swissgirl Group Title Construct a sequence of interpolating values $$Y_n, to,f(1 + \sqrt{10})$$, where $$f(x) = (1 + X^2)^{-1}$$ for $$-5 \leq X \leq 5$$, as follows: For each n = 1,2, ... ,10, let h = 10/n and $$Y_n = P_n(1 + \sqrt{10})$$, where Pn(x) is the interpolating polynomial for f(x) at the nodes $$x_0^n, x_1^n,…,x_n^n$$ and $$x_j^n= -5 + jh$$, for each j = 0, 1,2, ... ,n. Does the sequence {Y_n} appear to converge to $$f(1 + \sqrt{10})$$ How would i set up this sequence? one year ago one year ago

1. swissgirl Group Title

Every x has the formula of -5+jh

2. swissgirl Group Title

I am just not sure what i wld plug in for my j and h

3. swissgirl Group Title

Like i just need to figure out what my x's wld be but with these h's and j's I am getting confused

4. Mikael Group Title

@swissgirl Not that I ever dealt with interpolating polynomials much, BUT 1) They are definitely NOT unique - even I know of at least 2-3 completely different such interpolating polynomials - Lagrange, Bezier curves http://en.wikipedia.org/wiki/B%C3%A9zier_curve, and Chebyshev polynomials 2) They are very oscillating beasts - don't behave well when forced too much

5. swissgirl Group Title

I dont really need help finding the polynomials. There is a method for that

6. swissgirl Group Title

I am stuck finding my intial points the x's

7. mahmit2012 Group Title

it is always unique !

8. Mikael Group Title

Lagrange of specified degree IS unique . But if Not lagrange or not specific degree - MULTIPLIQUE !

9. mahmit2012 Group Title

My bamboo is going to be install, so I will tell you .

10. mahmit2012 Group Title

The different methods give a unique solution.

11. Mikael Group Title

Lagrange $\neq$ Chebyshev

12. Mikael Group Title

Same degree - is critically import

13. Mikael Group Title

I see now @swissgirl solved I think:

14. swissgirl Group Title

ohhh ya??????

15. Mikael Group Title

You find ur interp.-ing values by simple 1-st or 2-nd degree Taylor approxim. THEN you costruct your Lagrange polyn. or whatever

16. swissgirl Group Title

Read the question the x's are derived from the formula -5+jh

17. Mikael Group Title

Well I tried. Anyway , for me it very clear that the words "THE interpolating polynomial of degree 10" are ill defined.

18. mahmit2012 Group Title

Mikael Chebishov just gives you the fix points.

19. swissgirl Group Title

Ya maybe I am slow idk this question is confusing. Thanks @Mikael for trying :)

20. Mikael Group Title

So pls tell me - here you mean Lagrange ?

21. swissgirl Group Title

I guess cuz I need to use Neville's method

22. mahmit2012 Group Title

Mikael it is not different. The assumption gives fix points.

23. Mikael Group Title

I vaguely remember tha on compact interval they do converge in most norms to the function - unless of course the function has unbounded variation. And this may be here because of vertic asymptote

24. Mikael Group Title

no the functionis bounded and continuous ==> bounded variation

25. Mikael Group Title

They must converge to it

26. swissgirl Group Title

The sequence i dont think converges but you wld only be able to see that if u knew ur starting points

27. swissgirl Group Title

I posted the question on MSE maybe someone will have an answer

28. mahmit2012 Group Title

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29. mahmit2012 Group Title

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30. mahmit2012 Group Title

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31. mahmit2012 Group Title

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32. mahmit2012 Group Title

I guess e can not solve it directly. So I guess it is not going to be zero because f(x) at interval [-5,5] has no Tylor polynomial. and it just converge for interval with radios one around a fix point.

33. mahmit2012 Group Title

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34. mahmit2012 Group Title

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35. swissgirl Group Title

Thanks @mahmit2012