A model emission spectrum

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Computational Physics · Scientific programming for physicists

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

Problem

A model emission spectrum, with photon energy

E

in electronvolts and intensity in arbitrary units, is a smooth continuum plus one narrow Gaussian line of unit area,

f(E)=e^{-E/10}+\frac{1}{\sigma\sqrt{2\pi}}\exp\!\left[-\frac{(E-E_0)^2}{2\sigma^2}\right],

with

E_0=30

and

\sigma=0.001

. A physicist integrates

f

over

[0,50]

with a standard adaptive quadrature routine (21-point Gauss-Kronrod) at its default tolerance. It returns

I=9.932\,621

with an estimated absolute error of

1.1\times10^{-13}

, after

21

function evaluations and no warning. They report: 'the estimated error is fourteen orders of magnitude smaller than the answer, so the line contributes negligibly to the total emission.' ((a)) Find the error in this claim, and explain from the routine's sample points why it occurred. ((b)) Give the correct value of

I

from the closed forms of the two terms, and describe a check with breakpoints bracketing

E_0

that confirms it.

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