This paper presents fixed point and Bregman iterative algorithms for solving the nuclear norm minimization problem, which is a convex relaxation of the matrix rank minimization problem. The proposed algorithms, called FPCA (Fixed Point Continuation with Approximate SVD), are designed to efficiently solve large-scale matrix rank minimization problems. The FPCA algorithm uses an approximate singular value decomposition (SVD) to reduce computational complexity while maintaining high accuracy. The algorithm is shown to be significantly faster and more robust than existing methods such as SDPT3, particularly in recovering low-rank matrices from a small number of sampled elements. The paper also discusses the theoretical foundations of the algorithms, including their convergence properties and connections to compressed sensing. Numerical experiments on various problems, including matrix completion, online recommendation, DNA microarray data, and image inpainting, demonstrate the effectiveness and efficiency of the proposed algorithms. The results show that FPCA can recover matrices of moderate rank with high accuracy and low computational cost, making it a promising method for solving large-scale matrix rank minimization problems.This paper presents fixed point and Bregman iterative algorithms for solving the nuclear norm minimization problem, which is a convex relaxation of the matrix rank minimization problem. The proposed algorithms, called FPCA (Fixed Point Continuation with Approximate SVD), are designed to efficiently solve large-scale matrix rank minimization problems. The FPCA algorithm uses an approximate singular value decomposition (SVD) to reduce computational complexity while maintaining high accuracy. The algorithm is shown to be significantly faster and more robust than existing methods such as SDPT3, particularly in recovering low-rank matrices from a small number of sampled elements. The paper also discusses the theoretical foundations of the algorithms, including their convergence properties and connections to compressed sensing. Numerical experiments on various problems, including matrix completion, online recommendation, DNA microarray data, and image inpainting, demonstrate the effectiveness and efficiency of the proposed algorithms. The results show that FPCA can recover matrices of moderate rank with high accuracy and low computational cost, making it a promising method for solving large-scale matrix rank minimization problems.