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I agree with Frank that practically just making small differences and
following through the calculation, especially if it is automated is a
safe, easy and intuitive method for testing the errors. Indeed its
almost worth saying that any calculation you are going to rely on in
navigation you should always be asking "how different would that be if
I was slightly wrong", and that is especially easy when you have
software to do it.

However the advantage of doing calculus is that you can see how the
errors would vary over the full range of input values, for example if
a navigational method becomes much less accurate in certain
conditions. Also what you want for errors IS differences, but the
accuracy of approximation the derivative gives to the difference is
the error term in Taylor's theorem, so you can test if the derivative
gives an accurate enough bound on the error.

As for differentiating complicated formulae with lots of trig
functions in, if you dont trust your own calculus skills (and you are
close enough to land to have an internet connection) whip out your
smart phone and check it using Wolfram Alpha
https://www.wolframalpha.com/ (as many of my students do these days).
You can input the formula in a fairly free syntax and it is very
forgiving. Just remember that for navigation we use degrees while
calculus uses radians

For example you can type in d(tan (pi x/180 ) sin  (pi x/180 )) /dx to
find the derivative of tan x sin x for x in degrees

Hope this of help to some of you out there. Perhaps we should do a
worked example?

Bill

On 5 December 2016 at 08:13, Frank Reed  wrote:
> David C, you wrote:
> "It is many many years since I studied differential calculus so I doubt I
> could work out d lat/d t for the above formula."
>
> So don't differentiate! Simulate the small changes instead. A good
> simulation is almost always going to be more useful than a differential
> calculation (though a differential calculation will put you on the right
> track much more efficiently than "blind" simulation). There are many tools
> you can use to simulate the variability, but I would suggest starting with
> the USNO celestial navigation web tool:
> http://aa.usno.navy.mil/data/docs/celnavtable.php. It's old and it's
> old-fashioned, but it's all safe and reliable. Start with any initial
> conditions you like. Then to test "differential sensitivity", just bump
> those conditions up or down in the appropriate variable. You can get a
> handle on these things very quickly. How much does one second matter? Just
> try it out! This isn't cheating. Though differential analysis by proper
> calculus is important, in the real world this sort of thing is mostly done
> by what's technically called "finite differencing" today. Really that just
> means simulate it for slightly different input conditions.
>
> Frank Reed
>
> 



--
Professor of Applied Mathematics
http://www.maths.manchester.ac.uk/bl

   

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