Optimal Order of One-Point and Multipoint Iteration

Transcription

1 Optimal Order of One-Point and Multipoint Iteration H. T. RUNG AND J. F. TRAUB Carnegie-Mellon University, Pittsburgh, Pennsylvania ABSTRACT. The problem is to calculate a simple zero of a nonlinear function / by iteration. There is exhibited a family of iterations of order 2 n _ 1 which use n evaluations of / and no derivative evaluations, as well as a second family of iterations of order 2 n _ l based on n 1 evaluations of / and one of /'. In particular, w T ith four evaluations an iteration of eighth order is constructed. The best previous result for four evaluations was fifth order. It is proved that the optimal order of one general class of multipoint iterations is 2 n _ 1 and that an upper bound on the order of a multipoint iteration based on n evaluations of / (no derivatives) is 2\ It is conjectured that a multipoint iteration without memory based on n evaluations has optimal order 2 ~K n KEY WORDS AND PHRASES: computational complexity, iteration theory, optimal algorithms, nonlinear equations, multipoint iterations CR CATEGORIES: 5.10, 5.15, Introduction We deal with iterations for calculating simple zeros of a scalar function/. This problem is a prototype for many nonlinear numerical problems [11]. Newton-Raphson iteration is probably the most widely used algorithm for dealing with such problems. It is of second order and requires the evaluation of /and/, that is, it uses two evaluations. Consider an iteration consisting of two successive Xewton-Raphson iterates (composition of iterates). This iteration has fourth order and requires four evaluations, two of / and two of /. More generally, an iteration composed of n Newton iterates is of order 2 n and requires n evaluations of / and n evaluations of/, that is, 2n evaluations. We shall show that an iteration of order 2 n _ 1 may be constructed from just n evaluations of /. We exhibit a second type of iteration which requires n 1 evaluations of / and one evaluation of / to achieve order 2 n ~ l. In particular, with four evaluations we construct an iteration of eighth order. The best previous result [10, p. 196] for four evaluations was fifth order. Newton-Raphson iteration is an example of a one-point iteration. The basic optimality theorem for one-point iteration states that an analytic one-point iteration which is based on n evaluations is of order at most n. (This theorem was first stated by Traub [9; 10, Sec. 5.4]; we give an improved proof here.) We conjecture that a multipoint iteration based on n evaluations has optimal order 2 n _ 1. We prove that the optimal order of one important family of multipoint iterations is 2 n _ 1 and that an upper bound on the order of Copyright (c) 1974, Association for Computing Machinery, Inc. General permission to republish, but not for profit, all or part of this material is granted provided that ACM's copyright notice is given and that reference is made to the publication, to its date of issue, and to the fact that reprinting privileges were granted by permission of the Association for Computing Machinery. This research was supported in part by the NSF under Grant GJ and the Office of Naval Research under Contract N A , NR Authors' address: Department of Computer Science, Carnegie-Mellon University, Pittsburgh, PA Journal of the Association for Computing Machinery, Vol. 21, No. 4, October 1974, pp

2 644 H. T. KUNG AND J. F. TRAUB multipoint iteration based on n evaluations of / is 2 n. This upper bound is close to the conjectured optimal order of 2 n ~\ To compare various algorithms, we must define efficiency measures based on speed of convergence (order), cost of evaluating/ and its derivatives (problem cost), and the cost of forming the iteration (combinatory cost). We analyze efficiency in another paper [5]. We confine ourselves here to iterations without memory, deferring the analysis of iterations with memory to a future paper. We summarize the results of this paper. The class of problems and algorithms studied in this paper is defined in Section 2. Particular families of iterations are defined in Sections 3-5. The optimality theorem for one-point iterations is proven in Section 6. An optimal order theorem for one general class of multipoint iterations and an upper bound for the order of a second class are proven in Section 7. A general conjecture is stated in Section 8. Section 9 contains a small numerical example. The Appendix gives ALGOL programs for forming two families of multipoint iterations. 