On the Quadratic Convergence of the Levenberg-Marquardt Method without Nonsingularity Assumption
On the Quadratic Convergence of the Levenberg-Marquardt Method without Nonsingularity Assumption
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DOI：
10.1007/s00607-004-0083-1
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发表时间：
2005-02
期刊：
影响因子：
3.7
通讯作者：
Jinyan Fan;Ya-xiang Yuan
中科院分区：
文献类型：
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作者：
Jinyan Fan;Ya-xiang Yuan
最近，Yamashita和Fukushima[11]在没有非奇性假设的情况下，建立了Levenberg-MarQuardt方法的一个有趣的二次收敛结果。本文推广了Yamashita和Fukushima的结果，用μk=||F(Xk)||δ代替δ∈[1，2]，而不是μk=||F(Xk)||2作为勒文伯格-马夸特参数。如果||F(X)||给出了非线性方程组F(X)=0的一个局部误差界，则证明了新方法产生的序列{Xk}二次收敛到一个解，它比Yamashita和Fukushima给出的dist(Xk，X*)→0强.数值结果表明，该方法对奇异问题有较好的处理效果。
Recently, Yamashita and Fukushima [11] established an interesting quadratic convergence result for the Levenberg-Marquardt method without the nonsingularity assumption. This paper extends the result of Yamashita and Fukushima by usingμk=||F(xk)||δ, where δ∈[1,2], instead ofμk=||F(xk)||2as the Levenberg-Marquardt parameter. If ||F(x)|| provides a local error bound for the system of nonlinear equationsF(x)=0, it is shown that the sequence {xk} generated by the new method converges to a solution quadratically, which is stronger thandist(xk,X*)→0 given by Yamashita and Fukushima. Numerical results show that the method performs well for singular problems.