A composite Simpson's rule routine estimates .

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Computational Physics · Research software practice, verification and validation

Scientific Computing for Physicists, Ch Is the equation solved right? Testing, debugging and code verification

Problem

A composite Simpson's rule routine estimates

\int_0^{\pi/2}\cos x\,\mathrm{d}x = 1

. ((a)) Using step

h = \pi/64

, the Simpson error bound is

(b-a)h^4\max|f^{(4)}|/180

with

b-a = \pi/2

and, since

f(x)=\cos x

gives

f^{(4)}(x)=\cos x

\max|f^{(4)}|=1

. Compute the tolerance this gives, and say whether a unit test comparing the routine's result with

1

passes it, given that the measured error at this step is

3.2\times10^{-8}

. ((b)) The composite rule uses

33

function evaluations over

[0,\pi/2]

at this step. A property test checks linearity, that Simpson applied to

f+g

, with

f=\cos x

and

g=e^x

, equals the sum of Simpson applied to

f

and to

g

separately, to relative tolerance

10M\epsilon_{\rm mach}/2

with

M=33

evaluations and

\epsilon_{\rm mach}=2^{-52}

. Compute the tolerance, and say whether a measured relative difference of

3.7\times10^{-16}

passes it. ((c)) An order-of-accuracy test gives maximum errors

e(\pi/16) = 8.30\times10^{-6}

e(\pi/32) = 5.17\times10^{-7}

and

e(\pi/64) = 3.23\times10^{-8}

. Compute the observed order

\hat{p}

from each consecutive pair using

\hat{p} = \ln[e(h)/e(h/2)]/\ln 2

, and say whether the code is verified against Simpson's formal order

p_{\rm formal}=4

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