2. Definitions We define the ensemble of problems and algorithms. Let D = \f / is a real analytic function defined on an open interval // C R which contains a simple zero a f of / and / not vanish on //.}. Let 12 denote the set of functions 4> which maps every / D to < (/) with the following properties: 1. </>(/) is a function mapping c // into 1$,/ for some open subinterval 1+,/ containing a f. 2. </>(/)(<*/) = a s. 3. There exists an open subinterval C containing a f such that if Xi+i = <t>(f)( x i) then lim^x Xi = a f whenever x 0 /J,/. 4. For any <f>, there exist nonnegative integers k, do,, d k -i, and functions u j+ i(y 0 ) yi,, yso+i;, yi,, 2/3 y +i) of 1 + So (rf> + 1) variables for j = -1, k - 1 such that, for all f D, where z 0 = u 0 (x), does = z k, (2.1) z J+l = u^(x;f(zo), -,/ W o ) (*o); ;f(zj), j = 0,, k - 1. The assumption that / D is needed for theorems dealing with a class of iterations. Any particular <p can be applied to/having only a certain number of derivatives. In (2.2) we assume Uo(x) = x. In another paper [6], we prove that if <p 12 and if u 0 (x) is continuous, then u 0 (x) must be identically equal to x. If <f> 12, <f> is called an iteration without memory, since if the sequence j j,j is generated by Xi+i = <t>(f)(xi), Xi+i is computed using information only at the current point a\. In this paper we limit ourselves to iterations without memory. We classify iterations without memory. If k is the nonnegative integer in (2.1), then we say <t> is a k-point iteration. In particular, if k = 1, we call 0 a one-point iteration and if k > 1 and the value of k is not important, we call <j> a midtipoint iteration. (Similar definitions of one-point and multipoint iteration are given in [9; 10, Sec. 1.22].) If there exists p((f>) such that for any / t D, lim[(0(/)(a-) - a s )/(x - a f ) pw ] = S(4>, f) x-*a f exists for a constant S(4>, /) and S(<t>, /) ^ 0 for at least one / D, then 4> is said to have order of convergence (order) p(<f>) and asymptotic error constant S(4>,f).

3 Optimal Order of One-Point and Multipoint Iteration 645 This is perhaps the simplest definition of order of convergence. The results of this paper hold, with suitable modifications, for weaker definitions of order [7], but we feel that use of these definitions would obscure the proofs without substantially strengthening the results. Let Vi(<t>) denote the number of evaluations of f l) used to compute </>(/) (x). Then v{4>) = 2Zi>o Vi(<j>) is the total number of evaluations required by <j>(f)(x) per step. To simplify notation, we often use a, <t>, p, v iy v instead of a f, v(<f>), if there is no ambiguity. The following two examples illustrate the definitions. < (/)(x), p(<j>), i\(<t>), Example 2.1. (Newton-Raphson Iteration) < (/) (x) = x f(x)/f (x). This is a onepoint iteration with v 0 = 1, v\ = 1, v = 2, and p = 2. Example 2.2. z 0 = x, z x = z 0 - f(z 0 )/f (z 0 ), <t>(f)(x) =z 2 = Z l - {/(*i)/(zo)/[/(*i) -f(zo)?\ [/U.VAzo)]. This is a two-point iteration with vq = 2, v\ = 1, v = 3, and p = 4. (See Section 5.) 3. A Family of One-Point Iterations For / 6 D, let F be the inverse function to /. For every n, define 7>(/):// R, j = 1,, n, as follows: 7i(/) (x) = x and for n > 1, 7/+i(/)(*) = 7>(/)(*) + [(-1)Vj!]-[/(^)] j^ ( ; ) (/(^)) (3.1) for j = 1,, n 1. Note that F (j) (f(x)) can be expressed in terms of/ (, ) (x) for i = 1, 2,, j. It is easy to show that, for example, 7i = x, 72 = 7i ~ fix)/f(x), 73 = 72 ~ l/,, (x)/2/(x)][/(x)//(x)] 2. The family J7 n has been thoroughly studied [10, Sec. 5.1]. Its essential properties are summarized in THEOREM 3.1. Lety n be defined by (3.1). Then for n > 1, (1) y n 6 fi, andy n is a onepoint iteration; (2) p(y n ) = n, (3) v»(7 n ) = 1, i = 0,, n 1, v t -(7n) n I. Hence v(y n ) = n. = 0, t > Thus 7 requires the evaluation of / and its first n 1 derivatives. In Section 6 we shall show that, under a mild smoothness condition on the iteration, every one-point iteration of order n requires the evaluation of at least/and its first n 1 derivatives. 4. A Family of Multipoint Iterations We construct a family of multipoint iterations, \^n), which require the evaluation of f at n points, no evaluation of derivatives of f, and for which p(\[/ n ) = 2 n _ 1. For every n, define rf/jif) C I f > 7^.,/, j = 0,, n, as follows: \f/o(f)(x) = x and for n > 0, $i(f)( x ) x + fif(x), /3 a nonzero constant,j = (MO), ) for j = 1,, n 1, where (?., ( i/) is the inverse interpolatory polynomial for / at /(&(/) (x)), k = 0,, j. That is, Q>(i/) is the polynomial of degree at most j such that Q,(f(Mf)(*))) = Mf)M,k = o,...,j. The ypj{f), j = 1,, n, are well defined if That (4.2) holds for I+ jt j sufficiently small will be part of the proof of Theorem 4.1.

