ON THE CONVERGENCE OF HALLEY'S METHOD

Transcription

1 Reprinted trom the American Mathematical Monthly, Vol. 88, No. 7, August-September 98. ON THE CONVERGENCE OF HALLEY'S METHOD G. ALEFELD Fachbereich Mathematik, Technische Universität Berlin Strasse des 7. Juni 35, Berlin, West Germany. Introduction. A mimber of papers have been written about Halley's method, a third-order method for the solution of a nonlinear equation. (See, for example, [8].)For real-valued functions, this method is usually written as Xk+l=Xk- f(xk) f'(xk) -l f " ( ) f(xk), Xkk ;;;. O. (0) This method is also called the method of tangent hyperbolas, as in [3], because x k+ as given by (0) is the intercept with the x-axis of a hyperbola that is osculatory to the curve y = f( x) at x = Xk. Construction of the appropriate hyperbola, given f(xk)' f'(xk)' and f"(xk)' is an interesting exercise. Many of the authors writing on Halley's method have, in particular, been concemed with developing a convergence theorem similar to the so-called Newton-Kantorovich theorem. (See, for ~xample, [5, p. 4 ff].) The most far-reaching results can undoubtedly be found in [3], where also a comprehensive list of references is given. Error bounds, also, are given in [3]. These reflect for the first time the correct order of the error. In this note we add some new results on Halley's method for real functions. From aremark by G. H. Brown, Jr. [], Halley's method can be derived by applying Newton's method to the function g(x) =f(x)/if'(x). The adaptation of Theorem 7. of Ostrowski [7] to Halley's method gives us results on the existence and uniqueness of a zero and on the convergence to this

2 98] CLASSROOM NOTES 53 zero. In particular, we get error bounds that reflect correctly the order of the error. This method is demonstrated by a simple well-tried numerical example.. TheoreticalResults. THEOREM.Let f( x) be a real-valued lunction 0 the real variable x, and let I( Xo)f'( xo) some Xo. Furthermore let == Olor ' ( ) / " ( ) I(xo) == 0 Xo - Xo f'(xo). Deline and set I(xo) ho= - I(xo), f'(xo) --lf"(xo) f'(xo) XI= Xo + ho, [XO,XO+hO], ho>o Jo= { [xo+ho,xo], ho<o. For X E Jo let I have a continuous third derivative. Suppose that f' doesn't change sign in Jo and that with. we have and g(x) I(x) = Jf'(x) Ig"( x) ~ Mo lholmo~ Ig'(xo)l. Then starting with Xo the leasibility 0 Ha/ley's method is guaranteed. All Xk are contained in Jo, and the sequence {Xk} convergesto a zero x* oll (which is unique in Jo). Defining () () hk= - f(xk), ' ( ) / " ( ) I(Xk) XIc - Xk f'(xk) k~o, Jk= [Xk,XIc+hk], { [xk+hk,xk], hk>o hk<o' k~o, we have the error estimates Ig"(x) ~ Mk' xe Jk> k ~ 0, Mk-I IXHI-Xkl ~Ig'(xk)llxk-Xk-',. I Mk-I I IX*-Xk+II~Ig'(Xk) Xk-Xk-I, Mk-I Ix* - xkl ~ Ig'(Xk),lxk - xk-d ' k ~ I, (3) k ~ I, (4) k ~. (5) ~.

3 53 G. ALEFELD [Aug.-Sept. If and if then we have /f'(x) the coarser estimates f'(x) ~ N, x EJO' (6) f"'(x) 3 f"(x» f'(x) + ( f'(x) ) I~M, x EJo, (7) NM 3 IXk+J-Xkl~I_'I- \IIXk-Xk-l!' k ~, (3') * I,,;:: NM - 3 X - Xk+ I """ I _'I - \ Xk xk-l, k ~, (4') NM Ix* - Xkl ~ Ig'(Xk)!lxk - Xk-I, 3 k ~. (5') Proof. In some essential parts the proof is similar to that of Theorem f'(xo) =i=0, we can assurne thatf'(xo) > O. We then consider the function For x E Jo, g has a continuous from which Furthermore f(x) g(x) = /f'(x), X EJo. second derivative and we have ) = rp7\ ' ( ) -! ' ( f"(x )f(x) g x Vi 'x} / f'(x) 3'. f" ( x ) 3 f" (x) g"(x)=~g(x) [ - f'(x) + ( f'(x) ) ]. 7. in [7]. Because of (8) and, using (), we have Ig'(x) - g'(xo) =I~:g"(t) dtl ~ Ix - xolmo, from which we may estimate Ig'(xI) - g'(xo)! ~ IXI- xolmo= IholMo~ Ig'\xo)/,, Ig'(xo) Ig'(xJ) ~ Ig'(xo)I-lg'(xo) - g (xl) ~ -. (9) (0) Integrating by parts and using ho = - g(xo) " \, we get f XI (x,- x )g"(x) dx = g(x]) Xo

