Newton Rhapson question

Oct 2008
157
3
This is of four parts:

1] show that the equation ln2x = sin(x/2) has one root \(\displaystyle \alpha\) that lies between 0.6 and 0.8. I've done this and it was simple enough.

2] An iterative sequence, based on rearranging this equation, is given by

\(\displaystyle x_0 = 0.7\),

\(\displaystyle x_{n+1} = f(x_n)\)

where f(x) = \(\displaystyle \frac{1}{2}e^{sin(\frac{x}{2})}\).

evaluate f ' (0.7) and state why the result suggests that the sequence is convergent.

I have f ' (x) = 0.25\(\displaystyle e^{sin(\frac{x}{2})}cos(\frac{x}{2})\)

and when I put x=0.7 I get 0.331 to 3dp, and dividing this by 0.7 gives me 0.473 to 3dp. In the modulus, this gives me a value of less than 1 so this states that this is convergent.

My work above is uncertain and I'm not sure if I'm right.

3] Calculate \(\displaystyle x_1\) and \(\displaystyle x_2\) recording the answers as accurately as possible.

I've done this and I had values that were 0.2102871815 and then -0.2894231095, and seeing as the root is between 0.6 and 0.8 I'm not quite sure why this is. Either I've used the Newton-Rhapson method incorrectly or maybe I should do something else?

4] Round the value of \(\displaystyle x_2\) to three decimal places and determine whether this is the root correct to 3 decimal places.

I will be able to do this once I get a value that resembles something like what I'm looking for.

Can anyone help me with this iteration? Thanks if you can guide me in the right direction with this :)
 
Aug 2007
3,171
860
USA
First, why is this in "Algebra / PreAlgebra"?

1] show that the equation ln2x = sin(x/2) has one root \(\displaystyle \alpha\) that lies between 0.6 and 0.8. I've done this and it was simple enough.
How?

I have f ' (x) = 0.25\(\displaystyle e^{sin(\frac{x}{2})}cos(\frac{x}{2})\)
I am quite puzzled by this. Why did you switch to 0.25, rather than staying with 1/4?

this gives me a value of less than 1 so this states that this is convergent.
Reality check. Read the definition again. It WILL converge or is has a chance to converge?

I've done this and I had values that were 0.2102871815 and then -0.2894231095
Something funny, here. Are you simply evaluating \(\displaystyle x_{i} = f(x_{i-1})\) or are you trying to build \(\displaystyle x_{i} = x_{i-1} - \frac{g(x_{i-1})}{g'(x_{i-1})}\) for some function g that has not been specified in your post?

REALLY BIG NOTE: When you created f(x), you should have noticed a Domain change. Did you? What is it? Why does it matter? Will it make any difference?
 

awkward

MHF Hall of Honor
Mar 2008
934
409
This is of four parts:

1] show that the equation ln2x = sin(x/2) has one root \(\displaystyle \alpha\) that lies between 0.6 and 0.8. I've done this and it was simple enough.

2] An iterative sequence, based on rearranging this equation, is given by

\(\displaystyle x_0 = 0.7\),

\(\displaystyle x_{n+1} = f(x_n)\)

where f(x) = \(\displaystyle \frac{1}{2}e^{sin(\frac{x}{2})}\).

evaluate f ' (0.7) and state why the result suggests that the sequence is convergent.

I have f ' (x) = 0.25\(\displaystyle e^{sin(\frac{x}{2})}cos(\frac{x}{2})\)

and when I put x=0.7 I get 0.331 to 3dp, and dividing this by 0.7 gives me 0.473 to 3dp. In the modulus, this gives me a value of less than 1 so this states that this is convergent.

My work above is uncertain and I'm not sure if I'm right.

3] Calculate \(\displaystyle x_1\) and \(\displaystyle x_2\) recording the answers as accurately as possible.

I've done this and I had values that were 0.2102871815 and then -0.2894231095, and seeing as the root is between 0.6 and 0.8 I'm not quite sure why this is. Either I've used the Newton-Rhapson method incorrectly or maybe I should do something else?

4] Round the value of \(\displaystyle x_2\) to three decimal places and determine whether this is the root correct to 3 decimal places.

I will be able to do this once I get a value that resembles something like what I'm looking for.

Can anyone help me with this iteration? Thanks if you can guide me in the right direction with this :)
Your problem title indicates that you have misunderstood this problem. It's not about Newton-Raphson iteration. It's about fixed-point iteration (or maybe you know it by some other name).

Just start with
\(\displaystyle x_0 = 0.7\)
and then use
\(\displaystyle x_{n+1} = f(x_n)\)
to calculate \(\displaystyle x_1\) and \(\displaystyle x_2\).