# NavList:

## A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Wayfinding

**Re: Double Altitude - Sensitivity to Time Interval**

**From:**Bill Lionheart

**Date:**2016 Dec 5, 09:21 +0000

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 Reedwrote: > 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 > > View and reply to this message -- Professor of Applied Mathematics http://www.maths.manchester.ac.uk/bl