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Re: [Phys-l] quadratic uncertainty



Ludwik,

you mentioned that you did not understand the Matlab formalism.
In this you are one with a sizable proportion of the educated classes.
I looked out this URL which shows how engineers are taught:

Complex Numbers: Numerical Experiment (Quadratic Roots)

http://cnx.org/content/m21413/latest/

http://tinyurl.com/2amm4qx

...which is all very well, but does not get to the heart of your question.
Still it brings up the fascinating story of how an undergraduate student
approaches the issue of quadratic polynomials now, and then, i.e.....
how Gauss dropped out of Gottingen at 21, but was promised a student grant
if he would gain a PhD. Which he did, by acclamation, a few months later.

http://www.schillerinstitute.org/educ/pedagogy/gauss_fund_bmd0402.html

http://tinyurl.com/2axequ4

Here is an exposition, an engaging exposition purveyed in the NY Times:

NYTimes Opinionator today, from S Strogatz Cornell on March 7th

http://opinionator.blogs.nytimes.com/2010/03/07/finding-your-roots/

http://tinyurl.com/y98mtak

Despite the straight-forward look of these polynomials, I see that Aaronson puts
them to advantage in describing amplitudes, of the quantum kind:

PHY771 Lec 9 -- Quantum S Aaronson Stanford

http://www.scottaaronson.com/democritus/lec9.html

http://tinyurl.com/ykpa5n

An Example of Aaronson's lecture Idea.

http://academic.reed.edu/chemistry/alan/Research/Bond/BasicQM/PrAmp/pramp.html

http://tinyurl.com/27jjtbv

Still, when all is said and done, I have to share that this is a class of puzzle calculated to mislead the unwary:
particularly those who would have us think there is a 'one size fits all' for specifying uncertainties
in experimental evaluations.

To make it quite explicit: it is not difficult to specify the coefficients of a quadratic with huge uncertainties,
which nevertheless produce roots with modest uncertainties. On the other hand, there are quadratics
whose coefficients are so carefully crafted that even quite tight uncertainties just will not do.
The looked for root(s), disappear in a cloud of complexity, and polarity, that in some physical cases is,
well, unphysical.

So an appropriate response to getting word of such a puzzle, is to say:
if this is a physical experiment that requires an output of a particular kind, say a real positive output,
this specification is wrong:the required uncertainties need to be specified within bounds needed
to realise the physicality in question.
Here, real and positive(1): there, real and negative or positive(2). Elsewhere always complex (3).
Even rarely (I suppose) sometimes real and sometimes complex(4) - but rarely.....

So to follow on the four thousand year tradition of examining roots of quadratics, I will put a
little effort into specifying bounds on the coefficients which give the cited quadratic physical meaning
in cases 1,2,3,4 if they exist, as time permits.

Brian W


On 8/28/2010 11:43 PM, ludwik kowalski wrote:
Dear Brian,

I do not understand the format of your display.

Prompted by you, I wrote a standard Monte Carlo code. I also assumed that the uncertainties represent two standard deviations.

I assumed that x must be a real number. But I counted how often the calculated x was a complex number.

Here are my results, for 10000 cases


1) when I use + to calculate x

complex solutions occurred in 41.59% of cases
mean x was 1.1156, stand. deviation was 0.9420


2) when I use - to calculate x
complex solutions occurred in 40.92% of cases
mean x was 0.9670, stand. deviation was 0.8053

Suppose I assume that x represent the mass. In this case real but negative x must also be ignored. I did not count them but this can easily be done. (The percentage additional rejections of would be large, as one can guess by looking at standard deviations.

It is too late; good night.

Ludwik

===========================================

On Aug 28, 2010, at 6:05 PM, brian whatcott wrote:

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.

<http://i880.photobucket.com/albums/ac6/betwys/quadratic/quadroots.jpg>

Surprising...
Brian W

$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$
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.
$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$

_

Ludwik