4 646 H. T. KUNG AND J. F. TRAUB It is easy to show that, for example, ^3 = ^ - K+o)f(h)/U(h) - Mo)} {(*!- *o)/[/(*l) " /(Ml - (fc - *l)/[/(tf 2) ~ An ALGOL program (Program 1) is given in the Appendix for computing \p n for n > 4. Our interest in the family of iterations {\p n \ is due to the properties proved in THEOREM 4.1. Let \p n be defined by (4.1). Then for?i > 1, (1) ^ n 12, and \p n is an n-point iteration, (2) p(\f/ n ) = 2 n ~ l, (3) v 0 (\p n ) = n, *>i(^n) = 0, i > 0. Hence v(\(/ n ) = n. PROOF. We want to show that, for f (z D, lim(*» - «)/(* - a) 2n ~ l = n = 1,2,... (4.3) for constants <S(^,/). The proof is by induction on n. Since lim^ (^1 a)/(x a) = 1 + (4.3) holds for n = 1. Assume that (4.3) holds for n = 1,, m 1. From general interpolatory iteration theory [10, Ch. 4], we know that lim[(* m - a)/ II (*» - «)] = F m (/), (4.4) x-*a 0<n<m where Y m (f) = (-l) n+1 F im) (0)/(m\[F'(Q)] m ) and F is the inverse function of/. From (4.4) and the induction hypothesis, lim - a)/(x - a) 2 "' 1 ] = lim [(* m - a)/ J\ (^ - a)] [(* 0 - «)/(* - a)} x-+a x-+a 0<n<m J! M 1< n<m n - *)/{x - «) 2 ""'] = n sw.,f). l<n<m Hence S(\p m,f) = Y m (f)y[i<n< m S(\(/ n, f) and this completes the induction. From (4.3) one can easily show that 4sj(f),j = 2,, n, satisfies (4.2) for J^.,/ sufficiently small and hence is well defined. It can be verified that i/v 12. It is not difficult to show that (^, /) 0 for some / Z). Therefore, p(^ ) = 2 n _ 1. The fact that v 0 (yf/ n ) = v t (\f/ n ) = 0, z > 0 follows from the definition of i/v Q.E.D. The iteration i^2 is second order and is based on evaluations of / at x and x + &J(x). This iteration is given by Traub [10, Sec. 8.4]. The iteration i/^ uses n evaluations of / and is of order 2 n _ 1. For n > 2, no iterations with these properties were previously known. 5. A Second Family of Multipoint Iterations We now construct a second family of multipoint iterations, {u n }, such that p(oj n ) = 2 n _ 1 and v(co n ) = n. However, o? requires the evaluation of f at n 1 points and the evaluation of f at one point. For every n, define CO>(/) : I Uj,/ C 7/ > I Uj,j y j = 1,, n, as follows: COI(/)(x) = x and for n > 1, u> 2 (/)(x) = x -f(x)/f(x),) for j = 2,, n 1, where Rj(y) degree at most j such that i ; > (5-D «*,(/) (x) = 72,(0), ) is the inverse Hermite interpolatory polynomial of R J (f(x))=x, R/(f(x)) = l/f(x), >(/( «*(/)(*))) = u k (x),k = 2,...,j. (5.2) One can prove that <*>>(/), / = 2,, n, are well defined for I Ujtf easy to show that, for example, WI = x, o) 2 = WI - /(WI)//(COI), C0 3 = 0) 2 - «,)/(/( Wi) -/(C0 2 )] 2 }.[/(W!)//(C01)]. sufficiently small. It is An ALGOL program (Program 2) is given in the Appendix for computing o> 4 for «. > 4.

