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swissgirl

  • 2 years ago

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?

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  1. swissgirl
    • 2 years ago
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    Every x has the formula of -5+jh

  2. swissgirl
    • 2 years ago
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    I am just not sure what i wld plug in for my j and h

  3. swissgirl
    • 2 years ago
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    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
    • 2 years ago
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    @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
    • 2 years ago
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    I dont really need help finding the polynomials. There is a method for that

  6. swissgirl
    • 2 years ago
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    I am stuck finding my intial points the x's

  7. mahmit2012
    • 2 years ago
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    it is always unique !

  8. Mikael
    • 2 years ago
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    Lagrange of specified degree IS unique . But if Not lagrange or not specific degree - MULTIPLIQUE !

  9. mahmit2012
    • 2 years ago
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    My bamboo is going to be install, so I will tell you .

  10. mahmit2012
    • 2 years ago
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    The different methods give a unique solution.

  11. Mikael
    • 2 years ago
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    Lagrange \[ \neq \] Chebyshev

  12. Mikael
    • 2 years ago
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    Same degree - is critically import

  13. Mikael
    • 2 years ago
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    I see now @swissgirl solved I think:

  14. swissgirl
    • 2 years ago
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    ohhh ya??????

  15. Mikael
    • 2 years ago
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    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
    • 2 years ago
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    Read the question the x's are derived from the formula -5+jh

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

  18. mahmit2012
    • 2 years ago
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    Mikael Chebishov just gives you the fix points.

  19. swissgirl
    • 2 years ago
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    Ya maybe I am slow idk this question is confusing. Thanks @Mikael for trying :)

  20. Mikael
    • 2 years ago
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    So pls tell me - here you mean Lagrange ?

  21. swissgirl
    • 2 years ago
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    I guess cuz I need to use Neville's method

  22. mahmit2012
    • 2 years ago
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    Mikael it is not different. The assumption gives fix points.

  23. Mikael
    • 2 years ago
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    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
    • 2 years ago
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    no the functionis bounded and continuous ==> bounded variation

  25. Mikael
    • 2 years ago
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    They must converge to it

  26. swissgirl
    • 2 years ago
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    The sequence i dont think converges but you wld only be able to see that if u knew ur starting points

  27. swissgirl
    • 2 years ago
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    I posted the question on MSE maybe someone will have an answer

  28. mahmit2012
    • 2 years ago
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    |dw:1347835065357:dw|

  29. mahmit2012
    • 2 years ago
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    |dw:1347835285217:dw|

  30. mahmit2012
    • 2 years ago
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    |dw:1347835529207:dw|

  31. mahmit2012
    • 2 years ago
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    |dw:1347835583070:dw|

  32. mahmit2012
    • 2 years ago
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    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
    • 2 years ago
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    |dw:1347835972434:dw|

  34. mahmit2012
    • 2 years ago
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    |dw:1347836054752:dw|

  35. swissgirl
    • 2 years ago
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    Thanks @mahmit2012

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