Abstract: The Quasi-Newton method is widely regarded as one of the most effective methods to solve the problem of small scale optimization problem. They avoid the problem of solving the hession matrix by Newton method. In fact, the Quasi-newton method produces a symmetric matrix to approximate the hession matrix at each iteration. There are many Quasi-newton update formulas, frequently, such as BFGS update formula, SR1 update formula, DFP update formula and so on. Compared to other Quasi-Newton update formulas, SR1 Update formulas are more simpler and need less calculation. But most of time its convergence is improved on the strict condition of uniformly linearly independent. So this paper proposes a new modified SR1 Update formalus, and analyzes the convergence of the quasi-Newton algorithm based on the new modified formula SR1 under two different hypothesises respectively, including the condition of uniformly linearly independent and not uniformly linearly independent. By avoiding uniformly linearly independent and adding the assumption of positive-definite bounded hessian approximations, the new modified SR1 Update formalus is proved to be $n$+1 step $q$-superlinearly convergent. In addition, We also validate the reasonableness of assumptions and effectiveness of algorithms through experiments. The numerical results are reported to support that new update formulas proposed in the paper has better convergence rate in some conditions.

Key words: Quasi-Newton method, SR1, Quasi-Newton equation, algorithms