Simpson's rule is applied to a smooth integral with known exact value

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

Simpson's rule is applied to a smooth integral with known exact value, and the absolute errors measured are:N=8: 1.5×10−6,N=16: 9.4×10−8,N=32: 5.9×10−9,N=64: 3.7×10−10,N=128: 2.3×10−11,N=256: 1.5×10−12,N=512: 4.0×10−13,N=1024: 3.8×10−13.\begin{aligned}N=8:&\ 1.5\times10^{-6},\\N=16:&\ 9.4\times10^{-8},\\N=32:&\ 5.9\times10^{-9},\\N=64:&\ 3.7\times10^{-10},\\N=128:&\ 2.3\times10^{-11},\\N=256:&\ 1.5\times10^{-12},\\N=512:&\ 4.0\times10^{-13},\\N=1024:&\ 3.8\times10^{-13}.\end{aligned}Someone argues: "The ratios of successive errors are about 1616 up to N=256N=256, but then they fall to 3.753.75 and then to 1.051.05. Since Simpson's rule was derived as fourth order, the ratio should stay near 1616 forever; because it does not, the quadrature routine must contain a bug that appears only at large NN." Find the error in this reasoning, and state what is really happening in the table, including roughly where it happens and at what error level.

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