THE COMPLEXITY OF OBTAINING STARTING POINTS FOR SOLVING OPERATOR EQUATIONS BY NEWTON 1 S METHOD

The complexity of searching for good starting points for itera tions is studied. Global and non-asymptotic results are obtained: A useful lemma for proving lower bounds is identified, several op timal results are given for scalar equations, and upper bounds for operator equations are established by a new procedure for obtaining starting points for Newton's method.


INTRODUCTION
Algorithms for finding a root a of a nonlinear equation f(x) = 0 usually consist of two phases: 1. Search phase: Search for initial approximation(s) to a.
2. Iteration phase: Perform an iteration starting from the initial approximation(s) obtained in the search phase.
Most results in analytic computational complexity assume that good initial approximations are -available and deal with the iteration phase only. Since the complexity, i.e., the time, of the computation for solving f(x) = 0 is really the sum of the complexities of both the search and iteration phases, we propose to study both phases. Moreover, we observe that the complexities of the two phases are closely related.
The speed of convergence of the iteration at the iteration phase in general depends upon the initial approximation(s) obtained in the search phase. If we spend much time in the search phase so that "good 11 initial approximation(s) are obtained, then we may expect to reduce the time needed in the iteration phase. This observation will be made precise in this paper. On the other hand, if we do not spend much time in the search phase and initial approximation(s) obtained are not so "good", then the complexity of the iteration phase could be extremely large, even if the corresponding iteration still converges. Some good examples of the phenomenon can be found in Traub and Wozniakowski [75]. All these show that the complexity of the iteration phase depends upon that of the search phase. Hence we feel that it is necessary to include both phases in the complexity analysis. Through this approach we can also obtain the optimal decision on when the search phase should be switched to the iteration phase, since it can be found by minimizing the total complexity of the two phases.
In this paper, we shall assume that f satisfies some property (or conditions), and include in our analysis the time needed in both the search phase and iteration phase.
Note that it is necessary to assume f satisfies some property, since we have to make sure at least that there exists a root in the region to be searched. The general question we ask in the paper is how fast we can solve f(x) = 0 in the worst case, when f satisfies certain conditions.
In the following section we give the methodology to be used in the paper, which does not have the usual assumption that "good initial approximations are available 11 . Instead, we assume that some property of the function f is known, i.e., f satisfies certain conditions. A useful lemma for proving lower bounds on complexity, in our methodology, is also given in the section.
Section 3 gives several relatively simple results for f: R -* R. The main purpose of the section is to illustrate the techniques for proving lower bounds. One of the results shows that even if we know that M ^ f 1 (x) ^ m > 0 on an interval [a,b] and f(a)f(b) < 0, it is impossible to solve f(x) = 0 by a superlinearly convergent method. However, if in addition |f"| is known to be bounded by a constant on [a,b], then the problem can be solved superlinearly.
In Section 4 we give upper bounds on the complexity for solving certain operator equations f(x) = 0, where f maps from Banach spaces to Banach spaces (Theorem 4.3) 0 This section contains the main results of the paper. A procedure (Algorithm 4.2) is given for finding points in the region of convergence of Newton's method, for f satisfying certain natural conditions. The complexity of the procedure is estimated a priori (Theorem 4.2), and the optimal branching condition on when the search phase is switched to the iteration phase is also given. We believe that the idea of the procedure can be applied to other iterative methods for f satisfying various conditions. By a preliminary version (Algorithm 4.1) of the procedure, we also establish an existence theorem (Theorem 4.1) in Section 4. Summary and conclusions of the paper are given in the last section.  Ix-aJ + Ix-q^I ^ | C^-o^ | ^ 2e.