5 Optimal Order of One-Point and Multipoint Iteration 647 The basic properties of the family of iterations {a> ) is stated in the following theorem. The proof is omitted since it is similar to the proof of Theorem 4.1. THEOREM 5.1. Let O> N be defined by (5.1). Then for n > 2, (1) CO (E ft, and W is an (n I)-point iteration, (2) p(o> ) = 2 N _ L, (3) v 0 (o) n ) = n 1, I>I(O> ) = 1, t\(a> ) = 0, i > 1. Hence v(u> n ) = n. It is straightforward to show that S(+ n,f)/s(w,f) = [1 + &f(a)f'\ (5.3) If yp n is used, 0 should be chosen if possible so that 1 -f- /3/( a) is small. The iteration O> 3 uses two evaluations of / and one of / and p( O> 3 ) = 4. Another iteration with these properties is defined by Ostrowski [8, App. G] and a geometrical interpretation is given by Traub [10, Sec. 8.5]. King [3] gives a family of fourth-order methods based on two evaluations of / and one of /. Jarratt [2] constructs a fourth-order iteration based on one evaluation of / and two of f. The iteration O> uses n 1 evaluations of / and one of / and p(o)n) 2 N _ 1. For n > 3, no iterations with these properties were previously known. 6. The Optimal Order of One-Point Iterations By imposing a mild smoothness condition we can prove that one-point iterations of order n require the evaluation of / and at least its first n 1 derivatives. No such requirement holds for multipoint iterations. For example, the multipoint iteration \p n denned in Section 4 has order 2 n ~ l but requires no derivative evaluation. Let </> be a one-point iteration. Then from (2.1) and (2.2), *(/)<x) = «i(x,/(x), J^ix)), where Ui(y 0, yi,, yd 9 +i) is a multivariate function of do + 2 variables. In this section we drop the superscript on y i m The following theorem was first given by Traub [9; 10, Sec. 5.4]. We regard the proof given here as an improvement of Traub's proof. THEOREM 6.1. Let <f> be a one-point iteration of order p{<j>) and let Ui(y 0, yi,, 2/<* 0+I) be analytic with respect to yi at y\ 0. Then > 1, i = 0,, p(<t>) 1, and hence P(<t>) < *>(</>) PROOF. For f (z D, define T(f)(x) «= [y P (f)(x) - 4>(f)(x)}/f(x), (6.1) where y p is a member of the family of iterations defined in Section 3. To simplify notation, we write/ for/(x). Define o\ by y p = ^f^o 1 where <ii depends explicitly on f(x),, f (l) (x) [10, Sec. 5.1]. By the analyticity condition on <j>, <j> = ^*-o X<f*. Therefore from (6.1), T = 2 (<r, - X,)/- p - E (6.2) Since <f> and y p are of order p, (6.1) implies that lim T(Mx) = [S(y p,f) - S(4>, f)}/[/(a f )] p < OC for all f D. Hence it follows from (6.2) that <j x = X t, i = 0,,p - 1, V/G D. (6.3) Consider o- P_I = X P_I. We know that CR P_I depends explicitly on f'(x),, f (p v (x) and that the same must be true for X P _I. Assume Vj(<j>) = 0, for some j, 0 < j < p 1. Then Ui(y 0t yi,, I/D 0+I) does not depend on y j +i. This implies that (d p-1 /(92/I P ~ ) WI(yo,, 2/DO+I) i s independent of and that {d v ~ l /dyi^1) Ui(x, 0, f(x),,

