The Newton-Raphson
Method 12.3
Introduction
This Section is concerned with the problem of “root location”; i.e. finding those values of x which
satisfy an equation of the form f (x) = 0. An initial estimate of the root is found (for example by
drawing a graph of the function). This estimate is then improved using a technique known as the
Newton-Raphson method, which is based upon a knowledge of the tangent to the curve near the
root. It is an “iterative” method in that it can be used repeatedly to continually improve the accuracy
of the root.
• be able to differentiate simple functions
Prerequisites
• be able to sketch graphs
Before starting this Section you should . . .
'
$
• distinguish between simple and multiple roots
• estimate the root of an equation by drawing
Learning Outcomes a graph
On completion you should be able to . . . • employ the Newton-Raphson method to
improve the accuracy of a root
& %
38 HELM (2008):
Workbook 12: Applications of Differentiation
, ®
1. The Newton-Raphson method
We first remind the reader of some basic notation: If f (x) is a given function the value of x for
which f (x) = 0 is called a root of the equation or zero of the function. We also distinguish between
various types of roots: simple roots and multiple roots. Figures 21 - 23 illustrate some common
examples.
y y y
y = (x − 2)3
y = (x − 1)2
y = f (x)
x0 x x x
1 2
simple root double root triple root
Figure 21 Figure 22 Figure 23
More precisely; a root x0 is said to be:
df
a simple root if f (x0 ) = 0 and 6= 0.
dx x0
df d2 f
a double root if f (x0 ) = 0, = 0 and 6= 0, and so on.
dx x0 dx2 x0
In this Section we shall concentrate on the location of simple roots of a given function f (x).
Task
Given graphs of the functions (a) f (x) = x3 − 3x2 + 4, (b) f (x) = 1 + sin x
classify the roots into simple or multiple.
Your solution
(a) f (x) = x3 − 3x2 + 4: The negative root is: and the positive root is:
y
x=2 x
Answer
The negative root is simple and the positive root is double.
Your solution
(b) f (x) = 1 + sin x: Each root is a root
y
x
Answer
Each root is a double root.
HELM (2008): 39
Section 12.3: The Newton-Raphson Method
Method 12.3
Introduction
This Section is concerned with the problem of “root location”; i.e. finding those values of x which
satisfy an equation of the form f (x) = 0. An initial estimate of the root is found (for example by
drawing a graph of the function). This estimate is then improved using a technique known as the
Newton-Raphson method, which is based upon a knowledge of the tangent to the curve near the
root. It is an “iterative” method in that it can be used repeatedly to continually improve the accuracy
of the root.
• be able to differentiate simple functions
Prerequisites
• be able to sketch graphs
Before starting this Section you should . . .
'
$
• distinguish between simple and multiple roots
• estimate the root of an equation by drawing
Learning Outcomes a graph
On completion you should be able to . . . • employ the Newton-Raphson method to
improve the accuracy of a root
& %
38 HELM (2008):
Workbook 12: Applications of Differentiation
, ®
1. The Newton-Raphson method
We first remind the reader of some basic notation: If f (x) is a given function the value of x for
which f (x) = 0 is called a root of the equation or zero of the function. We also distinguish between
various types of roots: simple roots and multiple roots. Figures 21 - 23 illustrate some common
examples.
y y y
y = (x − 2)3
y = (x − 1)2
y = f (x)
x0 x x x
1 2
simple root double root triple root
Figure 21 Figure 22 Figure 23
More precisely; a root x0 is said to be:
df
a simple root if f (x0 ) = 0 and 6= 0.
dx x0
df d2 f
a double root if f (x0 ) = 0, = 0 and 6= 0, and so on.
dx x0 dx2 x0
In this Section we shall concentrate on the location of simple roots of a given function f (x).
Task
Given graphs of the functions (a) f (x) = x3 − 3x2 + 4, (b) f (x) = 1 + sin x
classify the roots into simple or multiple.
Your solution
(a) f (x) = x3 − 3x2 + 4: The negative root is: and the positive root is:
y
x=2 x
Answer
The negative root is simple and the positive root is double.
Your solution
(b) f (x) = 1 + sin x: Each root is a root
y
x
Answer
Each root is a double root.
HELM (2008): 39
Section 12.3: The Newton-Raphson Method
