Newton's transformation

[Vader07]

Hello everyone,


Derive a newton transform, F, whose iterates tend to converge to solutions of the equation

x^4 + 3 = 2/x

Then F(1) = ?


This how I started:

Step 1: (x^4 + 3 - 2x) / (4x^3 + 2x^-2)
Step 2: x^4 + 3 - 2x^-1
Step 3: (1 +3 -2) / (4 + 2)

Can anybody give some steps/advice for help

 

[galactus]

Multiply by x and get \(\displaystyle f(x)=x^{5}+3x-2\)

\(\displaystyle f'(x)=5x^{4}+3\)

By using Newton's method, you are finding the real solutions to \(\displaystyle x^{5}+3x-2=0\)

There is only one real solution to this. The other 4 are complex, so no need to worry about those.

Now, use Newton's method. Try an intial guess of \(\displaystyle x_{n}=.5\)

\(\displaystyle x_{n}-\frac{f(x)}{f'(x)}\)

Upon finding your first iteration, then sub that back in. Keep it up until the desried accuracy is obtained.

It converges pretty quick, so you will not need many iterations.