Abstract: For the trust region subproblems, it is modified by Bunch-Parlett method when the Hessian matrix is indefinite. In addition, symmetric positive-definite matrix is also constructed, and the stator problem is transformed into a positive-definite subproblem. The Heun third-order algorithm is given by using a new polygonal line to approximate the solution curve. Then, the feasibility of this algorithm is theoretically proved by analyzing the properties of path of Heun third-order polyline. Finally, the numerical experiments of two test-function show that the algorithm is effective.

Key words: trust region subproblems, differential equation model, indefinite matrix, Heun third-order algorithm

摘要： 针对信赖域子问题，当Hessian矩阵不正定时，利用Bunch-Parlett法对矩阵进行修正，构造了对称正定的矩阵，将不定子问题转化为正定子问题，用新的折线来逼近最优解曲线，给出了求解的Heun三阶算法。通过对Heun三阶折线路径性质的分析，理论上证明了算法的适定性。利用两个测试函数进行了数值实验，结果表明该算法有效。

关键词: 信赖域子问题, 微分方程模型, 不定矩阵, Heun三阶算法