By R. Schmidt

ISBN-10: 3540113967

ISBN-13: 9783540113966

ISBN-10: 3540390804

ISBN-13: 9783540390800

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Naturally, the question concerning the linear independence of the vector _d)+I can also be answered by considering k ka the value of (Idl+lll. I f the vector _o'i'+1 is linearly independent on the already sem k T k lected vectors--d), . . , d~ then the matrix ((D~+I) D~+1)-1 exists (see,. appendix A). ~. Now we show how i t is possible to clear up the question of the linear independence and to calcuk T k -1 k T k -1 late the matrixk((D~+l) D~+I) simultaneouslyon the basis~of the kmatrlx ((D~) Dl) if the vector di+ I is linearly independent on the vectors _d~.

2/2) Q = eTe . 2/3) we obtain m ~ instead of eq. 2/1). With respect to the eqs. 2/2) we can write for the gradient of the sum of squares Q from eq. 1/13) = := Z(x ) . 2/4) m With these new notations we have made all preparations necessary for the derivation of a new method for the solution of nonlinear least squares problems. 2. 2 is shown in the logic diagram 2. Depending on the choice of the matrices 3 R we obtain different minimizing methods. In the sequel we derive a new instruction for the determination of the matrix kR in each stage of the iteration.

2/11) we have shown that the sum of squares Q does not increase in each stage of the iteration provided that we only choose the step-length factor small enough. 2/8) we want to guarantee that the sum of squares Q decreases in each stage of the iteration. In order to achieve this we have to demand that the matrix I~k satisfies a further condim tion. By inspecting eq. 2/12) holds. 2/11) vanishes i f and only i f the vector ~(x_K) has no component in the subspace B. K which can be seen by inspecting eq.

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Advances in Nonlinear Parameter Optimization by R. Schmidt


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