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8, pp. 16–35. [2]IMSL Math/Library Users Manual (IMSL Inc., 2500 CityWest Boulevard, Houston TX 77042). [3]Ralston, A., and Rabinowitz, P. 1978, A First Course in Numerical Analysis, 2nd ed. (New York:McGraw-Hill), §8.9–8.13. [4]Adams, D.A. 1967, Communications of the ACM, vol. 10, pp. 655–658. [5]Johnson, L.W., and Riess, R.D. 1982, Numerical Analysis, 2nd ed. (Reading, MA: AddisonWesley), §4.4.3. [6]Henrici, P. 1974, Applied and Computational Complex Analysis, vol. 1 (New York: Wiley).Stoer, J., and Bulirsch, R.
1980, Introduction to Numerical Analysis (New York: Springer-Verlag),§§5.5–5.9.9.6 Newton-Raphson Method for NonlinearSystems of EquationsWe make an extreme, but wholly defensible, statement: There are no good, general methods for solving systems of more than one nonlinear equation. Furthermore,it is not hard to see why (very likely) there never will be any good, general methods:Consider the case of two dimensions, where we want to solve simultaneouslyf(x, y) = 0g(x, y) = 0(9.6.1)The functions f and g are two arbitrary functions, each of which has zerocontour lines that divide the (x, y) plane into regions where their respective functionis positive or negative.
These zero contour boundaries are of interest to us. Thesolutions that we seek are those points (if any) that are common to the zero contoursof f and g (see Figure 9.6.1). Unfortunately, the functions f and g have, in general,no relation to each other at all! There is nothing special about a common point fromeither f’s point of view, or from g’s. In order to find all common points, which areSample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5)Copyright (C) 1988-1992 by Cambridge University Press.Programs Copyright (C) 1988-1992 by Numerical Recipes Software.Permission is granted for internet users to make one paper copy for their own personal use. Further reproduction, or any copying of machinereadable files (including this one) to any servercomputer, is strictly prohibited.
To order Numerical Recipes books,diskettes, or CDROMsvisit website http://www.nr.com or call 1-800-872-7423 (North America only),or send email to trade@cup.cam.ac.uk (outside North America).This equation, if used with i ranging over the roots already polished, will prevent atentative root from spuriously hopping to another one’s true root. It is an exampleof so-called zero suppression as an alternative to true deflation.Muller’s method, which was described above, can also be useful at thepolishing stage..
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