Interpolating a function with a limited number of division operations

Question

Let $f(x)=-x^5$

Approximate the value of $f$ in $x=1$, interpolating with polynomial of second degree with nodes: $x_0=-2,\ x_1=0,\ x_2=2$.

Then approximate it with another polynominal, interpolating with nodes: $x_0=-2,\ x_1=0,\ x_2=2,\ x_3=-3,\ x_4=3,\ x_5=-4,\ x_6=4$

Calculate the error in both cases.

Of course this exercise is pretty trivial, but there is an additional condition: I can use division only 4 times doing it. Additionally I am supposed to use either Lagrange's or Newton's method of interpolation.

Can you please tell me what trick can I use to reduce the number of divisions?

Answer 1

Computing Newton's form of the interpolating polynomial of degree $2$ requires $3$ divisions to get the necessary divided differences. You do not need any divisions to evaluate the Newton form anywhere.

In the second case, there is a unique polynomial of degree at most 6 which passes through the 7 points $(x_i,f(x_i))$ that are given. While $f(x) = x^5$ is a polynomial of degree 5 it is also true that it is a polynomial of degree at most 6 and that it passes through the 7 points. We can conclude that it is identical to the requested interpolating polynomial. Therefore there is nothing to compute and no divisions are needed.