Please help:)
The iterative formula $x_{i+1}=a_0+a_1x_1^2$ $(a_0, a_1$ positive ) is being used to solve the equation $x=a_0+a_1x^2.$ What is the condition of convergence?

Question

Please help:)
The iterative formula $x_{i+1}=a_0+a_1x_1^2$ $(a_0, a_1$ positive ) is being used to solve the equation $x=a_0+a_1x^2.$ What is the condition of convergence?

anonymous · Answer

did you try couchy?

ajprincess · Answer

sorry didn't get u?

anonymous · Answer

couchy convergence test:
$$|x_{i+k}-x_{i}|<\epsilon$$

ajprincess · Answer

sorry i havnt learnt t.

anonymous · Answer

basicly it says that the far enough terms have distance less than any positive number

perl · Answer

i think here you assume that there is a limit

perl · Answer

one sec, let me work on it

perl · Answer

if limit exists, that means lim xn = L , then x_i+1 is roughly the same as x_i for large i

perl · Answer

so substitute L for xi and xi+1 
L = a0 + a1 L ^2, it is a quadratic

perl · Answer

use quadratic formula , a1 L^2 - L + a0 = 0,  where L = xi  as i -> oo,

perl · Answer

L = [-(-1) + - sqrt ( 1 - 4*a1*a0)] / ( 2*a1)

perl · Answer

this has real solutions only when the discriminant is at least positive

perl · Answer

so the condition for convergence is , when a1*a0 <= 1/4

perl · Answer

also you have a typo in your question

perl · Answer

so for instance, a0 = 1/2 and a1 = 1/3 converges for any x

perl · Answer

for any initial seed x

ajprincess · Answer

Can u plz tell me what is the typo in the question? @perl

perl · Answer

your iterative formula

perl · Answer

it should say |dw:1354436826022:dw|