Numerical Analysis | Part IB, 2003


f(0)a0f(1)+a1f(0)+a2f(1)=μ(f)f^{\prime \prime}(0) \approx a_{0} f(-1)+a_{1} f(0)+a_{2} f(1)=\mu(f)

be an approximation of the second derivative which is exact for fP2f \in \mathcal{P}_{2}, the set of polynomials of degree 2\leq 2, and let

e(f)=f(0)μ(f)e(f)=f^{\prime \prime}(0)-\mu(f)

be its error.

(a) Determine the coefficients a0,a1,a2a_{0}, a_{1}, a_{2}.

(b) Using the Peano kernel theorem prove that, for fC3[1,1]f \in C^{3}[-1,1], the set of threetimes continuously differentiable functions, the error satisfies the inequality

e(f)13maxx[1,1]f(x).|e(f)| \leq \frac{1}{3} \max _{x \in[-1,1]}\left|f^{\prime \prime \prime}(x)\right| .

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