THE COMPLEXITY OF OBTAINING STARTING POINTS FOR SOLVING OPERATOR EQUATIONS BY NEWTON'S METHOD

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This chapter discusses the complexity of obtaining starting points for solving operator equations by Newton's method. Most results in analytic computational complexity assume that good initial approximations are available and deal with the iteration phase only. As the complexity of the computation for solving f(x) = 0 is the sum of the complexities of both the search and iteration phases, it is important to study both phases. The speed of convergence of the iteration at the iteration phase depends upon the initial approximations obtained in the search phase. If too much time is spent in the search phase so that good initial approximations are obtained, then it can be expected to reduce the time needed in the iteration phase. However, if too much time is not spent in the search phase and initial approximations obtained are not so good, then the complexity of the iteration phase could be extremely large, even if the corresponding iteration still converges. The complexity of the iteration phase depends upon that of the search phase. Therefore, it is necessary to include both phases in the complexity analysis. Through this approach, the optimal decision can be obtained on when the search phase should be switched to the iteration phase, as it can be found by minimizing the total complexity of the two phases.