Editing Asymptotic analysis (section)

=== Asymptotic versus Numerical Analysis ===
De Bruijn illustrates the use of asymptotics in the following dialog between Dr. N.A., a Numerical Analyst, and Dr. A.A., an Asymptotic Analyst:
<blockquote>N.A.: I want to evaluate my function <math>f(x)</math> for large values of <math>x</math>, with a relative error of at most 1%.

A.A.: <math>f(x)=x^{-1}+\mathrm O(x^{-2}) \qquad (x\to\infty)</math>.

N.A.: I am sorry, I don't understand.

A.A.: <math>|f(x)-x^{-1}|<8x^{-2} \qquad (x>10^4).</math>

N.A.: But my value of <math>x</math> is only 100.

A.A.: Why did you not say so? My evaluations give<blockquote><math>|f(x)-x^{-1}|<57000x^{-2} \qquad (x>100).</math></blockquote>

N.A.: This is no news to me. I know already that <math>0<f(100)<1</math>.

A.A.: I can gain a little on some of my estimates. Now I find that<blockquote><math>|f(x)-x^{-1}|<20x^{-2} \qquad (x>100).</math></blockquote>

N.A.: I asked for 1%, not for 20%.

A.A.: It is almost the best thing I possibly can get. Why don't you take larger values of <math>x</math>?

N.A.: !!!  I think it's better to ask my electronic computing machine.

Machine:  f(100) = 0.01137 42259 34008 67153

A.A.: Haven't I told you so? My estimate of 20% was not far off from the 14% of the real error.

N.A.: !!! . . .  !

Some days later, Miss N.A. wants to know the value of f(1000), but her machine would take a month of computation to give the answer. She returns to her Asymptotic Colleague, and gets a fully satisfactory reply.<ref>{{Cite book |last=Bruijn |first=Nicolaas Govert de |title=Asymptotic methods in analysis |date=1981 |publisher=Dover publ |isbn=978-0-486-64221-5 |series=Dover books on advanced mathematics |location=New York |pages=19}}</ref></blockquote>