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Inverse interpolation and Neville $ algorithm (P.142-144 of Cheney-Kincaid)....

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Inverse interpolation and Neville $ algorithm (P.142-144 of Cheney-Kincaid).

Inverse interpolation and Neville $ algorithm (P.142-144 of Cheney-Kincaid).



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A different kind of approximation When approximating a function $f$ using a Taylor polynomial, we use information about $f$ and its derivative at one point. An alternative approach (called interpolation) uses information about $f$ at several different points. Suppose we wish to approximate $f(x)=\sin x$ on the interval $[0, \pi]$. a. Write the (quadratic) Taylor polynomial $p_{2}$ for $f$ centered at $\pi / 2$. b. Now consider a quadratic interpolating polynomial $q(x)=a x^{2}+b x+c .$ The coefficients $a, b,$ and $c$ are chosen such that the following conditions are satisfied: $$q(0)=f(0), q\left(\frac{\pi}{2}\right)=f\left(\frac{\pi}{2}\right), \text { and } q(\pi)=f(\pi)$$ Show that $q(x)=-\frac{4}{\pi^{2}} x^{2}+\frac{4}{\pi} x$ c. Graph $f, p_{2},$ and $q$ on $[0, \pi]$ d. Find the error in approximating $f(x)=\sin x$ at the points $\frac{\pi}{4}$ $\frac{\pi}{2}, \frac{3 \pi}{4},$ and $\pi$ using $p_{2}$ and $q$. e. Which function, $p_{2}$ or $q$, is a better approximation to $f$ on $[0, \pi] ?$ Explain.


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