonlinear eigenvalue problems with
eigenvector-dependency (NEPv). Using a tangent-angle matrix as an intermediate
measure for approximation error, we establish new formulas for two fundamental
quantities that optimally characterize the local convergence of the plain SCF:
the local contraction factor and the local average contraction factor. In
comparison with previously established results, new convergence rate estimates
provide much sharper bounds on the convergence speed. As an application, we
extend the convergence analysis to a popular SCF variant -- the level-shifted
SCF. The effectiveness of the convergence rate estimates is demonstrated
numerically for NEPv arising from solving the Kohn-Sham equation in electronic
structure calculation and the Gross-Pitaevskii equation in the modeling of
Bose-Einstein condensation.
