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An interesting result of Chapter 2 is Theorem 2. It is well known that the conclusion of Theorem 2. It is used several times in Chapter 3 : for example, see Theorem 3. In Chapter 3 we prove some theorems which provide a theoretical foundation for the algorithms described in Chapters 4 and 5. In particular, we show when the algorithms will converge superlinearly, and what the order i.

Algorithms for Minimization Without Derivatives

For these results the effect of rounding errors is ignored. The reader who is mainly interested in the practical applications of our results might omit Chapter 3 , except for the numerical examples Section 3. For the applications in Chapters 4 and 5 , the most important results are Theorem 3.

These numbers are well known, but our assumptions about the differentiability of f are weaker than those of previous authors, e. From a mathematical point of view, the most interesting result of Chapter 3 is Theorem 3. Jarratt and Kowalik and Osborne assume that. However, even for such a simple function as. We should point out that this exceptional case is unlikely to occur: an interesting conjecture is that the set of starting points for which it occurs has measure zero. The practical conclusion to be drawn from Theorem 3. In Section 3. Finally, some numerical examples, illustrating both the accelerated and unaccelerated processes, are given in Section 3.

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In Chapter 4 we describe an algorithm for finding a zero of a function which changes sign in a given interval. The algorithm is based on a combination of successive linear interpolation and bisection, in much the same way as "Dekker's algorithm" van Wijngaarden, Zonneveld, and Dijkstra ; Wilkinson ; Peters and Wilkinson ; and Dekker Our algorithm never converges much more slowly than bisection, whereas Dekker's algorithm may converge extremely slowly in certain cases.

Examples are given in Section 4. It is well known that bisection is the optimal algorithm, in a minimax sense, for finding zeros of functions which change sign in an interval.

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We only consider sequential algorithms: see Robbins , Wilde , and Section 4. The motivation for both our algorithm and Dekker's is that bisection is not optimal if the class of allowable functions is suitably restricted. For example, it is not optimal for convex functions Bellman and Dreyfus , Gross and Johnson , or for C1 functions with simple zeros. Both our algorithm and Dekker's exhibit superlinear convergence to a simple zero of a C 1 function, for eventually only linear interpolations are performed and the theorems of Chapter 3 are applicable.

Thus, convergence is usually much faster than for bisection. Our algorithm incorporates inverse quadratic interpolation as well as linear interpolation, so it is often slightly faster than Dekker's algorithm on well-behaved functions: see Section 4. An algorithm for finding a local minimum of a function of one variable is described in Chapter 5. The algorithm combines golden section search Bellman , Kiefer , Wilde , Witzgall and successive parabolic interpolation, in the same way as bisection and successive linear interpolation are combined in the zero-finding algorithm of Chapter 4.

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