Mathematical Methods for Physics and Engineering : A Comprehensive Guide

NUMERICAL METHODS

nxn f(xn)

1 1.7 5.42

2 1.544 18 1.01

3 1.506 86 2. 28 × 10 −^1

4 1.497 92 5. 37 × 10 −^2

5 1.495 78 1. 28 × 10 −^2

6 1.495 27 3. 11 × 10 −^3

7 1.495 14 7. 34 × 10 −^4

8 1.495 12 1. 76 × 10 −^4

Table 27.1 Successive approximations to the root of (27.1) using the method

of rearrangement.

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

11.0 − 4 .0000 1.7 5.4186 1.2973 − 2. 6916

2 1.2973 − 2 .6916 1.7 5.4186 1.4310 − 1. 0957

3 1.4310 − 1 .0957 1.7 5.4186 1.4762 − 0. 3482

4 1.4762 − 0 .3482 1.7 5.4186 1.4897 − 0. 1016

5 1.4897 − 0 .1016 1.7 5.4186 1.4936 − 0. 0289

6 1.4936 − 0 .0289 1.7 5.4186 1.4947 − 0. 0082

Table 27.2 Successive approximations to the root of (27.1) using linear

interpolation.

successive values can be found. These are recorded in table 27.1. Although not

strictly necessary, the value off(xn)≡x^5 n− 2 x^2 n−3 is also shown at each stage.

It will be seen thatx 7 and all laterxnagree with the precise answer (27.3) to

within one part in 10^4. However,f(xn)andxn−ξare both reduced by a factor

of only about 4 for each iteration; thus a large number of iterations would be

needed to produce a very accurate answer. The factor 4 is, of course, specific

to this particular problem and would be different for a different equation. The

successive values ofxnare shown in graph (a) of figure 27.2.

27.1.2 Linear interpolation

In this approach two values,A 1 andB 1 ,ofxare chosen withA 1 <B 1 and

such thatf(A 1 )andf(B 1 ) have opposite signs. The chord joining the two points

(A 1 ,f(A 1 )) and (B 1 ,f(B 1 )) is then notionally constructed, as illustrated in graph

(b) of figure 27.2, and the valuex 1 at which the chord cuts thex-axis is determined

by theinterpolation formula

xn=

Anf(Bn)−Bnf(An)

f(Bn)−f(An)

, (27.8)