Sequence of Interpolating Values

Hero

New member
Construct a sequence of interpolating values $$y_n$$ to $$f(1 +\sqrt{10})$$, where $$f(x)= \frac{1}{1+x^2 }$$ for $$−5≤x≤5$$, as follows: For each $$n = 1,2,…,10$$, let $$h =\frac{10}{n}$$ and $$y_n= P_n (1+\sqrt{10})$$, where $$P_n(x)$$ is the interpolating polynomial for $$f(x)$$ at nodes $$x_0^n,x_1^n,…,x_n^n=−5+jh$$, for each $$j=0,1,2,…,n$$. Does the sequence $${y_n }$$ appear to converge to $$f(1+\sqrt{10} )$$?

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Sudharaka

Well-known member
MHB Math Helper
Construct a sequence of interpolating values $$y_n$$ to $$f(1 +\sqrt{10})$$, where $$f(x)= \frac{1}{1+x^2 }$$ for $$−5≤x≤5$$, as follows: For each $$n = 1,2,…,10$$, let $$h =\frac{10}{n}$$ and $$y_n= P_n (1+\sqrt{10})$$, where $$P_n(x)$$ is the interpolating polynomial for $$f(x)$$ at nodes $$x_0^n,x_1^n,…,x_n^n=−5+jh$$, for each $$j=0,1,2,…,n$$. Does the sequence $${y_n }$$ appear to converge to $$f(1+\sqrt{10} )$$?
Hi Hero,

I am not very clear about your question. Do you have to construct interpolating polynomials for each, $$\frac{10}{n}$$ where $$n=1,2,\cdots,10$$ separately?

Kind Regards,
Sudharaka.

chisigma

Well-known member
Construct a sequence of interpolating values $$y_n$$ to $$f(1 +\sqrt{10})$$, where $$f(x)= \frac{1}{1+x^2 }$$ for $$−5≤x≤5$$, as follows: For each $$n = 1,2,…,10$$, let $$h =\frac{10}{n}$$ and $$y_n= P_n (1+\sqrt{10})$$, where $$P_n(x)$$ is the interpolating polynomial for $$f(x)$$ at nodes $$x_0^n,x_1^n,…,x_n^n=−5+jh$$, for each $$j=0,1,2,…,n$$. Does the sequence $${y_n }$$ appear to converge to $$f(1+\sqrt{10} )$$?
That is a classical 'example' of 'divergence' of a interpolating polynomial with equidistant points that was 'discovered' by the German Mathematician C.D.T. Runge. See...

Runge's phenomenon - Wikipedia, the free encyclopedia

The 'error function' is given by...

$\displaystyle f(x)-p(x)= \frac{f^{(n+1)}(\xi)}{(n+1)!}\ \prod_{k=1}^{n+1} (x-x_{k})$ (1)

... where $\xi$ is in the definition interval of f(*). In general is the behavior of the error at the edges of interval that causes divergence...

Kind regards

$\chi$ $\sigma$

chisigma

Well-known member
That is a classical 'example' of 'divergence' of a interpolating polynomial with equidistant points that was 'discovered' by the German Mathematician C.D.T. Runge. See...

Runge's phenomenon - Wikipedia, the free encyclopedia

The 'error function' is given by...

$\displaystyle f(x)-p(x)= \frac{f^{(n+1)}(\xi)}{(n+1)!}\ \prod_{k=1}^{n+1} (x-x_{k})$ (1)

... where $\xi$ is in the definition interval of f(*). In general is the behavior of the error at the edges of interval that causes divergence...

Kind regards

$\chi$ $\sigma$
A good method to avoid the ‘Runde’s phenomenon’is to avoid to use equidistant point and to interpolate in the so called ‘Chebysheff-Gauss-Lobatto’ points given by …

$\displaystyle x_{k}= - \cos \frac{k\ \pi}{n}\,\ k=0,1,…,n$ (1)

For the details see…

Chebyshev Interpolation

... where a very interesting 'animation' at the end of section 5.1 shows the better performance of the CGL points approach respect to the 'spontaneous' equidistant points approach...

Kind regards

$\chi$ $\sigma$