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In Part I of this paper, we introduced a two dimensional eigenvalue problem(2DEVP) of a matrix pair and investigated its fundamental theory such asexistence, variational characterization and number of 2D-eigenvalues. In PartII, we proposed a Rayleigh quotient iteration (RQI)-like algorithm (2DRQI) forcomputing a 2D-eigentriplet of the 2DEVP near a prescribed point, and discussedapplications of 2DEVP and 2DRQI for solving the minimax problem of Rayleighquotients, and computing the distance to instability. In this third part, wepresent convergence analysis of the 2DRQI. We show that under some mildconditions, the 2DRQI is locally quadratically convergent for computing anonsingular 2D-eigentriplet.

2D Eigenvalue Problem III: Convergence Analysis of the 2D Rayleigh Quotient Iteration