6 648 H. T. RUNG AND J. F. TRAUB f d \x)) is independent o f H e n c e XP-I = (l/(p - 1)0 (d p ' l /dyr l ) u{x,o t f(x),,/ W (x)) is independent of / 0 ) (x), which is a contradiction. Therefore i>.(0) > 1, t = 1,, p 1. Next we show i> o (0) > 1. Suppose this is false. Then X, = 0, i > 0 and from (6.3), a x : = 0, i = 1,, p 1, which is a contradiction. Q.E.D. COROLLARY 6.1. Let y n be defined by (3.1). Then y n achieves the optimal order of any one-point iteration <t> for which v{4>) = n and which satisfies the analyticity condition of the theorem. Remark. The analyticity condition is not restrictive. For example, it includes all rational iterations and all iterations defined by simple zeros of polynomials with analytic coefficients. 7. Two Optimal Order Theorems for Multipoint Iterations We prove an optimal order theorem for one important class of iterations and prove a fairly tight upper bound for the maximal order of a second class of iterations. Our first class consists of all iterations such that for j = 0,, k 1, z J+ i appearing in (2.2) is given by a Hermite interpolatory iteration based on the points z 0,, Zj. If 4> belongs to this class, we say it is a Hermite interpolatory k-point iteration. The order of 0 may be computed as follows. From Traub [10, Sec. 4.2], where the d, are as in (2.2). Hence 2*1 - a = 0[(z } - a)" i+l (z, - a) d +1 ] «(/)(*) - a = z k - a = 0[(* - a)"}, where p(<f>) = (do + 1) n*-i (d, + 2). It is easily verified that v(<t>) = ( < * ; + 1). We wish to choose k, d 0, -, d k -i such that for v(<j>) fixed, p(</>) is maximized. The choice of k and the di are given by THEOREM 7.1. Let dj > 0, k > 1 be integers. Let v{<j>) = 2IyIo (d } + 1) = n be fixed. Thenp{<f>) = (do + 1) ITy==i ( ^' + 2) is maximized exactly when or k = n, dj = 0, j = 0,, n - 1 (7.1) k = n - 1, d 0 = 1, dj = 0, j = 1,, n - 2. (7.2) PROOF. Since d } + 1 < n, k < n, there are only finitely many cases and the maximum exists. Let the maximum of p be achieved at d j} j = 0,, k 1. We show first that dj = 0, j 1,, k 1. Assume that d r = m, m > 1, for some r, r = 1,, k 1. Define 3 ;, j = 0,, k + m 1 as 3y = dj, j = 0,, k 1, j ^ r, 3 r = 0, 3j = 0, j = k,, k + m 1. Then we can verify that and Jfc+M-L ic-1 ;=o 1) = (<*>+ 1) = n j=o Jfc+M-1-1 ( ) n + 2) > (& + 1 ) n ^j + 2). This contradiction proves that d } = 0, j = 1,, k 1. A similar argument may be used to prove d 0 < 1. If <? 0 = 0, k = n while if <? 0 = 1, k = n 1, which completes the proof. Q.E.D.

