Fourier ptychography (FP) is a computational imaging technique that can achieve large field-of-view (FOV) and high-resolution (HR) at the same time. This technique sequentially captures a set of low-resolution (LR) images describing different spatial spectrum bands of the sample, and then stitches these spectrum bands together in the Fourier domain to reconstruct the entire HR spatial spectrum, including both amplitudes and phases. The sequential Gauss-Newton method, which has quadratic order of convergence, provides the best trade-offs between robustness and computational cost. Here, we incorporate Newton-type updating rules with cubic convergence mechanism by employing interpolatory quadrature rule on inverse function of the gradient and demonstrate our proposed algorithm achieves cubic convergence. Compared to Gerchberg-Saxton approach and Gauss-Newton method, the real experimental results show that our cubic convergent algorithm produces similar HR images with approximately 90% and 40% reduction in runtime, respectively.