Question

Let . If we use Accelerated Newton-Raphson method to approximate the root of the equation , which of the following(s) is/are ture:

(I)  is multiple root of order

(II) Accelerated Newton-Raphson formula is :

(III) The sequence  obtained by the Accelerated Newton-Raphson method converge to the root  quadratically.

Answer 1

Answer Option (b) : (II) and (III)

Detailed Solution :

Given ,

Accelerated Newton-Raphson Method :

To find the root of the equation   .

First we can clearly see that x =0 is root of f(x) = 0

So let us first find the order of the root x=0 .

We see that  

So we can see that  

Therefore   is a multiple root of    of order   .

So  

   is a multiple root   of order    .

Therefore option (i) is not true

Formula for Accelerated Newton -Raphson Method is  

i.e  

So now

  

Then  

Therefore We have Accelerated Newton - Raphson formula for finding root of   as :

Simplifying we get

An finally

We can cancel out   from numerator and denominator assuming   for any k

Therefore we get

Therefore Accelerated Newton - Raphson formula for finding root of   is:

.....................(**)

Therefore we can see that option (ii) matches with the formula we have obtained .

Hence option (ii) is true .

Also the newton-raphson method has quadratic order of convergence .

And it's order of convergence does not change by using accelerated newton-raphson method .

Therefore Accelerated newton-raphson method has quadratic order of convergence .

Hence we get , the sequence   obtained by the Accelerated newton-raphson method converges to the root     quadratically .

So the option (iii) is also true ,

Therefore answer is :

Option (b) : (II) and (III)

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