Four-point one-sided second-derivative formula

At unit spacing, exactness through cubic polynomials gives

f^{''} (- 1) \approx 2 f (- 1) - 5 f (0) + 4 f (1) - f (2) .

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- Sharp Peano-kernel constant for the four-point endpoint second derivative Four-point one-sided second-derivative formula

Sharp Peano-kernel constant for the four-point endpoint second derivative

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If the fourth-order Peano kernel of the four-point endpoint formula is nonnegative, its sharp sup-norm error constant is its integral. Evaluating the error on

(x + 1)^{4} /24

gives

c = \int_{- 1}^{2} K (t) d t = \frac{11}{12} .

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Past exam of the mathematics course of the University of Cambridge / 2024 / ib / Paper 3 / 17A / Solution

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