Newton's Method

[aeh49]

Please help me figure out how to obtain the procedure x<sub>2 = (x1 + k/x1)2 from Newton's method applied to
f(x) = x-squared - k.



Also, By Newton's method applied to f(x) = x<sup>n - k, n a positive integer, show that if x1 is an approximation to
k^(1/n), then

x2 = [(n-1)x1 + k/(x1)^(n-1)]/n

is a better aproximation, and so on.



Thank you!</sup></sub>

 

[stapel]

The only "Newton's Method" with which I'm familiar involves calculus. What method are you using in algebra?

Please include the complete formula or algorithm, being sure to define all terms and variables.

Thank you!

 

[tkhunny]

I have seen it introduced in algebra as more or less a secret formula. It is a rational introduction to iterative methods. On the other hand, it would not surprise me to see "Newton's Method" used as a generic term for an interative solution.

In any case, let's see if it works.

Let's take k = 2

We're trying to solve x^2 - 2 = 0 -- We're hunting for the Square Root of 2.

Start with x1 = 1

(1 + 2/1)^2 = 3^2 = 9 = x2

(9 + 2/9)^2 = 85+

It looks like we're getting nowhere. Are you SURE you wrote it correctly?

 

[Deleted member 4993]

There is Picard's alogrithm of finding square roots - which is modified Newton's method. But I cannot quite decipher the original post - with all those [ sub]s and [ sup]s hanging around. The first problem looks like Piccard's method.

Say you want to find ?k

then

x[sub:192gfnlo]2[/sub:192gfnlo] = (x[sub:192gfnlo]1[/sub:192gfnlo] + k/x[sub:192gfnlo]1[/sub:192gfnlo])/2

following Newton-Raphson's method

f(x) = x[sup:192gfnlo]2[/sup:192gfnlo] - k <<< we want to find the roots of this equation

f'(x) = 2x

then

x[sub:192gfnlo]n[/sub:192gfnlo] = x[sub:192gfnlo]o[/sub:192gfnlo] - f(x[sub:192gfnlo]o[/sub:192gfnlo])/f'(x[sub:192gfnlo]o[/sub:192gfnlo]) <<< Newton-Raphson

= x[sub:192gfnlo]o[/sub:192gfnlo] - (x[sub:192gfnlo]o[/sub:192gfnlo][sup:192gfnlo]2[/sup:192gfnlo] - k)/(2x[sub:192gfnlo]o[/sub:192gfnlo])

= (x[sub:192gfnlo]o[/sub:192gfnlo] + k/x[sub:192gfnlo]o[/sub:192gfnlo])/(2) <<< Piccard

say you want to find ?95

x[sub:192gfnlo]o[/sub:192gfnlo] = 9

x[sub:192gfnlo]1[/sub:192gfnlo] = (9 + 95/9)/2 = 9.25

x[sub:192gfnlo]2[/sub:192gfnlo] = (9.25 + 95/9.25)/2 = 9.760135135

x[sub:192gfnlo]3[/sub:192gfnlo] = (9.760135135 + 95/9.760135135)/2 = 9.746803462

x[sub:192gfnlo]4[/sub:192gfnlo] = (9.746803462+ 95/9.746803462)/2 = 9.746794345

x[sub:192gfnlo]5[/sub:192gfnlo] = (9.746794345+ 95/9.746794345)/2 = 9.746794345 <<<< There we are..

 

[tkhunny]

Ah! Division. Didn't think of that. I wondered what that '2' was at the end. Good work.