Mathematical Methods for Physics and Engineering : A Comprehensive Guide

NUMERICAL METHODS

nAn f(An) Bn f(Bn) xn f(xn)

1 1.0000 − 4 .0000 1.7000 5.4186 1.3500 − 2. 1610

2 1.3500 − 2 .1610 1.7000 5.4186 1.5250 0. 5968

3 1.3500 − 2 .1610 1.5250 0.5968 1.4375 − 0. 9946

4 1.4375 − 0 .9946 1.5250 0.5968 1.4813 − 0. 2573

5 1.4813 − 0 .2573 1.5250 0.5968 1.5031 0. 1544

6 1.4813 − 0 .2573 1.5031 0.1544 1.4922 − 0. 0552

7 1.4922 − 0 .0552 1.5031 0.1544 1.4977 0. 0487

8 1.4922 − 0 .0552 1.4977 0.0487 1.4949 − 0. 0085

Table 27.3 Successive approximations to the root of (27.1) using binary

chopping.

27.1.3 Binary chopping

Again two values ofx,A 1 andB 1 , that straddle the root are chosen, such that

A 1 <B 1 andf(A 1 )andf(B 1 ) have opposite signs. The interval between them is

then halved by forming

xn=^12 (An+Bn), (27.9)

withn=1,andf(x 1 ) is evaluated. It should be noted thatx 1 is determined

solely byA 1 andB 1 , and not by the values off(A 1 )andf(B 1 ) as in the linear

interpolation method. Nowx 1 is used to replace eitherA 1 orB 1 , depending on

which off(A 1 )orf(B 1 ) has the same sign asf(x 1 ), i.e. iff(A 1 )andf(x 1 ) have the

same sign thenx 1 replacesA 1. The process isthen repeated to obtainx 2 ,x 3 ,etc.

This has been carried through in table 27.3 for our standard equation (27.1)

and is illustrated in figure 27.2(c). The entries have been rounded to four places

of decimals. It is suggested that the reader follows through the sequential replace-

ments of theAnandBnin the table and correlates the first few of these with

graph (c) of figure 27.2.

Clearly, the accuracy with whichξis known in this approach increases by only

a factor of 2 at each step, but this accuracy is predictable at the outset of the

calculation and (unlessf(x) has very violent behaviour nearx=ξ)arangeofx

in whichξlies can be safely stated at any stage. At the stage reached in the last

row of table 27.3 it may be stated that 1. 4949 <ξ< 1 .4977. Thus binary chopping

gives a simple approximation method (it involves less multiplication than linear

interpolation, for example) that is predictable and relatively safe, although its

convergence is slow.

27.1.4 Newton–Raphson method

The Newton–Raphson (NR) procedure is somewhat similar to the interpolation

method, but, as will be seen, has one distinct advantage over the latter. Instead