A postdoc evaluates the damped oscillatory integral K= _0^ 20 e^ -x/5 (50x) ,dx with…

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Scientific Computing for Physicists, Ch The numerical toolkit closed: backward error and the audit of a library routine

Problem

A postdoc evaluates the damped oscillatory integral

K=\int_0^{20} e^{-x/5}\sin(50x)\,dx

with a standard adaptive quadrature routine at its default tolerance,

1.49\times10^{-8}

. It returns

K=0.019\,860

with an estimated absolute error of

0.17

and a warning that the maximum number of subdivisions (

50

) has been reached. To audit it, they tighten the tolerance to

10^{-12}

, get the same digits, the same error estimate and the same warning back, and conclude: 'the same answer under a tolerance more than ten thousand times tighter settles it;

K=0.019\,860

.' ((a)) Find the error in this conclusion. ((b)) Name an independent check that would complete the audit of

K

, and state what it shows.

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