7 Optimal Order of One-Point and Multipoint Iteration 649 COROLLARY 7.1. Let <j> be a Hermite interpolatory iteration with v(<f>) = n. Then p{<t>) < 2 n ~\ Note that defined in Section 4, is an instance of (7.1) while OJ, defined in Section 5, is an instance of (7.2). (Both {\f/ n} and {a> n } are based on inverse interpolation. There are two other families of iterations based on direct interpolation.) Thus we have COROLLARY 7.2. Let \p n and o) n be defined by (4.1) and (5.1), respectively. Then \p n and CO have optimal order for Hermite interpolatory iteration with n evaluations. The second theorem of this section gives an upper bound on the order achievable for any multipoint iteration using values of/ only (and no derivatives). THEOREM 7.2. Let <f> be a multipoint iteration with Vq(4>) =?i, v l (<f>) = 0, i > 0. Then p(<f>) < 2 N. PROOF. For / 6 A let x 0 be a starting point such that if x l+ i = <j>(f) (x x ) then lim^ x, = a f. From (2.2), for each i, denote z in = Xi and z i n + j +i = (z in, f(z in ),, /(z t n + >)) for / = 0,, n 1. Then lim (z ( l + 1 ) n - a)/(z in - cc) p = S(<j>,f). (7.3) Hence lim(- log z i n - a\ l l i ) = p. (7.4) Brent, Winograd, and Wolfe [1, (6.1)] show that there exist an / ^ D and a sequence ZI) such that hm(- log z t - a\) l l i < 2. (7.5) By (7.4) and (7.5) it follows that p < 2 N. Q.E.D. In Section 4, we constructed an iteration \[/ n such that v 0 (\l/ n ) = n, r t (^ ) = 0, i > 0 and p(ypn) = 2 n ~ l. Hence the upper bound of Theorem 7.2 is within a factor of two of the order of that iteration. We conjecture in the next section that p = 2 N _ 1 is optimal. 8. A Conjecture CONJECTURE 8.1. Let 0 be an iteration (with no memory) with v(<t>) = n. Then p(4>) < 2 N _ 1. (8.1) This extends a conjecture of Traub [11] which states (8.1) for n = 2, Numerical Example Let f(x) = x 3 + ln(l + x) where In denotes the logarithm to the natural base. Hence a = 0. Starting at x 0 = 10" 1 and 10~ 2, we compute Xi by iterations yp n and u) n, n = 3, 4, 5. For comparison we also use as many steps of the Newton-Raphson iteration as necessary to bring the error to about 10" 1 6. Calculations were done in double precision arithmetic on a DEC PDP-10 computer. About 16 digits are available in double precision. Results are summarized in Examples 1-3. The parameter J3 that appears in \p n was chosen f$ =.2 which makes the asymptotic error constant of yp n for this problem near unity. The asymptotic error constants of co n, n = 3, 4, 5, and the Newton-Raphson iteration are also near unity for this problem. Recall that p(\p n ) = p(o) n ) = 2 N 1 and p that for Newton-Raphson iteration, p = 2. We expect Xi = x 0 to hold and this is numerically verified in the examples. From (5.3), we expect ^n(zo)/u n (Xo) = (' 8 ) 2n " 2 > t 9 ' 1 ) and (9.1) is numerically verified in the examples for x 0 = 10~ 2. The examples illustrate the advantage of \p n and CO N over the repeated use of Newton- Raphson iteration. Starting with x 0 = 10 _ 1, w 5 (^o) calculates the zero to "full accuracy" 1 Note added in proof. Kung and Traub [61 have established this conjecture for n = 2.

