Simpson’s rule for monomial degree > 4

Question

Does there exist a monic $x^n$, n > 4, for which Simpson’s rule is exact? If not, why?

$$ S(f) = \frac{b-a}{6}f(a) + \frac{2(b-a)}{3}f(\frac{a+b}{2})+ \frac{b-a}{6}f(b)$$

Answer 1

If you take, for sake of simplicity, $a=0, b=1$ (and $f(x)=x^n$) you would require that $$ \frac 16 f(0) + \frac 23 f(1/2) + \frac 16 f(1) = \frac{1}{n+1} $$

i.e.

$$ \frac 23 (1/2)^n+\frac 16 = \frac{1}{n+1}\Leftrightarrow \frac{1}{2^{n-2}}=\dfrac{5-n}{n+1} $$

However, if $n>4$ this last equality is impossible.

Answer 2

Simpson's rule approximates a function, f(x), when integrating $\int_a^b f(x)dx$, by dividing the interval from a to b an odd number of subintervals, then taking the quadratic that passes through each set of three points. If f is quadratic function the Simpson's rule would approximate it by itself so give an exact answer.

If f(x) is a function of "quadratic type", say $ax^4+ bx^2+ c$, where we can write $y= x^2$ to become $ay^2+ by+ C$, that will Simpson's rule would give an exact value for that.

As for monic polynomials, $x^8$ or $x^{12}$ will work.