一类非线性方程组奇异解的计算方法及其应用

针对一类特殊的非线性方程组雅克比矩阵奇异的问题，提出了一种基于对偶空间的牛顿迭代方法。给出了一个显式的计算对偶空间的公式，在此基础上利用对偶空间作用于原方程组构造新的方程，使扩充后的方程组在近似值点的雅可比矩阵满秩，从而恢复牛顿迭代算法的二次收敛性。实验结果表明，改进后的算法一般迭代3次计算精度就可以达到10^(-15)。所提算法丰富了代数几何中关于理想的对偶空间理论，也为工程应用中的数值计算提供了一种新方法。

Calculation method for singular solutions of a class of nonlinear equations and its application

To resolve the peculiar problem of the Jacobian matrix for a special class of nonlinear equations, an improved Newton mtheod was proposed based on the dual space. This paper proposed an explicit formula to compute the dual space of an ideal in a point through polynomial multiplication, and constructed augmented equations using the dual space. Meanwhile, the Jacobian matrix of augmented equations at initial point was full rank, and then the algorithm recovered quadratical convergence of Newton's iteration. The experimental results show that after three iterations, the accuracy of computation can achieve 10^(-15). The proposed method further enriches the theories of the dual space of ideal in algebra geometry and provides a new method for the numerical calculation in engineering applications.