4 98] CLASSROOM NOTES 533 and therefore Ig(X\) ~ ( I) Because of (0), the feasibility of Halley's method (0) is guaranteed Since ~lhoimo- x=x\+hl- for k = I and we have we have from (0) and () hl = - g(xj) g'(x\ ) Finally in a way similar to that in [7] and Ihd ~ IholMo Ig'(xo)-. lhllmo~ Ig'(xl)l, () I \hd ~ "lhol- From these inequalities it follows that X lies in Jo and JI is contained in Jo; that is, we have MI ~ Mo- Therefore () can be replaced by For the sequence as computed by Halley's method it holds in general that hdmj~ Ig'(xl)l- (3) Xk+ = Xk + hk (4) g(xk) hk = - g'(xk) - Therefore (3) shows that our assumptions remain true if we replace Xoby XI and ho by hl, and by Xk and hk, respectively, in general. The convergence to a zero x* that is unique in Jo can now be proved as in [7]. To prove the error estimates we start with (): We have and, therefore, using (5), ( ) I~ "lhol Mo Ig XI g(xl). I,,;::: Mo - ' IX xd-lhll"""~g'(xi) I """Ig'(xl)llxl xol- This is (3) for k =. For k > I the assertion is proved by mathematical induction. We omit the details. Since Ix* - xhd ~ hk we immediately get (4) from (3).. Finally we have Ix* - xkl ~ IXHI- xkl + IXk+l- x*l, and (5) is therefore proved by using (3) and (4). Applying the mean-value theorem we have, for xe Jk-I' f(x) f"(x) Ig"(X) =! _f'''(x) + 3 I Vf'(x) [ f'(x) " ( f'(x) ) ] I~NMlx-x*I~NMlxk-Xk-d. (5)

5 534 G. ALEFELD [Aug.-Sept. Replacing, therefore, in (3), (4), and (5) Mk-I by the upper bound NMlxk - xk-d, we establish (3'), (4') and (5'). 0 Without going into details of the proof we remark that the estimation (5) can further be improved. If we define then - t I k- Ig'(xk),lhkIMk, k~o,. I Mk-I, lxk-xk-d, Ix -xkl~.; + - tk g ( Xk )l k ~. (5") This last can be proved in exactly the same manner as the corresponding inequality for Newton's method (see [4, p. 34 ff]). Since one can easily show that (5") can be replaced by IX.-Xkl~ ~- tk-i tk ~ - tk-i F + - tk-i - tk-i, k ~, Mk-I Ig'(Xk),lxk- xk-d. Since tk ~ 0, (5") and (5"') are asymptotically better than (5) by a factor of. 3. Numerical Example. In order to compare OUTtheorem with other results, we consider the simple well-tried example (see, for example, [3, p. 453, Tabelle ]) Wehave As in [3], we choose Xo = and get For X E Jo we have Furthermore f(x) = X3-0. rex) = 3X, f"(x) = 6x, f"'(x) = 6. g(xo) t ho - - ' , g ( Xo ) Xl = Xo+ ho= , andjo = [, ]. hence the main inequality () holds. We have g"(x) ~ Mo = Ig'(xo) = ; Ig'(XI) = Using this value in (5) for k = we have M Ix. - xd ~ I_"'~ \Ix) - XOl= (5"') tan numerical values have been computed using a HP pocket calculator.

6 98] CLASSROOM NOTES 535 The actual error is Ix. - xd ::::: t The error estimation reflects the order of the actual error and is only three times as large as the actual error. For hl we get hl = Therefore we have JI = [x I' XI + hi ] = [ ,.55037], and, for x E J, hence g"(x) E [ , ], MI = F or x we get the value X = , and therefore Using (5) for k =, we finally have The actual error for Xis Ig'(X) = M Ix. - x ~ Ig'(~k)llx- x,~ 8.78X 0-. Ix. - x:::::.93 X 0-. Therefore the actual error is overestimated only by a factor of 3. (5"') gives us the estimate Ix. - x:::::4.39 X 0-. None of the error estimates that are to be found in Tabelle in [3,p. 453] givea smaller bound for the error than (5"'). 4. Conclusion. H we compare the assumptions of our theorem with other results (especially with those given in [3]), then it seems that the most drastic assumption is the requirement that f' doesn't change sign in Jo. We need this assumption, however, in order to define g( x) in the whole interval Jo. Hone wants to use the more precise error estimate (5), then one has to compute the bound Mk-I for the second derivative of g. There is no essential difficulty in doing this since one can get these bounds very simply by using interval arithmetic in (8). See,for example, [,p. 8 ff], or [6, p. 6 ff], or [9]. Tbe same is true for the inequality (5') and the bounds M and N appearing there. t The """, " sign means here and in the sequel that the number is rounded upwards in the usual manner.

7 536 PROBLEMS AND SOLUTIONS [Aug.-Sept. In conclusion we remark that the most important applications of Halley's method are to nonlinear equations in Banach spaces. See, for instance, the discussion and examples in [3]. The generalization of the results of this paper to this general case and some numerical examples will be discussed in another paper. 5. Acknowledgment. The author wants to express his thanks to Professor Jon Rokne for reading the manuscript. Thanks also to the Department of Mathematics and to the Department of Computer Science of the University of Calgary, Alberta, for the hospitality during the academic year 979/80 when this paper was wriuen. References. G. Alefeld and J. Herzberger, Einführung in die Intervallrechnung, Bibliographisches Institut, Mannheim, G. H. Brown, Jr., On Halley's variation of Newton's method, this MONTIILY,84 (977) B. Döring, Ei)ige Sätze über das Verfahren der tangierenden Hyperbeln in Banach-Räumen, Aplikace matematiky, 5 (970) B. Döring, Über das Newtonsche Näherungsverfahren, Math.-Phys. Sem.-Ber., 6 (963) M. Ortega and W. C. Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables, Academic Press, New York and London, L. Rall, Computational Solution of Nonlinear Operator Equations, Wiley, New York, A. M. Ostrowski, Solutions of Equations in Euclidean and Banach Spaces, Academic Press, New York and London, J. F. Traub, Iterative Methods for the Solution of Equations, Prentice-Hall, Englewood Cliffs, N.J., R. E. Moore, Methods and Applications of Interval Analysis, SIAM Publications, Philadelphia, 979.