By R. Schmidt
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Extra resources for Advances in Nonlinear Parameter Optimization
We must show that the matrix ~k given by eq. 2/26) really satisfies the eqs. 2/14). m From eq. 2128) From eq. 2/29) eq. 2130) , " So we have shown that the above specified matrix ITk satisfies the eqs. 2/14). k from eq. 2/5) by m 40 utilizing the matrix _~< from eq. 2/26) the resulting minimizing method complies with the requirements 1 to 3 from section 4,1. Another method which also complies with the requirements I to 3 is proposed in . This method is very expensive because i t is necessary to determine the eigenvalues of the matrix (Dk)TD3 and the corresponding eigenvectors.
I)TD~,I] -1= ~ y e s JFinish the setectionI , procedure,, I Logic diagram 7: Selection of l linearly independent vectors and simultaneous calcuk TDk lation of the matrix (,D L-I) --l) -i • 51 b founded and independent on the arbitrary numeration of the vectors~. But before we occupy ourselves with these criteria we want to derive an important relation for orthogonal projectors which is interesting in the sequel . Assume we have selected l linearly independent vectors d~, . . , d~. Besides i t is k -~k -ik possible to find a further vector d_l+l so that the vectors dl , .
The slope of the function Q(~k+~_r~) at the point kx is then given analogously to eq. 1/2) by . ~. ~l - where~ is the direction vector determined on the basis of l selected vectors~. I f we now add a further vector~+1 which is linearly independent on the already selected vectors kd. 1t5) ' we have IleCx_k)112 = IleCx_k)"ll 2 + Ilel×k)~ll 2 . 3/29) we obtain lle(xk)tk -- 112 < lle(xk)tkll 2 . 1/7) 5 Because of eq. ,,. ~) at the point _xk under certain circumt stances by adding a further linearly independent vector~+l.