Searching for some help with the following questions:
Given the $3$ numbers $0.1329, 1.543, 23.21$,
$1$ ) Add the $3$ numbers in both ascending and descending order rounding all calculations to $4$ digits.
My workings: I am a little confused with the ascending and descending order, wouldn't these two calculations be equivalent?
Ascending $0.1329+1.543+23.21 = 24.89$
Descending $23.21 + 1.543+ 0.1329 = 24.89$
$2$) Compute the relative errors:
What relative errors do i need to calculate? if i was to calculate the erors for each number individiually wouldnt they be $0$ since they lost no accucary when rounded to 4 digits.
$3$) Which is more accurate and why?
Best Answer
Your second sum is wrong (you forgot to round to $4$ digits): $23.21 + 1.543 = 24.753 \rightarrow 24.75$ and $24.75 + 0.1329 = 24.8829 \rightarrow 24.88.$
This intermediate round is the reason why the order of summation may give different results.
The true sum is $s=24.8859.$ Compute the relative error: Ascending $(s-24.89)/s = -0.00016475$ and decending $(s-24.88)/s = 0.000237082.$
Therefore the ascending sum is more accurate.