564 Chapter 20Numerical methods
The Newton–Raphson method
The Newton–Raphson method, also called
simply Newton’s method, is the most famous
method of finding the zeros of a function.
1Unlike the bisection method, it requires the
evaluation of the derivativef′(x)as well as the
functionf(x)itself. The method is illustrated
in Figure 20.2.
Given an approximate solutionx
0off(x) 1 = 10 ,
the tangent to the curve atx
0is extended until
it crosses the x-axis, at pointx
1. The gradient
of the tangent is
(20.9)
so that
(20.10)
Thenx
1is the new estimate of the zero. The process is repeated withx
0replaced byx
1to givex
2, and so on:
(20.11)
EXAMPLE 20.6Newton–Raphson for
To find writef(x) 1 = 1 x
21 − 151 = 10. Then f′(x) 1 = 12 x, so that the Newton–Raphson
formula is
Table 20.3 shows the computation using all the figures on a standard 10-digit
calculator, starting withx
01 = 13.
xx
x
x
nnnn+=−
−
125
2
5 ,
5
xx
fx
fx
n
nnnn+=−
′
=,,,,
10123
()
()
...
xx
fx
fx
1000=−
′
()
()
fx
fx
xx
′=
−
()
()
00011Numerical methods for the solution of equations have their origins in the determination of square and
cube roots by the Babylonians and by the Chinese. Chapter 4 of the Jiuzhang suanshu(Nine chapters on the
mathematical art) of the early Han dynasty, around 200 BC, describes a method of finding square roots that is
similar to Newton’s method. Jia Xian generalized the method for the solution of polynomial equations in the 11th
century (he also described the construction and uses of the Pascal triangle), and the first detailed account was given
in the Shuchu jiuzhang(Mathematical treatise in nine sections) of 1247 by Qin Jiushao (c.1202–1261). Newton’s
method, described in the Methodus fluxionumof 1671 but not published until 1736, is essentially the Chinese
method for polynomials. Joseph Raphson (1648–1715), who ‘was one of the few people whom Newton allowed to
see his mathematical papers’, published the method in 1690.
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x
0x
1x
2x
y
f(x
0)
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Figure 20.2