By R. Schmidt
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Extra resources for Advances in Nonlinear Parameter Optimization
The principal components of this selection procedure are shown in the logic diagram 7. At the beginning of the selection procedure i t is checked whether a vector ~ is equal to the null vector respectively whether its absolute value is less than an upper bound n. In this case i t is linearly dependent so that i t does not participate in the selection procedure. Remembering the selection procedure as described above we remark that the selection of the vectors for the building up of the m a t r i x ~ depends on the random sequence of the vectors d~.
2/16), when paying attention to numerical problems. If ~k is not great enough i t w i l l be necessary to return to Marquardt's method and that means to increase ~k adequately. The problem of assigning a rank to the matrix l~ also results from another reason which is connected with the sensitivity of the components of the direction vector r ~ I L due to disturbances in the system of linear equations from which the direction vector r k is obtained for the different minimizing methods. 5. For these and the above=mentioned reasons we have to derive five requirements for an always applicable method for the solution of nonlinear least squares problems.
Depending on these requirements we can develop a strategy for the selection of the next vector. e speak of a selection according to the criterion "maximal change of the slope". In order to apply the selection procedure on the basis of this criterion without great numerical effort i t is necessary to derive a relation between II~(xk)~k II 2 --ul+i and ll~(x3)~kll 2. 3/18). 1/9) --- we obtain after a short calculation ~k The notation,. _VIQ(xk ) _ in eq. °f the gradient vector vQ(x~). We see that i t is possible to calculate the value of A~ for each vec- 56 t o r ~ + l to be examined practically without any supplementary effort because the scalar a~ and the vector u_~ have to be determined for the calculation of the matrix ((~+I)T_D~+I)'1 anyhow.
Advances in Nonlinear Parameter Optimization by R. Schmidt