Here's what Matlab made of the puzzle. It was first necessary to
decide what meaning to apply to the uncertainty numbers associated with
the parameters, and what form of that uncertainty to select. I did not
presume to link the variability between the three parameters.
I arbitrarily chose the uncertainty to represent a rather high
confidence of capturing the variability so I specified the uncertainty
as two SDs of a normal distribution, and took a hundred samples of each
parameter. Matlab has a root extractor built in, and it provided results
as complex numbers, which plot as red 'x' symbols in a circle in the
complex plane.
These are the four lines I entered on the command line.
>> a=normrnd(1,0.00005,[1 100]);
>> b=normrnd(-2.08,0.005,[1 100]);
>> c=normrnd(1.08, 0.005,[1 100]);
>> plot(roots([a b c]),'xr');
This is the plot depiction as a jpg on a free image server.Forgive the ads.
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Here's a little puzzle with some seasonal relevance:
We need to find a good value for x
/and for the uncertainty associated with x/
given that:
a x2 + b x + c = 0 [1]
a = 1 ± .0001
b = -2.08 ± .01
c = 1.08 ± .01
This was mentioned in connection with the annual "sig figs"
donnybrook on the chemistry list. There are a thousand people
on that list, and so far nobody has come up with a solution.
One person came kinda close, but no cigar.
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