Gauss-Newton法恰2阶收敛性及其有效实现

Exactly Quadratic Convergence and Efficient Implementation of Gauss-Newton Method

研究求解零残差非线性最小二乘问题的算法。给出了保证Gauss-Newton法恰2阶收敛的条件，在此基础上构造了利用条件预化共轭梯度法求解Gauss-Newton方程的新的有效算法。新算法与传统的使用Choleski技术的Gauss-Newton法具有相同的收敛速率，但在求解Gauss-Newton方程组时减少了代数运算的计算量。如维数n＝200时，其计算量大体可减少35％，且当n趋于无穷时，两者的计算量之比以In2／Inn的速度趋于零。

The methods to solve the nonlinear least squares problem with zero vesidual arediscussed. A sufficient condition ensuring the Gauss-Newton method quadraticallyconvergent exactly is given. Based on it, a new efficient implementation of preconditionedconjugate gradient is put forward to solve the Gauss-Newton equation and save the cost oncomputation with the same exactly quadratic convergence to the traditional choleskifactorization. The ratio of computation will decrease 35% when n=200 and reduce to zero atthe rate of ln 2/ln n when n is infinite.