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Multiple Roots And Convergence
If the given equation f(x) = 0 has multiple roots at a point say 'x = s' then the order of convergence decreases for any iterative method.
For example consider the error equation of Newton's method for any transcendental equation f(x) = 0 with multiple roots at 'x = s' :
  ei+1 = (1-1/m)ei +        1  ei2   f(m+1)(s)  + O(ei3)
m2(m+1)   f(m)(s)
if m = 1 (i.e., s is only a simple root) then the coefficient of ei is zero and coefficient of ei2 is not equal to zero and hence the scheme is of second order.
If m ¹ 1 then the coefficient of ei itself is not equal to zero and hence the scheme is only of first order.
[Proof]
Modified Newton's Method :
case(i):

If we know the multiplicity ' m ' of the root at any point in advance then the Newton's method can be modified as 

xi+1= xi - a  fi                       i = 0, 1, 2 . . .
f 'i
where a is an arbitrary parameter to be determined. Then the error equation for this modified scheme is (please follow the same steps of the above proof with afi/f 'i in place of  fi/f 'i)
ei+1 = (1-a/m)ei +        a  ei2   f(m+1)(s)  + O(ei3)
m2(m+1)   f(m)(s)
=> if a = m then the coefficient of the ei becomes zero and coefficient of ei2 is not equal to zero hence the scheme is again of second order.
case(ii) :

If the multiplicity of the root is not known in advance then we use the following procedure. 

If f(x) = 0 has a root at  x = s with multiplicity m(>1) then f'(x) = 0 has the same root at x = s with multiplicity (m-1). Hence the function  h(x) = f(x)/f'(x) has a simple root at x = s. Now the Newton's method can be modified as

xi+1= xi - 
   h(xi)
 i = 0,1,2...
 h '(xi)

 
      = xi - 
fi fi'
  
fi'2 - fi fi'' 
This is again of second order but the drawback of this scheme is at each iteration three function f, f' and f'' are to be evaluated.
Similarly the Secant method can be modified as
xi+1=   xi-1 fi f 'i-1 -  xi fi-1 fi'
          i = 0, 1, 2... 
     fi f 'i-1 -  fi-1 fi'
Finding the multiplicity of a root 'm' :
Consider the equation f(x) = 0 which has a root at x = s with multiplicity m (>1).
Let xn-1, xn, xn+1 are respectively, the n-1, n  and n+1th approximations of s and en-1, en and en+1 are the errors involved in the approximation. Since the root is of multiplicity m > 1 the error equal is en+1 = cen.

Consider

 xn+1 - xn      =     (xn+1+ s) - (xn+ s)      =   en+1 - en
  xn - xn-1    (xn+ s) - (xn-1+ s)  en - en-1

 
                                 =     (c -1)en              (... en+1 = cen =>en-1 = 1/c en)
 
     (1-1/c)en

                                                          =  c
 

i.e., if the multiplicity m is greater than one then the ratio defined above is a constant for any n.
Also this constant must be approximately equal to (1-1/m) since for Newton's method for m > 1 the error at (n+1) iteration is approximately equal to en+1 ~ (1-1/m)en


                                                  => c = 1-1/m

                                                   => m = 1/(1-c)
 

Example : Consider the equation x2-2x+1 = 0
This has a root at x = 1 with multiplicity two (Double root).
The Newton's equation for this is 
xi+1= xi - 
 xi2 - 2xi +1
 i = 0,1,2...
2xi - 2
The iterative values are
i
0
1
2
3
4
5
6
xi
0
0.5
0.75
0.875
0.937
0.969
0.984
its very clear that the iterative scheme is converging very slowly. Let us consider the ratios

 
 x3 - x2  =   0.875 - 0.75  = 0.49
  x2 - x1  0.75 - 0.5

 
 x4 - x3  =   0.937 - 0.875  = 0.5
  x3 - x2  0.875 - 0.75

 
 x5 - x4  =   0.969 - 0.937  = 0.516
  x4-x3  0.937 - 0.875
i.e., all these ratios are approximately equal to 0.5
Hence the multiplicity of the root  m  =  1/(1-c) =  1/(1-0.5)  =  2
Now the modified Newton's method is
xi+1= xi - 2 
  xi2 - 2xi +1
i = 0,1,2...
     2xi - 2
Now with x0 = 0
x1= 0 - 2*  1   = 1
 2
is the required root.
Solution of Transcendental Equations | Solution of Linear System of Algebraic Equations | Interpolation & Curve Fitting
Numerical Differentiation & Integration | Numerical Solution of Ordinary Differential Equations
Numerical Solution of Partial Differential Equations