Abstract: The progressive iterative approximation (PIA) is a classical geometric iterative method for data fitting, which is simple to operate and has explicit representation. However, the classical PIA suffers from slow convergence rate. To address this issue, we propose a family of single-step nonstationary accelerated PIA methods by integrating the high-order convergence iterative algorithm of inverse matrix with the classical PIA method. Firstly, the given data points are parameterized by uniform or chord length parameterization method. Then, the accelerated PIA algorithm is employed to adjust the control points iteratively, and generates a sequence of fitting curves (surfaces). The limit of the generated sequence of curves (surfaces) is theoretically guaranteed to interpolate the original data points. Experimental results on fitting regular curves (surfaces), scattered data points and noisy scattered data points demonstrate that compared with the classical PIA algorithm, the proposed accelerated PIA requires an average reduction of 84.75% in the number of iterations and an average reduction of 65.53% in running time.