8 650 H. T. KUNG AND J. F. TRAUB at a cost of four evaluations of / and one of/. Four Newton-Raphson iterations are required with a cost of four evaluations of / and four of /. The difference is significant when the evaluation of / is expensive. This observation takes only the cost of / into account. A more complete analysis based on efficiency measure considerations is given by Kung and Traub [5]. Example 1. x 0 KT 1 IN- 2 Example 2. x l = ^(XO).21 X 10~ 4.27 X 10-8 xi = Mxo) -.80 X 10" X 10" 1 8 xi = * 5(X 0) --27 X 10~ 1 6 x 0 10" 1 10" 2 X! = UACRO).30 X 10"*.42 X 10~ 8 xi = oitixo) -.15 X 10" X 10~ 1 6 xi = A> 5(XO) -.24 X 10" 1 6 Example 3. Let x i+ i = <t>(xi)> where </> denotes Newton-Raphson iteration. x Q ICR 1 I0" 2 Appendix xi -.26 X 10" X 10~ 4 x X 10"' -.11 X HT 8 Xi -.54 X HT X 10" 17 x< -.31 X 10-" Program 1 and Program 2, which are adapted from a result of Krogh [4], compute \p n (f)(x) (with the parameter 0) and u n (f)(x) for n > 4. Program 1 comment THIS PROGRAM COMPUTES \fi n(j)(x) (WITH PARAMETER 0) FOR A GIVEN FUNCTION / AND A GIVEN VALUE x; integer i, k, N; real x, h, r, PSI; real array v[0 : n 1, 0 : N 1], PI[0 : n 11; v[0, 0] := x) h := 0 X f(x); v[0, 1] := v[0, 0] + h; v[l, 1] := h/(f(v[0, 1]) - /(i,[0,0])); R := v[l, 1] X f(v{0, 0]); v[0, 2] := v[0, 0] - r;»[1,21 := r/(f(v[0, 0]) - /(i,[0, 2])); PI [2] : = /W0,0]) X /M0, 1]); v[2, 2] := Ml, 1] - T,[L, 2])/(f(v[0, 1]) - /M0, 2])); PSI := v[q, 2] -J- PI[2] X v[2, 2]; for k : = 3 step 1 until N 1 do v[0, k] : = PSI; for i : = 0 step 1 until k 1 do T,[I + 1, fc] := (T;[I, I] - v[i, k])/(j(v[q, i]) - /M0, *])); PI[*J := /M0, k - 1]) X PI[* - 1]; PSI : = PSI -F PI[&] X v[k, k);

9 Optimal Order of One-Point and Multipoint Iteration 651 Program 2 comment This program computes un(j)(x) for a given function / and a given value z; comment /' is denoted by/p; integer i, k y n; real x, nil, omega, d; real array v[0 : n 1, 0 : n 1], pi[0 : n 1]; t>[0, 0] := x\ v[0, 1] := x\ nil : = /(»[0, 01)//p(v[0, 0]); v[0, 2] := i,[0, 0] - nil; d : = f(v[0, 0]) - /(t,[0, 2]); v[l, 2] := nll/d; v[2, 2] := v[l, 2] X /MO, 2])/d; omega := v[0, 2] - /MO, 0]) X v[2, 2]; tr[l, 1] := nll//m0, 0]); v[2, 2] := -v[2, 2}/f(v[0, 0]); pi[2] : = /(v[0, 0]) X /M0, 0J); for k : = 3 step 1 until n 1 do t;[0, k] : = omega; for i : = 0 step 1 until A; 1 do w[i + 1, k] : = i] - t;[i, A:])/(/(^[0, i]) - f(v[0, k])); pi[*j = -f(v[0, k - 1]) X pi[/c - 1]; omega := omega + pi [A] X v[k, k]; REFERENCES 1. BRENT, R., WINOGRAD, S., AND WOLFE, P. Optimal iterative processes for root-finding. Numer. Math. 20 (1973), JARRATT, P. Some efficient fourth order multipoint methods for solving equations. BIT 9 (1969), KING, R. R. A family of fourth-order methods for nonlinear equations. SI AM J'. Numer. Anal. 10 (1973), KROGH, F. T. Efficient algorithms for polynomial interpolation and numerical differentiation. Math. Corny. 24 (1970), KUNG, H. T. 7 AND TRAUB, J. F. Computational complexity of one-point and multipoint iteration. Report, Computer Sci. Dep., Carnegie-Mellon U., Pittsburgh, Pa. To appear in Complexity of Real Computation, R. Karp, Ed., American Mathematical Society, Providence, R. I. 6. KUNG, H. T., AND TRAUB, J. F. Optimal order for iterations using two evaluations. Report, Computer Sci. Dep., Carnegie-Mellon U., Pittsburgh, Pa., To appear in SI AM J. Numer. Anal. 7. ORTEGA, J. M., AND RHEINBOLDT, W. C. Iterative Solution of Nonlinear Equations in Several Variables. Academic Press, New York, OSTROWSKI, A. M. Solution of Equations and Systems of Equations, 2nd ed. Academic Press, New York, TRAUB, J. F. On functional iteration and the calculation of roots. Proc. 16th Nat. ACM Conf., 5A-1 (1961), pp TRAUB, J. F. Iterative Methods for the Solution of Equations. Prentice-Hall, Englewood Cliffs, N. J., TRAUB, J. F. Computational complexity of iterative processes. SIAM J. Comput. 1 (1972), RECEIVED APRIL 1973; REVISED SEPTEMBER 1973 Journal of the Association for Computing Machinery, Vol. 21, No. 4, October 1974