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Buckethead

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- #1

Buckethead

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- #2

fresh_42

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Have a look at the graphs at the end of the page: http://www.wolframalpha.com/input/?i=Taylor+f(x)=ln(x^3+1)

- #3

anorlunda

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Order can refer to order of magnitude. Therefore first order effects give the right answer to within 10%, second order to within 1% and so on.

- #4

Buckethead

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Is it just guesswork as to when a solution is sufficient in the number of orders used? For example I saw a discussion where an effect was not seen at first order but was seen at second order. So in general one should solve to as high an order as practical until you get diminishing returns?

I also seen things such as "an exact solution could not be found" and I imagine this just refers to the order at which the equation was solved and does not refer to a failure at being able to solve the equation?

- #5

fresh_42

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Not quite. It is driven e.g. by the accuracy of measurements. It makes no sense to compute ##10## digits if you can only measure ##2##. Or it is given by the purpose. Earth's surface is curved, nevertheless we get along well with flat street maps. The error is just too small. But you better hope your pilot doesn't use flat directions on a trans-Atlantic flight.Is it just guesswork as to when a solution is sufficient in the number of orders used?

Ideally yes, but first order approximations are linear approximations, i.e. tangents. Those are far easier to calculate and really often sufficiently close at small distances, cp. the examples above. They at least carry the tendency.For example I saw a discussion where an effect was not seen at first order but was seen at second order. So in general one should solve to as high an order as practical until you get diminishing returns?

No, that's a computational remark. E.g. solve ##x^5+c_1x^4+c_2x^3+c_3x^2+c_4x+c_5=0## can usually not be done via formulas, but only by numerical and thus approximation procedures. And there are a lot more problems, where we don't have closed forms for and only numerical approaches. The reason is that natural processes are often far more complicated than could be described by simple equations, so that algorithms will be necessary - and the computer has no number for ##\pi##, only an approximation.I also seen things such as "an exact solution could not be found" and I imagine this just refers to the order at which the equation was solved and does not refer to a failure at being able to solve the equation?

The series I mentioned above are exact, even if they are infinitely long. So in those cases we can work with them and make no mistake. But they are often not nice to handle, so that they are cut after some steps. In the example above, the error for ##x=0.1## in the second order approximation is at the tenth digit, and thus irrelevant for most applications.

- #6

Buckethead

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Great! Thank you so much for the detailed answer. Much appreciated.

- #7

sophiecentaur

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You have to remember that many of the mathematical models that are used to describe the Physical World do not involve simple polynomials. There are many models that involve trigonometrical and exponential functions etc etc.. When we talk about second (or other) order effects from these models, we are actually approximating them with a simple polynomial ( a quadratic or higher order).Can someone explain in lay terms, what first order and second order refer to?

This is something that people to take for granted but it can be confusing for someone who isn't familiar with the Taylor Expansion and other mathematical tricks. (Also, you need to bear in mind that it cannot always be done this way.)

- #8

Buckethead

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Thank you for that reminder. It is easy to forget that mathematical models of reality are just that and not the actual mechanisms by which reality operates and are therefore always going to be approximations. Excellent!You have to remember that many of the mathematical models that are used to describe the Physical World do not involve simple polynomials.

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