Mathematical Methods for Physics and Engineering - Matematica.NET

27.1 ALGEBRAIC AND TRANSCENDENTAL EQUATIONS6(a)6(b)44x 1x 22x 22−2−4642−2−41.0 1.2 1.4 1.6ξ(c)x 3x 3x 14x 3 x 42ξ1.0 1.2 1.4x1.61.0 1.2 1.4 1.62x2−2ξx 1−41.0−2−4(d)61.2x 11.4x 3ξ1.6Figure 27.2 Graphical illustrations of the iteration methods discussed inthe text: (a) rearrangement; (b) linear interpolation; (c) binary chopping;(d) Newton–Raphson.with n =1.Next,f(x 1 ) is evaluated and the process repeated after replacingeither A 1 or B 1 by x 1 , according to whether f(x 1 ) has the same sign as f(A 1 )orf(B 1 ), respectively. In figure 27.2(b), A 1 is the one replaced.As can be seen in the particular example that we are considering, with thismethod there is a tendency, if the curvature of f(x) is of constant sign nearthe root, for one of the two ends of the successive chords to remain unchanged.Starting with the initial values A 1 = 1 and B 1 =1.7, the results of the firstfive iterations using (27.8) are given in table 27.2 and indicated in graph (b) offigure 27.2. As with the rearrangement method, the improvement in accuracy,as measured by f(x n )andx n − ξ, is a fairly constant factor at each iteration(approximately 3 in this case), and for our particular example there is little tochoose between the two. Both tend to their limiting value of ξ monotonically,from either higher or lower values, and this makes it difficult to estimate limitswithin which ξ can safely be presumed to lie. The next method to be describedgives at any stage a range of values within which ξ is known to lie.989