- #1
MrCreamer
- 6
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Homework Statement
The goal is to approximate the number [itex]\frac{-1+\sqrt{5}}{2}[/itex] using linearization methods.
Homework Equations
This number is a solution to x[itex]^{2}[/itex]=1-x
The Attempt at a Solution
I was told to use f(x)= x[itex]^{2}[/itex]+x-1 with the Newton method to find x[itex]_{1}[/itex],x[itex]_{2}[/itex],x[itex]_{3}[/itex],x[itex]_{4}[/itex] at x[itex]_{0}[/itex]=2;
The general equation was [itex]\frac{x^{2}_{n}+1}{2x_{n}+1}[/itex] thus yielding:
x[itex]_{1}[/itex] = 1
x[itex]_{2}[/itex] = [itex]\frac{2}{3}[/itex]
x[itex]_{3}[/itex] = [itex]\frac{13}{21}[/itex]
x[itex]_{4}[/itex] = [itex]\frac{610}{987}[/itex]
Using my calculator [itex]\frac{610}{987}[/itex] is extremely close to [itex]\frac{-1+\sqrt{5}}{2}[/itex].