## ajprincess Group Title 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? one year ago one year ago

1. myko Group Title

did you try couchy?

2. ajprincess Group Title

sorry didn't get u?

3. myko Group Title

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

4. ajprincess Group Title

sorry i havnt learnt t.

5. myko Group Title

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

6. perl Group Title

i think here you assume that there is a limit

7. perl Group Title

one sec, let me work on it

8. perl Group Title

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

9. perl Group Title

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

10. perl Group Title

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

11. perl Group Title

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

12. perl Group Title

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

13. perl Group Title

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

14. perl Group Title

also you have a typo in your question

15. perl Group Title

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

16. perl Group Title

for any initial seed x

17. ajprincess Group Title

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

18. perl Group Title

19. perl Group Title

it should say |dw:1354436826022:dw|

20. perl Group Title

makes sense?

21. ajprincess Group Title

ya it does. Thanxx a lottt.

22. perl Group Title

I take that back that it converges for all x,

23. perl Group Title

your seed has to be close enough to the x intercept (zero) for the iteration to converge

24. perl Group Title

you know what, i dont think i answered the question. i answered what it will converge to in the case that it does converge

25. ajprincess Group Title

ohhh k. Thanksss a lottt for helping me.

26. perl Group Title

I think i have a condition in my book.

27. perl Group Title

the derivative has to be less than 1 . so 2a1*x < 1 , so x < 1/ (2*a1)

28. mahmit2012 Group Title

|dw:1354453952416:dw|

29. mahmit2012 Group Title

|dw:1354454207481:dw|

30. mahmit2012 Group Title

|dw:1354454557959:dw|

31. mahmit2012 Group Title

|dw:1354454728910:dw|

32. mahmit2012 Group Title

|dw:1354454813862:dw|

33. mahmit2012 Group Title

|dw:1354455134554:dw|

34. ajprincess Group Title

Thanksss a lottttt.