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Find the interpolating polynomial of degree $3$ that interpolates $f(x) = x^3$ at the nodes $x_0=0, x_1 = 1, x_2=2, x_3 = 3$.

Here are my workings below

The basic Lagrange polynomials are:

$$L_0(x) = \frac{(x-1)(x-2)(x-3)}{(0-1)(0-2)(0-3)}$$

$$L_1(x) = \frac{(x-0)(x-2)(x-3)}{(1-0)(1-2)(1-3)}$$

$$L_2(x) = \frac{(x-0)(x-1)(x-3)}{(2-0)(2-1)(2-3)}$$

$$L_3(x) = \frac{(x-0)(x-1)(x-2)}{(3-0)(3-1)(3-2)}$$

Then the interpolating polynomial is:

$$P(x) = L_0(x)+(1)^3L_1(x)+(2)^3L_2(x)+(3)^3L_3(x)$$

Am I allowed to find the interpolating polynomial using these basic lagrange polynomials? and is my $P(x)$ correct? I wasn't sure if the first term should be $L_0(x)$?

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  • $\begingroup$ You would get $$P(x) = 0+ 0.5 (x-3.) (x-2.) (x+0.)-4. (x-3.) (x-1.) (x+0.)+4.5 (x-2.) (x-1.) (x+0.) = x^3$$ $\endgroup$ – Moo Oct 30 '18 at 16:20
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    $\begingroup$ You're allowed to do Lagrange interpolation, and you're probably allowed to do all sorts of complicated things, but you'd be better off if you just observe that the given $f(x)=x^3$ is already a polynomial of degree 3. $\endgroup$ – Andreas Blass Oct 30 '18 at 16:33
  • $\begingroup$ sorry I made a mistake, you are correct $\endgroup$ – mt12345 Oct 30 '18 at 16:58
  • $\begingroup$ @Moo sure, if you could add what error you can expect when using this interpolating polynomial to approximae $f(x) = x^3$ at $x=\frac{1}{2}$ that would be great $\endgroup$ – mt12345 Oct 30 '18 at 21:07
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    $\begingroup$ It is already there. Regards $\endgroup$ – Moo Oct 30 '18 at 21:42
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Your calculations are correct.

The final result is

$P_3(x) = 0.0 + 0.5 (x-3) (x-2) (x+0)-4 (x-3) (x-1) (x+0)+4.5 (x-2) (x-1) (x+0) = x^3$

The formula for the error bound is given by:

$$E_n(x) = {f^{(n+1)}(\xi(x)) \over (n+1)!} \times (x-x_0)(x-x_1)...(x-x_n)$$

So, we have

$$E_3(x) = {f^{(4)}(\xi(x)) \over 4!} \times (x-0)(x-1)(x-2)(x-3)$$

The fourth derivative of $f(x) = x^3$ is zero, so $E_3(x) = 0$.

The reason for this is if $f(x) = $ polynomial of degree $M$ where $M \le N$, then $$f^{(n)}(x) = 0 \implies E_n(x) = 0 ~\forall~ x$$

Therefore $P_3(x)$ is an exact representation of $f(x) = x^3$.

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  • $\begingroup$ one question, did you use my lagrange calculations above to find $p_3(x)$ or did you use another formula? $\endgroup$ – mt12345 Oct 30 '18 at 21:44
  • $\begingroup$ I used the same approach, but used code I wrote to get my result, but yours is the same (I verified it). There are other approaches too. You can check your result using Wolfram Alpha also. $\endgroup$ – Moo Oct 30 '18 at 21:46
  • $\begingroup$ okay I didn't know if I could use lagrange to get that polynomial or not $\endgroup$ – mt12345 Oct 30 '18 at 21:51

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