ajprincess
  • ajprincess
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?
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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schrodinger
  • schrodinger
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anonymous
  • anonymous
did you try couchy?
ajprincess
  • ajprincess
sorry didn't get u?
anonymous
  • anonymous
couchy convergence test: \[|x_{i+k}-x_{i}|<\epsilon\]

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More answers

ajprincess
  • ajprincess
sorry i havnt learnt t.
anonymous
  • anonymous
basicly it says that the far enough terms have distance less than any positive number
perl
  • perl
i think here you assume that there is a limit
perl
  • perl
one sec, let me work on it
perl
  • perl
if limit exists, that means lim xn = L , then x_i+1 is roughly the same as x_i for large i
perl
  • perl
so substitute L for xi and xi+1 L = a0 + a1 L ^2, it is a quadratic
perl
  • perl
use quadratic formula , a1 L^2 - L + a0 = 0, where L = xi as i -> oo,
perl
  • perl
L = [-(-1) + - sqrt ( 1 - 4*a1*a0)] / ( 2*a1)
perl
  • perl
this has real solutions only when the discriminant is at least positive
perl
  • perl
so the condition for convergence is , when a1*a0 <= 1/4
perl
  • perl
also you have a typo in your question
perl
  • perl
so for instance, a0 = 1/2 and a1 = 1/3 converges for any x
perl
  • perl
for any initial seed x
ajprincess
  • ajprincess
Can u plz tell me what is the typo in the question? @perl
perl
  • perl
your iterative formula
perl
  • perl
it should say |dw:1354436826022:dw|
perl
  • perl
makes sense?
ajprincess
  • ajprincess
ya it does. Thanxx a lottt.
perl
  • perl
I take that back that it converges for all x,
perl
  • perl
your seed has to be close enough to the x intercept (zero) for the iteration to converge
perl
  • perl
you know what, i dont think i answered the question. i answered what it will converge to in the case that it does converge
ajprincess
  • ajprincess
ohhh k. Thanksss a lottt for helping me.
perl
  • perl
I think i have a condition in my book.
perl
  • perl
the derivative has to be less than 1 . so 2a1*x < 1 , so x < 1/ (2*a1)
anonymous
  • anonymous
|dw:1354453952416:dw|
anonymous
  • anonymous
|dw:1354454207481:dw|
anonymous
  • anonymous
|dw:1354454557959:dw|
anonymous
  • anonymous
|dw:1354454728910:dw|
anonymous
  • anonymous
|dw:1354454813862:dw|
anonymous
  • anonymous
|dw:1354455134554:dw|
ajprincess
  • ajprincess
Thanksss a lottttt.

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