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Name: Kimberly M.
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Question:
I was curious to know why Newton's method of computing a square root works (ie. square root of 10= (5+2)/2...3.5+(10/3.5)/2...and so on. I understand how to get the solution but I do not know why it >works. Also, I do not understand how to find the solutions of a quadratic solution...using x1=xo-(axo^2+bxo+c)/(2axo+b)...could you give me an example of what this means using numbers?



Replies:
Let us do the first part, first. It is easier to "see". Problem: Find X = sqrt(b).

This means: X^2 = b. That is, upper case 'letter ex' is the sqrt(b)

If I were lucky and guessed "the answer" on my first try, then I could write that: (X^2 +b)/2 = (b+b)/2 = 2b/2 = b. Very smart, but not very interesting, and not very probable (or in the case of irrational numbers, not even possible).

But I am not so smart so I "guess" another number, call it 'xo' lower case 'letter ex sub zero'. Then, (xo^2 + b)/2 will be larger than xo^2 /2 but less than (X^2 + b) /2 = (b+b)/2 = b. In fact it will be about half way in between, since it is the mean (the average of xo^2 and b).

So then I will take this "new" estimate of sqrt(b) and call it 'x1'. Now since x < x1< b. It cannot be greater than 'b' because it is the average of xo, which is smaller than 'b', and "the answer" b. In effect I have "trapped" the answer between 'xo' and 'b'.

Not having anything better to do I will take the new guess 'x1' and find the mean of it and 'b'. Or rather I'll take the average of the square of 'x1' and 'b' since that is what I know. My new guess then is 'x2' = (x1^2 + b)/2. This second guess is larger than 'x1' but still smaller than 'b' because it is the mean (average) of x1 and b. Symbolically, x1 < x2 < b. And so on.

I think it is easier to see what is happening if you take a perfect square. Let us say we want sqrt(9). Of course, the answer is 3 but let us see how Newton's method works:

I will guess a value of 2. Then my next guess is (2^2+9)/2 = 13/2 =6.5. But sqrt(6.5)=2.55 That is better, but let us try again: (6.5 + 9)/ 2 = 15.5 /2 = 7.75. But sqrt(7.75)=2.78 That is still better but let us try again: (7.75 + 9) /2 = 8.375. But again sqrt(8.375)=2.894. That is even better but let us try again: (8.375+9)/2 = 8.6875. And sqrt(8.6875)=2.947

You can see that we are quickly converging on the answer: sqrt(9) = 3.

The "answer" to your second part is a bit more complicated if you have not had any calculus. It will take me some more thought and time to phrase it in non-calculus terms, since it involves slopes of curves, but I will give it a shot.

Vince Calder



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