Past exam of the mathematics course of the University of Cambridge / 2016 / ib / Paper 3 / 19D (Numerical Analysis)

The Lagrange interpolation polynomial basis for nodes $0,2,3$ givesEach polynomial is one at its own node and zero at the other two. For a polynomial $f$ of degree at most two, the difference between $f$ and this interpolation polynomial has degree at most two and three distinct roots, so it is identically zero. This also proves uniqueness of the quadratic coefficients.

Evaluating the Lagrange interpolation polynomial coefficients at five, and their derivatives at one, givesBoth rules are exact for every polynomial of degree at most two. For $f(x)=x^3$, the first gives $95$, while the true value is $125$; the second gives $4$, while the true derivative is $3$. Thus the respective approximation errors, estimate minus true value, are $-30$ and $1$. There is no contradiction: a cubic lies outside the exactness class. The first formula extrapolates beyond the interpolation nodes, so its error can be substantial.

The Peano kernel theorem says that if a continuous linear error functional $L$ annihilates all polynomials of degree less than $r$ and permits the required interchange with integration, then for $f\in C^r[a,b]$,It follows by applying $L$ to the integral-remainder Taylor formula with integral remainder. Here choose $r=2$ andThis functional annihilates constants and linear functions (indeed quadratics too), and its Peano kernel isThus ${\int }_0^3|K|d\theta =1/2$ and $\max |K|=1/3$. The integral representation givesFor the first inequality, continuous functions $f^{\prime \prime }$ with $\Vert f^{\prime \prime }{\Vert }_{\infty }\le 1$ can approximate $\mathrm{sgn}K$ away from its zeros, making $L(f)$ arbitrarily close to $\int |K|$. Such derivatives integrate twice to admissible $C^2$ functions, proving no smaller $\alpha$ works. For the second, use continuous $f^{\prime \prime }$ of $L^1$ norm one concentrated near $\theta =1$ or $2$, where $|K|=1/3$, with the sign of $K$ there. This makes $|L(f)|$ approach $1/3$, proving optimality of $\beta$.

For $f=x^3$, $L(f)=2$, $\max |f^{\prime \prime }|=18$ and $\Vert f^{\prime \prime }{\Vert }_1=27$. Both right-hand bounds equal nine, so both inequalities hold.