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?

Question

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?

swissgirl · Answer

Every x has the formula of -5+jh

swissgirl · Answer

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

swissgirl · Answer

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

anonymous · Answer

@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

swissgirl · Answer

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

swissgirl · Answer

I am stuck finding my intial points the x's

anonymous · Answer

it is always unique !

anonymous · Answer

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

anonymous · Answer

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

anonymous · Answer

The different methods give a unique solution.

anonymous · Answer

Lagrange $$ \neq $$ Chebyshev

anonymous · Answer

Same degree - is critically import

anonymous · Answer

I see now @swissgirl  solved I think:

swissgirl · Answer

ohhh ya??????

anonymous · Answer

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

swissgirl · Answer

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

anonymous · Answer

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

anonymous · Answer

Mikael Chebishov just gives you the fix points.

swissgirl · Answer

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

anonymous · Answer

So pls tell me - here you mean Lagrange ?

swissgirl · Answer

I guess cuz I need to use Neville's method

anonymous · Answer

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

anonymous · Answer

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

anonymous · Answer

no the functionis bounded and continuous ==> bounded variation

anonymous · Answer

They must converge to it

swissgirl · Answer

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

swissgirl · Answer

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

anonymous · Answer

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