Download e-book for iPad: Advances in Nonlinear Parameter Optimization by R. Schmidt
By R. Schmidt
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Additional info for Advances in Nonlinear Parameter Optimization
1 for a 41 method for the solution of nonlinear least squares problems. The meaning of the solutionL~ from eq. 2/34) is discussed in details in appendix D. 1. For that purpose we must take into account that Hartley's method is only applicable i f the column vectors of the matrix Dk are linearly independent in each stage of the iteration. Assume that the matrix DK has this property. That means we have n linearly independent column vectors in each stage of the iteration, Setting l m = n in eq. k) .
The tool of equivalent function evaluations has been introduced by Box . When determining the number of equivalent function evaluations we do not only count the number of function evaluations but also the number of derivative evaluations. That means the evaluation of a gradient vector with n components is equivalent to n function evaluations. Applying the rough classification together with this criterion we obtain a sequence of the performance of the methods for each test problem, 28 Bard's investigations show that the four special minimizing methods perform best in eight out of ten test problems.
2/32) for the matrix C_ k. 2/33) By inspecting eq. 2/33) we recognize that only the f i r s t l components of the direction vector r k have to be determined. The other n-1 components are fixed to zero. The f i r s t l components of the direction vector r k are gathered up in the vector . 2/33) this vector r~ is given by = O +e_ -xk) - k TD_'i) k - l ( kD_'i)Te(xk) = ((D_'i) -- . 2/35) Eq. 2/35) represents a system of linear equations for the determination of the components of the vector ~ . This system of linear equations has to be solved only once in each stage of the iteration.