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Newton Raphson

It’s Christmas! An important day because it marks the anniversary of the birth of a man who changed the world! That’s right, Isaac Newton! (Ok, now I know at least one of you is about to comment with “but since the calendar changed from the Julian to the Gregorian his birthday should be moved to January 6th. I can see your point, but no one suggests moving Christmas so why would we move Newton’s birthday? Anyway, I tend to celebrate both.)

Recently a couple of people have asked me about the Newton – Raphson method for finding roots of equations, and why it works, so in keeping with the festive spirit here is a brief overview of why. Incidentally, I’m working this up into a help sheet for my further maths class so any additional input would be great.

Newton Raphson

A nice little numerical method if finding the root of an equation. You start with an approximation  (often referred to as Xo) and then you take away the ratio of f (Xo)/f’ (Xo) to get a better approximation. You keep going until you get an approximation which is correct to a suitable degree of accuracy.

But why? What is this witchcraft and why does it work?!

No, it’s not witchcraft,  and it’s relatively simple and based on our old favourite “right angled triangle trigonometry” see, I told you triangles were the saviours of everything….

Let’s start with a sketch (always a good start):

image

Here is a curve, as you can see I’ve drawn a tangent to it at C, a perpendicular from C to the x axis (which meets it at B) and labelled the point where the tangent intercepts the x axis as A. Already we have our right angled triangle!

Now we all know that f’ (Xo) or f'(B) will give us the gradient of the tangent to the curve at that point(C in this case). The gradient of that line is the same as the tangent ratio of the angle CAB, as you can see from the sketch (as the gradient is difference in y / difference in x). The opposite side in this case is f (Xo) so the adjacent side is f (Xo) / f’ (Xo) – ie is the opposite side over the tan ratio.

This shows us what’s going in here, we are taking away the adjacent side each time and getting closer to the actual root (when it does converge that is!).

So, a festive look at Newton Raphson. Merry Christmas.

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