The trapezium rule is applied to (exact)

PHYSICS · Subjective · Fermi University Physics · Mathematical Methods for Physics · Numerical and computational mathematics · Numerical and computational mathematics

Computational Physics, Ch Differentiating and integrating numbers

Problem

The trapezium rule is applied to

\int_0^1 x^4\,\mathrm{d}x = 0.2

(exact), giving these errors:

\begin{aligned}N=4:&\ 0.020703,\\N=8:&\ 0.005200,\\N=16:&\ 0.001300,\\N=32:&\ 0.000325.\end{aligned}

Someone argues: "Compare

N=4

with

N=16

: the error falls from

0.020703

0.001300

, a ratio of

0.020703/0.001300=15.9\approx16

. The trapezium rule was derived as second order, but this ratio of

16

shows it is actually behaving as a fourth-order method here, so there is a bug in the code that is accidentally buying two extra orders of accuracy." Find the error in this reasoning, and state what the table actually shows about the order of the trapezium rule.

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