METHODOLOGY AND
Hence either | x-a^ | ^ € or jx-o^l ^ e. •

SOME RESULTS ON REAL VALUED FUNCTIONS OF ONE VARIABLE
In this section we shall give several relatively easy results to illustrate the concepts given in the preceding section, and the use of Lemma 2.1. We consider f: [a,b]cR-*R.
For simplicity we assume that each function or derivative evaluation takes one unit of time and the time needed for other operations can be ignored.
Theorem 3.1. If f: [a,b~\ -* R satisfies the following properties: (3.1) f is continuous on fa,t>], and Proof. It is clear that by binary search we have that L 4 * (b-a)/2 1 ' r Let cp be any algorithm using i evaluations.
Algorithm 3.1 below constructs f^, such that (2.T) and b], and assume the first evaluation is at Xg.
for x € and set SL «-m.
4. Apply algorithm cp to function c(x) and compute the next approxima t ion.
5. If algorithm cp has not terminated, set m to be the point where the next evaluation takes place and go to step 2.
It is straightforward to check that r~j£ £ (b-a)/^* and that the distance between any zero of and any zero of f 2 is ^ It is also easy to see that (2.1), (3.1) and (3.2) hold for f^, Hence by Lemma 2.1, we have A i £ [(b-a)/2* +^] -6.
Since 6 can he chosen arbitrarily small, we have shown Theorem 3.1 establishes that binary search is optimal for finding a zero of f satisfying (3.1) and (3.2). The result is well-known. The theorem is included here because its proof is instructive. By slightly modifying the proof of Theorem 3.1, we obtain the following result: Theorem 3.2. If f: [a,b] ~> R satisfies the following properties : Proof. The proof is the same as that of Theorem 3.1, except that the functions u, v are now defined as and the functions f^, f^ have to be smoothed so that they satisfy (3.3). • One can similarly prove the following two theorems.
Theorem 3.3. If f: [a,b1 -» R satisfies the following properties: f'(x) £M for all x g ra.bl, and By Theorems 3.2 and 3.3, we know that even if f f is bounded above or bounded below, we still cannot do better than binary search in the worst case sense.
Hence their algorithm is better than binary search when j-j ^ However, by Theorem 3.4, we know that the problem cannot be solved superlinearly, even when f 1 is known to be bounded above and below by some constants. In order to assure that the problem can be solved superlinearly we have to make further assumptions on the function f. A natural way is to assume that |f n | is bounded. This leads to the following The proof is based on'the following algorithm.
The algorithm takes f satisfying the conditions of Theo-

4.
(It will be shown later that x i is a good starting point for approximating a zero, denoted by x^+ 1 , of f^.) Apply Newton 1 s method to f^,, starting from x^ to find .
They trivially hold for i = 0.
Further, by (4.11), we have S (x.) c S. (x n ). Hence x. is a r 1 ZT U 1 good starting point for approximating the zero x^ of f^.
We now assume that (4.10) and (4.11) hold and h^ < J.
Further by (4.11), S R (X I ) c S 2R (X Q ). Hence is a good starting point for approximating a.
It remains to show that the loop starting from step 2 is finite. Suppose that h Q £ j. Since X. g (0,1) for all i, we    Proof. The proof is based on the following algorithm, which

CRITICAL FACTORS IN CANCER IMMUNOLOGY
is adapted from Algorithm 4.1.
Suppose that (4.21) and (4.22) hold. Then   In the table, 6g is the 6 in which minimizes N(6) + T(6,e), i.e., R(e) = T(6 Q ) + T(6 Q ,€). Suppose, for example, that h Q = 9 or h Q ^ 9. Then by Algorithm 4.2 with 6 = .159 and by (4.27), we know that the search phase can be done in 255 Newton steps and the iteration phase in 5 Newton steps. Hence a root can be located within a ball of radius 10 ^r by 260 Newton steps.

SUMMARY AND CONCLUSIONS
The search and iteration phases should be studied togeth- Is Newton's method optimal or close to*optimal, in terms of the numbers of function and derivative evolutions required to approximate the root to within a given tolerance?

ACKNOWLEDGMENTS
The author would like to thank J. F. Traub for his comments and G. Baudet, D. Heller, A. Werschulz for checking a draft of this paper.