1.10 Multiple Roots
Definition. We say that r is a root of multiplicity m of a function f(x) if
f (r )  f (r )  ...  f ( m1) (r )  0 and f ( m) (r )  0 .
Example 1. r = 0 is a root of multiplicity 2 of f(x) = x2.
Example2. r =1 is a root of multiplicity 4 of f(x) = x4 - 4x3 + 6x2- 4x+ 1.
Example 3. r = 0 is a root of multiplicity 2 of f(x) = x sin x.
Definition. (Multiple and simple roots )
Let r be a root of multiplicity m of a function f.
(1) If m = 1, then r is called a simple root.
(2) If m > 1, then r is called a multiple root.
Remark: Multiple roots have disadvantages on the accuracy of the
calculations, and the convergence to the root with the methods we
study before is slower in the case of multiple root than that for a simple
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root. We can clarify these disadvantages geometrically by the following
figures:
Note that functions with a multiple root have graphs near the x-axis on
each sides of the root for a nearly far distance. In a simple root, we can
decide the accuracy from the value of f ( xn ) , when we approximate the
root r by xn , if it is large, then the approximation is not good. In the
case of a multiple root, the value of f ( xn ) may be small, as we noticed
in the previous graphs, even though xn is not a good approximation.
Also, we can clarify these disadvantages algebraically by the following
example
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Example 4. Let f ( x)  x  1 , and g ( x)  ( x  1) 4 . Then, r = 1 is a
simple root for f and a multiple root for g . If we take x0 = 1.1 as an
approximation for the root r with error E=0.1, then we find f ( x0 )  0.1
and g ( x0 )  0.0001  0 . In the case of the function f , f ( x0 ) gives a
good approximation of the error, while g ( x0 ) in the function g , gives
a false approximation of the error. In the function g , we may think
that 1.1 is a good approximation to the root with accuracy agree to 4
digits, which is not true.
How can we find a multiple root?
Some of the methods we have studied always work with multiple roots,
and some of them may or may not work. The bisection and false position
methods do not work to get multiple roots because the function may not
change sign at a multiple root. Secant, Newton's, and fixed-point methods
work to get multiple roots, but the convergence is slower than that for a
simple root. Another disadvantage of using Newton's or secant
method to find a multiple root is that f (r ) (or its approximation) will be
near zero. Thus almost all computers will find f (r )  0 because of the
imprecise arithmetic, and hence we will have division
by zero.
Remedies for multiple roots
We can solve the problem of multiple roots by two methods :
First method :This method can be applied to all previous methods for
finding roots of nonlinear equations. The general idea in this method is to
convert the multiple root of the function f to a simple root to another
function g , and this idea can be done by two ways:
1- Notice that if r is a root for the function f with multiplicity m,
then r is a simple root for the function g ( x)  f ( m1) ( x) . Thus we
can apply any of the previous methods with the function g to find
the root r which is also a root for the function f . The difficulty
in this method is that we must know the multiplicity m of the root
for the function f which is often unknown.
2- If r is a root for the function f ( simple or multiple), then r is a
simple root for the function  ( x) 
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f ( x)
. Then we can apply any
f ( x)
of the previous methods with the function  (x ) to find the root r
which is also a root for the function f . The advantage of this
way is that we do not need to know the multiplicity m of the root
for the function f , but the disadvantage is that we need to
calculate f (x ) , which may be difficult especially if we apply
Newton's method for  (x ) , we need to calculate  (x ) .
Second method : This method can be applied only to Newton's method.
It is called The Modified Newton's Method , in which we use the
f ( xn )
modified Newton's formula: xn 1  xn  m
, where m is the
f ( xn )
multiplicity of the root.
Example 5. Let f ( x)  ( x  1)(e( x1)  1) .
a) Find the multiplicity of the root r =1.
b) Use Newton's method before and after modification to find the root
with x0 = 2.
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Nearly Multiple Roots
A problem that is related to multiple roots is a function that has two roots
or more roots very close to each other, as in the last function in the figure
above in page 2. In this case also Newton's method converges slower if
we start outside the interval containing the roots. If we start with a value
between two almost equal roots, Newton's method may diverge.
Remark. The modifications of Newton's method do not work here, often
an infinite loop occurs.
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