Compressive sensing based on SCAD has good theoretical properties for sparse signal reconstruction with noise. It is vital to study this kind of algorithms. The iterative thresholding algorithm is one of the most efficient algorithms to solve the problem of compressed sensing. In this paper, we study the convergence of the iterative thresholding algorithm for compressive sensing based on SCAD. We give some sufficient conditions on the convergence of the iterative thresholding algorithm. We prove that the algorithm is convergent with exponentially decaying error. Furthermore, we study the convergence of an improved iterative thresholding SCAD algorithm based on an approximate message passing algorithm.