The trapezium rule with four equal intervals ( ) applied to gives .

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Numerical and computational mathematics · Numerical and computational mathematics

Error, Chance and the Machine, Ch Integration: error expansions, Gauss rules and adaptivity

Problem

The trapezium rule with four equal intervals (h=0.25h=0.25) applied to ∫12dxx\int_1^2 \dfrac{\mathrm{d}x}{x} gives T4=0.69702381T_4 = 0.69702381. ((a)) Use the leading Euler-Maclaurin term h212[f′(b)−f′(a)]\dfrac{h^2}{12}\left[f'(b)-f'(a)\right] to predict the error T4−IT_4 - I. ((b)) Give the Euler-Maclaurin-corrected estimate of the integral. ((c)) Compare the corrected estimate with the exact value ln⁡2=0.69314718\ln 2 = 0.69314718 and comment on the size of the remaining error relative to the uncorrected error. ((d)) Now add the next term, −h4720[f′′′(b)−f′′′(a)]-\dfrac{h^4}{720}\left[f'''(b)-f'''(a)\right]. Give its value and the new estimate, and say how the remaining error compares with the size of that term.

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