Newtons Method

Miss Snowy

Can someone point out to me how you derive Newtons Formula?

Derive Newton’s formula: xn + 1 = xn – ( f(xn)/f'(xn)).

Use diagram(s) to illustrate the derivation.


I understand how to do newtons method just I'm not sure how you go about explaining how to derive it?!

LeixlipRed

Ok, the idea is that you start with an approximation x_n for the root of some polynomial f. You find the derivative of that polynomial and the tangent line to the polynomial at (x_n, f(x_n)) will definitely cut the x-axis closer to the root than x_n did. The key is in the picture, you need to pick an x_n, mark (x_n, f(x_n)), draw it's tangent and the point it cuts the x-axis is the point x_(n+1), you need to derive an equation for that.

All you need then is to recall the equation of the line formula and that the slope here will be given by f'(x_n).

LeixlipRed

Sorry, should mention you can just let f be any polynomial, so just use a nice quadratic or something for your picture.

Miss Snowy

Sorry,

Any chance you could guide me with an example? I don't fully understand what you are saying?

Maths is not my strong point and find it hard to grasp.

LeixlipRed

Check out the wikipedia page:

http://en.wikipedia.org/wiki/Newton%27s_method

Unfortunately it's a proof that requires visual cues to really grasp it and I've no scanner to draw pics! Have a look and post back!

Miss Snowy

See my problem is understanding the various letters and symbols. So Wikipedia goes over my head a little.

- Start with an approximation and the derivative. So I have a polynomial function and I pick a number to guess the root?

- I also need to work out the tangent line.

Then graph all of the above? Is that not pretty much doing the full newtons method to determine the roots are (the full table where you work it out) and then graph the results?

Is there any visual example of this online? Iv only seen a bunch of text or different formula from Newton!

LeixlipRed

There's a visual example on the page I just linked to, did you scroll down? You can't prove this using numbers, the proof must be in general. The graph is where you start, you need to graph what happens when you pick a random x and get a tangent at the point (x, f(x)). What notation are you struggling with?

LeixlipRed

Here is the gif from the wiki page, this shows exactly what's happening.

http://en.wikipedia.org/wiki/File:NewtonIteration_Ani.gif

Miss Snowy

Do you just explain how newtons methods works and use that graph?

LeixlipRed

The question asks you to derive the formula. i.e. you need to create the formula from scratch. There's no shortcut here I'm afraid. You need to explain exactly what you don't understand and we can help you.

Miss Snowy

Is the description the part that explains how the formula is produced?

So I need to explain how the formula was created from scratch, correct?

I understand the graph is showing guesses and drawing them out on the graph until the point where it cuts x-axis.

Sorry I am dragging this out, a little hard to explain where I am going wrong on an internet forum I guess for someone with poor math skills!

LeixlipRed

You need to find the x value of the next guess, look back at my original post in this thread and look at the image and see can you see what I mean. You need a equation in which you can isolate x_(n+1) and that equation is the intersection of the tangent line and the x-axis (i.e. you need to find the equation of the tangent line and sub in 0 for y).

LeixlipRed

You can also derive it from the definition of derivative but not sure which is less or more confusing for you.

Sully

These any use Leixli?

See attached. Not sure if that answers the question or not tho.#

Also, a link on Yahoo - second answer gives a fair bit of detail but no graph. Seems a good explanation.

http://answers.yahoo.com/question/index?qid=20110116164729AAFgatV

timbrophy

You might find the following java applet of some help:

http://www.ul.ie/cemtl/Applets/SciMaths/newton.html

Miss Snowy

LeixlipRed wrote: »

You need to find the x value of the next guess, look back at my original post in this thread and look at the image and see can you see what I mean. You need a equation in which you can isolate x_(n+1) and that equation is the intersection of the tangent line and the x-axis (i.e. you need to find the equation of the tangent line and sub in 0 for y).

See I can understand the graph and what its doing. But surely I need more then just a graph -- do I not need to break the formula down and explain it?

LeixlipRed

Yes, that's what I'm trying to say. When I say "find" I mean "find an expression for". I don't mean locate visually on a page where the next guess is. Can you go back to my first post and try break that down and see if you can figure out what steps you need to take to derive the formula, you need to be specific about what exactly it is you can't get!

Miss Snowy

I have made an attempt at doing the question. Would I be right? See attached..