Abstract: In this paper, we obtain a necessary and sufficient condition, when the coefficient matrix A of the equation Ax=f considered is an T(1, 1) matrix,a sufficient condition, when A is an T(1, 2) or T(2, 1) matrix for the convergence of BPSD method.we also obtain the optimum parameters and the optimum rate of convergence of BPSD method, when A is an T(1, 1) matrix and a necessary and sufficient condition, when A is postive definite and we point out that the necessary and sufficient condition in 1 and 9 is only sufficient. The derivation is simple and interesting. Our method can also be used to prove the convergence of some other iterative methods.