Multiplicity of a Root
First Modified Newton’s Method
Second Modified Newton’s Method
So far we discussed about the function which has
simple root.
Now we will discuss about the function which has
multiple roots.
A root is called a simple root if it is distinct,
otherwise roots that are of the same order of
magnitude are called multiple.
Definition (Order of a Root)

The equation f(x) = 0 has a root α of order m, if there
exists a continuous function h(x), and
f(x) can be expressed as the product
f ( x)=(x
( x− α)mh(x),
 ) h( x) where --------------.
h( )  0
f(x)
m
So h(x) can be used to obtain the remaining roots of
f(x) = 0. It is called polynomial deflation.
A root of order m = 1 is called a simple root and
if m > 1 it is called multiple root.
In particular, a root of order m = 2 is sometimes
called a double root, and so on.
The behavior of the graph of f(x) near a root of
multiplicity m (m = 1, 2, 3) is shown in the last
Figure.
It can be seen that when α is a root of odd
multiplicity, the graph of f(x) will cross the x-axis
at (α, 0); and when α has even multiplicity the graph
will be tangent to but will not cross the x-axis
at (α, 0).
Moreover, the higher the value of m the flatter the
graph will be near the point (α, 0).
Sometime it is more difficult to deal with the
 concerning about the order of the
Definition ----root.
We will use the following Lemma which will
illuminate these concepts.
Lemma
Assume that function f(x) and its derivatives --------f ( x), f ( x), f ( x), f
( x)
-----------------------------------------------------------are defined and continuous on an interval about
x = α.
Then f(x) = 0 has a root α of order m if and only if
( m1)
f ( )  f ( )  f ( )  f ( m1) ( )  0,
f ( m) ( )  0
f ( x)  x  4 x  2 x  16 x  5x  20 x  12
6
5
4
3
 ( x  1)3 ( x  2) 2 ( x  3)
2
f ( x)  x  x  21x  45
3
2
Usually we don’t know in advance that an equation
has multiple roots, although we might suspect
it from sketching the graph.
Many problems which leads to multiple roots, are in
fact ill-posed.
The methods we discussed so far cannot be
guaranteed to converge efficiently for all problems.
In particular, when a given function has a multiple
root which we require, the methods we have
described will either not converge at all or converge
more slowly.
For example, the Newton’s method converges very
fast to simple root but converges more slowly when
used for functions involving multiple roots.
Multiplicity of a Root
First Modified Newton’s Method
Second Modified Newton’s Method
If we wish to determine a root of known multiplicity
m for the equation f(x) = 0, then the first
Newton’s modified method (also called the
Schroeder’s method) may be used. It has the form
f ( xn )
xn 1  xn  m.
; n  0,1,2,.....
f ( xn )
It is assumed that we have an initial approximation --x0 .
The similarity to the Newton’s method is obvious
and like the Newton’s method it converges very fast
for the multiple roots.
The major disadvantage of this method is that the
multiplicity of the root must be known in advance
and this is generally not the case in practice.
Multiplicity of a Root
First Modified Newton’s Method
Second Modified Newton’s Method
An alternative approach to this problem that does
not require any knowledge of the multiplicity of
the root is to replace the function f(x) in the
equation by q(x), where
f ( x)
q ( x) 
.
f ( x)
One can show that q(x) has only a simple root at
x = α.
Thus the Newton’s method applied to
find a root of q(x) will avoid any problems of
multiple roots.
If
Then
Thus
f ( x)  ( x   ) h( x),
m
m 1
m

f ( x)  m( x   ) h( x)  ( x   ) h( x)
( x   ) h( x )
q ( x) 
mh( x)  ( x   )h( x)
Obviously we find that q(x) has the root α to
multiplicity one.
So with this modification, the Newton’s method
becomes
q ( xn )
xn 1  xn 
,
q( xn )
which gives
f ( xn ) f ( xn )
xn1  xn 
, n  0,1,2,... 
2
 f ( xn )   f ( xn ) f ( xn )
 is known as the second
This iterative formula (2.26)
modified Newton’s method.
The disadvantage of this method is that we must
calculate a further higher derivative.
A similar modification can be made to the secant
method.