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.

翻译：在本论文的第一部分中，我们引入了矩阵对的二维特征值问题（2DEVP），并探讨了其基本理论，如存在性、变分表征及二维特征值的数量。在第二部分中，我们提出了一种类似于瑞利商迭代（RQI）的算法（2DRQI），用于计算2DEVP在指定点附近的二维特征三元组，并讨论了2DEVP和2DRQI在求解瑞利商极小极大问题以及计算到不稳定性距离中的应用。在第三部分中，我们给出了2DRQI的收敛性分析。我们证明，在温和条件下，2DRQI在计算非奇异二维特征三元组时局部二次收敛。