Assessment of Newton`s Method

Newton's method seems to converge rapidly for a simple roots. In this section, we seek to understand the behavior of Newton's method for simple and multiple roots. The convergence behavior of Newton's method is quite different for simple vs multiple roots, see Figure 1.4.1.

  
Figure 1.3: Single and double root

Using the definition on the method in Equation 1.3

If is near , then

Now and for a simple root, so

It is seen that if is near a simple root, the error in is roughly a constant multiple of the square of the error in . This is called quadratic convergence.

If an iteration is such that

the method is said to converge at rate r with constant . It has been argued that for a simple root, Newton's method converges at the rate r = 2. Unfortunately it is not the case when computing a multiple root. If is near a root of multiplicity m > 1 , then

This implies that

This expression shows for a root of multiplicity m, Newton's method is only linearly convergent with constant . Much is made of the quadratic convergence of Newton's method, but it is quadratically convergent only for simple roots.

Even for simple roots, quadratic convergence is observed only when sufficiently close to a root. An example of Wilkinson illustrates this. Consider the problem

It has a simple root at . Consider two cases for starting values:

1. start with :
   Then

   

   Hence, a reasonable guess for the root has lead to much worse approximation.

2. is much larger than 1:
   then

   

   But to similar accuracy:

   

   which says that Newton's method crawls back to the root at 1 at a very slow rate.