368 Chapter 9.Root Finding and Nonlinear Sets of Equations http://www.nr.com or call 1-800-872- read able files Permission is (including this one) granted fori internet -7423(North America to any server computer, tusers to make one paper 1988-1992 by Cambridge University Press. from NUMERICAL RECIPES IN C: THE 是 Figure 9.4.4.The complex z plane with real and imaginary components in the range (-2,2).The black region is the set of points from which Newton's method converges to the root z 1 of the equation 23-1 =0.Its shape is fractal. strictly proh Programs root.But that means that in the neighborhood of an extremum there must be a tiny, Copyright (C) perhaps distorted,copy of the basin of convergence-a kind of"one-bounce away" to dir copy.Similar logic shows that there can be "two-bounce"copies,"three-bounce" copies,and so on.A fractal thus emerges. Notice that,for equation(9.4.8),almost the whole real axis is in the domain of ART OF SCIENTIFIC COMPUTING(ISBN 0-521 convergence for the root z=1.We say "almost"because of the peculiar discrete points on the negative real axis whose convergence is indeterminate (see figure). What happens if you start Newton's method from one of these points?(Try it.) .Further reproduction, 1988-1992 by Numerical Recipes -431085 CITED REFERENCES AND FURTHER READING: (outside Acton,F.S.1970,Numerica/Methods That Work,1990,corrected edition (Washington:Mathe- matical Association of America),Chapter 2. North Software. Ralston,A.,and Rabinowitz,P.1978,A First Course in Numerical Analysis,2nd ed.(New York: McGraw-Hill),88.4. Ortega,J.,and Rheinboldt,W.1970,Iterative Solution of Nonlinear Equations in Several Vari- ables (New York:Academic Press). visit website machine Mandelbrot,B.B.1983,The Fracta/Geometry of Nature(San Francisco:W.H.Freeman). Peitgen,H.-O.,and Saupe,D.(eds.)1988,The Science of Fractal /mages (New York:Springer- Verlag).368 Chapter 9. Root Finding and Nonlinear Sets of Equations Permission is granted for internet users to make one paper copy for their own personal use. Further reproduction, or any copyin Copyright (C) 1988-1992 by Cambridge University Press. Programs Copyright (C) 1988-1992 by Numerical Recipes Software. Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5) g of machine￾readable files (including this one) to any server computer, is strictly prohibited. To order Numerical Recipes books or CDROMs, visit website http://www.nr.com or call 1-800-872-7423 (North America only), or send email to directcustserv@cambridge.org (outside North America). Figure 9.4.4. The complex z plane with real and imaginary components in the range (−2, 2). The black region is the set of points from which Newton’s method converges to the root z = 1 of the equation z3 − 1=0. Its shape is fractal. root. But that means that in the neighborhood of an extremum there must be a tiny, perhaps distorted, copy of the basin of convergence — a kind of “one-bounce away” copy. Similar logic shows that there can be “two-bounce” copies, “three-bounce” copies, and so on. A fractal thus emerges. Notice that, for equation (9.4.8), almost the whole real axis is in the domain of convergence for the root z = 1. We say “almost” because of the peculiar discrete points on the negative real axis whose convergence is indeterminate (see figure). What happens if you start Newton’s method from one of these points? (Try it.) CITED REFERENCES AND FURTHER READING: Acton, F.S. 1970, Numerical Methods That Work; 1990, corrected edition (Washington: Mathe￾matical Association of America), Chapter 2. Ralston, A., and Rabinowitz, P. 1978, A First Course in Numerical Analysis, 2nd ed. (New York: McGraw-Hill), §8.4. Ortega, J., and Rheinboldt, W. 1970, Iterative Solution of Nonlinear Equations in Several Vari￾ables (New York: Academic Press). Mandelbrot, B.B. 1983, The Fractal Geometry of Nature (San Francisco: W.H. Freeman). Peitgen, H.-O., and Saupe, D. (eds.) 1988, The Science of Fractal Images (New York: Springer￾Verlag)