中文摘要：

构造一个正则同伦函数来解算Jacob ian矩阵秩亏或者严重病态的不适定非线性最小二乘问题;建立不适定非线性最小二乘的正则同伦迭代格式,提出以残差平方和值大小为判断准则的两种正则参数选取的新策略：直接搜索法和区间划分法。对经典的非线性最小二乘问题进行解算,结果表明该方法是适用的;非线性秩亏自由网平差算例表明,正则同伦法不仅可以降低迭代矩阵的条件数,而且使得整个迭代过程中条件数波动较小,并可得到稳定的较小范数解。

英文摘要：

A regularization homotopy function is constructed for solving the ill-posed nonlinear least squares problem whose Jacobian matrix is rank-deficient or very ill-conditioned.A regularization homotopy iterative formula is established for ill-posed nonlinear least squares problem.The iterative process is derived in detial and the continuity and convergence conditions are given as well.Two new regularization parameter selecting strategies are proposed,which called as direct search method and interval division method.The calculation results of nonlinear least squares problems show that the method is correctly and applicable.Calculation results of nonlinear adjustment of free networks with rank deficiency examples also show that the method not only decrease the iterative matrix condition number,but also make the condition number small fluctuation in the total iterative process,so that a smaller stable norm solution can be obtained.