## ajprincess 3 years ago 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?

1. myko

did you try couchy?

2. ajprincess

sorry didn't get u?

3. myko

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

4. ajprincess

sorry i havnt learnt t.

5. myko

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

6. perl

i think here you assume that there is a limit

7. perl

one sec, let me work on it

8. perl

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

9. perl

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

10. perl

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

11. perl

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

12. perl

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

13. perl

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

14. perl

also you have a typo in your question

15. perl

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

16. perl

for any initial seed x

17. ajprincess

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

18. perl

19. perl

it should say |dw:1354436826022:dw|

20. perl

makes sense?

21. ajprincess

ya it does. Thanxx a lottt.

22. perl

I take that back that it converges for all x,

23. perl

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

24. 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

25. ajprincess

ohhh k. Thanksss a lottt for helping me.

26. perl

I think i have a condition in my book.

27. perl

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

28. mahmit2012

|dw:1354453952416:dw|

29. mahmit2012

|dw:1354454207481:dw|

30. mahmit2012

|dw:1354454557959:dw|

31. mahmit2012

|dw:1354454728910:dw|

32. mahmit2012

|dw:1354454813862:dw|

33. mahmit2012

|dw:1354455134554:dw|

34. ajprincess

Thanksss a lottttt.