  In Part I of this paper, we introduced a two dimensional eigenvalue problem
(2DEVP) of a matrix pair and investigated its fundamental theory such as
existence, variational characterization and number of 2D-eigenvalues. In Part
II, we proposed a Rayleigh quotient iteration (RQI)-like algorithm (2DRQI) for
computing a 2D-eigentriplet of the 2DEVP near a prescribed point, and discussed
applications of 2DEVP and 2DRQI for solving the minimax problem of Rayleigh
quotients, and computing the distance to instability. In this third part, we
present convergence analysis of the 2DRQI. We show that under some mild
conditions, the 2DRQI is locally quadratically convergent for computing a
nonsingular 2D-eigentriplet.
