Iterative Solution of Nonlinear Systems of Equations: by O. Axelsson (auth.), Rainer Ansorge, Theodor Meis, Willi

By O. Axelsson (auth.), Rainer Ansorge, Theodor Meis, Willi Törnig (eds.)

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Additional resources for Iterative Solution of Nonlinear Systems of Equations: Proceedings of a Meeting held at Oberwolfach, Germany, Jan. 31–Feb. 5, 1982

Example text

7b) becomes AX' _< ~C hk I (]_2 but violated for k=1. 7b) for k=], too. In this case one can proceed as follows (cf. Fig. 1) with ] instead of I. As usual the values ]~ are those at X~_ I • Fig. )_2). Fi~. 8 Thus, the smaller step width AX is used for the levels k=O and k=1, only. For k>2 the intermediate values Uk(X ) (k=O,l; ~ odd) are neglected. The modified al- gorithm results in approximations indicated in Fig. 7. l) even. 2 k different for all k. This approach leads to the results depicted in Fig.

Second, solutions of a level k might have no counterpart at another level. , in Fig. 9 there are two solutions Uo(%*) in the lower part of the branch related to no solution u1(%*) at level I. 2 Change of Parameter A very simple remedy is the introduction of a new parameter t such that tur- ning points vanish in the t-diagram. , in Fig. 9 the ordinate can serve as t. If again the reformulated problem can be written as Lk(Uk(t),t) = O, we regain the situation of §§ 2 - 4. ) described by ~k(Uk(t),t) = O.

3) is not O(A%h~) I is proportional A% proportional to p --i for i ÷ ~ [equivalent large as possible: . 7b) holds. %, the new nested as the nested work of the nested iteration iteration " the work is O(ipl). Assume That means, iteration to a fixed ~h~, one has to choose ~ " p < I/2. 7b). 6 Let p ! ]/2. Then the computational work (leading to a fixed accuracy) i. 20). |) applied to i/A%. 20). 3. 1 are met. 1). 6, the constand C 2 is well-known and the can- 37 contraction number p can be observed in course of the computation.

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