The matrix completion problem is to recover the matrix from its partially known samples. A recent convex relaxation of the rank minimization problem minimizes the nuclear norm instead of the rank of the matrix. In this paper, we use a smooth function—Hyperbolic Tangent function to approximate the rank function, and then using gradient projection method to minimize it. Our algorithm is named as Hyperbolic Tangent function Approximation algorithm (HTA). We report numerical results for solving randomly generated matrix completion problems and image reconstruction. The numerical results suggest that significant improvement be achieved by our algorithm when compared to the previous ones. Numerical results show that accuracy of HTA is higher than that of SVT and FPC, and the requisite number of sampling to recover a matrix is typically reduced. Meanwhile we can see the power of HTA algorithm for missing data estimate in images.