Second Degree Polynomial Interpolation, error related

Question

We want to create a table of the exponential integral function $$E_{1}(x)=\int_{x}^{\infty}\frac{e^{-t}}{t}dt, x>0$$ over the interval $x \in [1,10]$ with stepsize $h$. How large can $h$ be if a user of your table expects to obtain values at arbitrary x-locations with an absolute error $\leq 10^{-8}$ when using second degree polynomial interpolation?

I know that I will use the error formula for second degree polynomial interpolation and bound the terms.

I will need to compute $f^{'}(x), f^{''}(x),f^{'''}(x)$

BUT I'm honestly not even sure what I am trying to find here, some stepsize $h$? How does this relate to my equation?

Answer 1

Hints: I will map it out, please fill in the details.

The error formula for second degree polynomial interpolation is given by:

$$\tag 1 |P_2(x) - f(x)| \le \dfrac{|(x-x_0)(x-x_1)(x-x_2)|}{3!}~\mbox{max}_{a \le x \le b} |f^{(3)}(x)|$$

Since we are using three points, we can use equal spacing and take $x_0 = -h, x_1 = 0, x_2 = h$.

Now we need to do three things:

• Bound the term $|(x-x_0)(x-x_1)(x-x_2)|$ (in other words, find the max of a cubic in terms of $h$), and
• Find $\mbox{max}_{a \le x \le b} |f^{(3)}(x)| = |E_1^{(3)}(x)|$ (the third derivative under the integral of $E_1(x)$) over $a = 1, b = 10$.
• Using the previous two results in $(1)$ gives us a function in terms of $h$ and we set it $\le 10^{-8}$ and solve for $h$.

Aside: Here are some nice notes by Keith Conrad on differentiation under the integral sign, but it seems like you understand that.