1. Multiplicity of a Root First Modified Newton’s Method
Second Modified Newton’s Method

2. 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.

3. 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 − α)mh(x), where
So h(x) can be used to obtain the remaining roots of f(x) = 0. It is called polynomial deflation.

4. 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.

6. 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).

7. We will use the following Lemma which will illuminate these concepts.
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 Definition concerning about the order of the root.
We will use the following Lemma which will illuminate these concepts. 

8. Lemma
Assume that function f(x) and its derivatives
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

11. 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.

12. 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.

13. For example, the Newton’s method converges very fast to simple root but converges more slowly when used for functions involving multiple roots.

14. Multiplicity of a Root First Modified Newton’s Method
Second Modified Newton’s Method

15. 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
It is assumed that we have an initial approximation -- .

16. 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.

17. Multiplicity of a Root First Modified Newton’s Method
Second Modified Newton’s Method

18. 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

19. One can show that q(x) has only a simple root at
Thus the Newton’s method applied to
find a root of q(x) will avoid any problems of multiple roots.

20. If
Then
Thus
Obviously we find that q(x) has the root α to multiplicity one.

21. So with this modification, the Newton’s method becomes
which gives
This iterative formula (2.26) is known as the second modified Newton’s method.  

22. The disadvantage of this method is that we must calculate a further higher derivative.
A similar modification can be made to the secant method.