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中文摘要：
基于一类特殊矩阵的幂级数展开，推导了渐进迭代逼近方法和代数插值方法的等价性．在此基础上，针对PIA方法中因病态矩阵而导致收敛速度过慢的问题，通过矩阵QR分解引入变换矩阵，再优化迭代矩阵的谱半径，来加速PIA方法收敛；相对于因不同的参数化而导致计算效率的不确定性问题，采用向心加速参数化、优化配置矩阵来确保计算效率．最后通过数值实例验证了理论推导的正确性和文中方法的有效性．
英文摘要：
Based on the expansion of the matrix power series of a class of special matrices, we deduce the equivalence of the progressive iterative approximation method and the algebraic interpolation method. Furthermore, due to the slow convergence rate of the PIA method caused by the ill- conditioned collocation matrix, we present a new method to speed up the convergence rate of this method, which uses the QR method to decompose collocation matrix, and then uses a transform matrix to optimize the spectral radius of the collocation matrix. Due to the uncertain computational efficiency caused by the different parameterization methods, we propose to choose the centripetal method to insure the numerical efficiency by optimizing the collocation matrix. Some numerical examples are given to show that the theoretical reasoning is correct and the methods in this paper are effective.
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