$2\mathrm{.}7$ Exercises

$87$

Exercise $2\mathrm{.}19$: Explore round$-$off errors from a large number of inverse operations

Maybe you have tried to hit the square root key on a calculator multiple times and then squared the number again an equal number of times. These set of inverse mathematical operations should of course bring you back to the starting value for the computations, but this does not always happen. To avoid tedious pressing of calculator keys, we can let a computer automate the process. Here is an appropriate program:

from math import sqrt

for $n$ in range $(1,60)$ :

$r=2\mathrm{.}0$

for $i$ in range$($n$)$:

$r=\sqrt[\phantom{2}]{r}$

for $i$ in range$($n$)$:

$r=r****2$

print $\%$ d times sqrt and $****2$:$\%\mathrm{.}16f,\%(n,r)$

Explain with words what the program does. Then run the program. Round$-$off errors are here completely destroying the calculations when $n$ is large enough! Investigate the case when we come back to $1$ instead of $2$ by fixing an $n$ value where this happens and printing out $r$ in both for loops over i$.$ Can you now explain why we come back to $1$ and not $2?$

Filename: repeated$\_$sqrt$.$

Exercise $2\mathrm{.}20$: Explore what zero can be on a computer

Type in the following code and run it:

eps$=1\mathrm{.}0$

while $1\mathrm{.}0,1\mathrm{.}0+$ eps:

print $\text{'}..........\text{'},$ eps

eps $=ep\frac{s}{2\mathrm{.}0}$

print 'final eps:$\text{'},$ eps

Explain with words what the code is doing, line by line. Then examine the output. How can it be that the "equation" $11+$ eps is not true? Or in other words, that a number of approximately size ${10}^{-16}($the final eps value when the loop terminates$)$ gives the same result as if eps were zero?

Filename: machine$\_$zero.

Remarks The nonzero eps value computed above is called machine epsilon or machine zero and is an important parameter to know, especially when certain math$-$