The Chemistry Maths Book, Second Edition

20.3 Solution of ordinary equations 565

Table 20.3 The square root of 5

nx

n

f(x

n

) 2 f′(x

n

) x

n+ 1

0 3 0.66666 6666 2.33333 3333

1 2.33333 3333 0.09523 8095 2.23809 5238

2 2.23809 5238 0.00202 6343 2.23606 8896

3 2.23606 8896 0.00000 0918 2.23606 7978

4 2.23606 7978 0

The accuracy of the calculator is exhausted after 4 iterations, and the result is in error

by only 1 in the 10th significant figure.

0 Exercises 12–15

Newton–Raphson is derived formally from the Taylor series expansion of a

continuous function about a point. Let xbe the true value of a zero off(x), andx

n

an

approximate value such thatx 1 = 1 x

n

1 + 1 ε

n

. Then

(20.12)

The terms inε

n

2

and higher can be neglected whenε

n

is small enough. Then, since

f(x) 1 = 10 ,

(20.13)

and a new estimate isx

n+ 1

1 = 1 x

n

1 + 1 ε

n

, as in (20.11).

It can also be shown from the Taylor series that, when the errorsε

n

ofx

n

andε

n+ 1

ofx

n+ 1

are small enough,

(20.14)

This explains the rapid convergence of Newton–Raphson. The error decreases

quadratically, and the number of significant figures approximately doublesat each

iteration. The method is called a second-order iteration process, in contrast to the

bisection method which, with , is a first-order process.

The Newton–Raphson method is not a bracketing method and, as shown by

equations (20.13) and (20.14), can fail, for example, when|f′(x

n

)|is small (or zero).

Most problems are avoided either by a suitable choice of initial point x

0

or by

combining Newton–Raphson with bisection. Thus, a bisection step is taken whenever

Newton–Raphson goes out of bounds or fails to converge.

εε

nn+

≈

1

1

2

εε

nn

n

n

fx

fx

+

=−

′′

′

1

2

2

()

()

ε

n

n

n

fx

fx

=−

′

()

()

fx fx fx f x f x

nn n n n

n

n

() ( ) () ()=+= +′+′′+εε ()

ε

2

2