Abstract: In order to get an effective algorithm to solve the rank-1 approximation of the symmetric positive semidefinite matrix. A new iterative algorithm is constructed based on BFGS method. The rank-1 approximation of the symmetric positive semidefinite matrix is transformed into an unconstrained optimization problem by using X=YY^T,Y∈Rⁿ to characterize the feasible set, and then the unconstrained optimization problem is solved by BFGS method. Finally, two numerical examples are given. Numerical experiments show that this algorithm is effective and has application value.