AN ALGORITHM FOR LEAST-SQUARES ESTIMATION OF NONLINEAR PARAMETERS
AN ALGORITHM FOR LEAST-SQUARES ESTIMATION OF NONLINEAR PARAMETERS
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DOI：
10.1137/0111030
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发表时间：
1963-01-01
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通讯作者：
MARQUARDT, DW
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作者：
MARQUARDT, DW
导言。大多数用于非线性参数最小二乘估计的算法都集中在两种方法中的任何一种。一方面，模型可以扩展为泰勒级数，并在局部线性的假设下对每次迭代计算的几个参数进行修正。另一方面，使用了对最陡下降法的各种修改。这两种方法都经常搁浅，泰勒级数方法由于连续迭代的发散而搁浅，最陡下降(或梯度)方法由于在最初几次迭代后收敛极其缓慢而搁浅。本文提出了一种最大邻域方法，它实际上是在泰勒级数方法和梯度法之间进行最优内插，该内插是基于最大邻域，在该邻域中，所生成的泰勒级数能够很好地表示非线性模型。将所得结果推广到求解一组非线性代数方程的问题。假设要与数据进行拟合的模型是
Introduction. Most algorithms for the least-squares estimation of non-linear parameters have centered about either of two approaches. On the one hand, the model may be expanded as a Taylor series and corrections to the several parameters calculated at each iteration on the assumption of local linearity. On the other hand, various modifications of the method of steepest-descent have been used. Both methods not infrequently run aground, the Taylor series method because of divergence of thesuccessive iterates, the steepest-descent (or gradient) methods because of agonizingly slow convergence after the first few iterations. In this paper a maximum neighborhood method is developed which, in effect, performs an optimum interpolation between the Taylor series method and the gradient method, the interpolation being based upon the maximum neighborhood in which thetruncated Taylor series gives an adequate representation of the nonlinear model. The resultsare extended to the problem of solving a set of nonlinear algebraic equations.Statement of problem. Let the model to be fitted to the data be