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Welcome back.

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And now we will continue our comparison of the results given by Oyler methods compared to the new commands

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IVP that is provided by Python or more precisely by Typekit.

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So here we will go now to the pendulum and we will consider straight away the most difficult case of

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the pendulum for the last one that we had considered previously, which was this jammed and driven pendulum

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which gave us this results.

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And you can see I just copy the code from this cell and pasted it at the back or at the bottom of our

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notebook.

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We're on it.

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And here we get this result for theta over tea.

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Now, for the new result, we of course, have to once again consider this syntax very similar to this

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one.

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So let me copy this.

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These two cells, this one as well.

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And let's change them according to our needs.

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So our differential equation will be pretty similar to this one.

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So we could just copy this here.

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But of course, we have to make sure that we make this one correct.

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So I write to Peter and then I write C to see the one like because it there's no list with two elements

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and this one will be the zero element of SEATO.

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And this one we can leave as it is.

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And then like previously for the free fall, we have to provide also the other components.

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So I write.

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Peter, one, come.

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So this is really exactly the same way we changed the ordinary differential equation for the fall.

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But now we have just adapted to the other differential equation of the pendulum.

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So really nothing new here.

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Then we need to starting values to zero zero, which was two and three to zero one zero, which was

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zero.

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And now we just have to change you to values to fit them to our previous calculations.

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So if it's zero zero and ft one zero and.

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You know what?

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What should we use here, maybe?

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I think we did it like this.

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Let's compare.

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Yeah, we are 200 times steps at a total time of 100.

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So that's exactly what we have now here.

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And I think this should work already.

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Now we can just go ahead and change this one to theto and run it.

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So it looks pretty confusing at first, but we can just wait here not to connect the dots.

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And you see, now it looks much better.

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And I will write here that this is supposed to be read and I want to compare it to the previous solution,

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which was this one.

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So we can just copy this because these solution variables are still, you know, haven't been deleted,

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they're still valid, so we can still use them, of course.

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And so we can compare the different results.

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And so the new results are indirect ones.

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The old ones are the blue ones.

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And you see in general, it looks pretty similar, but in details they will differ a bit.

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And so we have discussed this earlier during a quarter, 45 methods can be pretty much different compared

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to the old methods, not only in terms of efficiency, where it is much more efficient and much faster,

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but also in terms of accuracy.

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And you see this here, for example.

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And now for the remaining part, I just want to show you that we can compare different methods and I

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just want to show you that they differ a tiny bit and just want to introduce to you to different methods.

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So here's what I found on this website where the sniper module integrates thoughts of undescribed is

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explained.

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There it is written.

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That's the optional argument.

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Methods can have the following values we have used so far our K five, but there are also other methods

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and we will explore them now in the following.

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So the good thing is, you're really that we can just use the same syntax as before by writing solution

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underscore R.K. 45 and then the integrates of IVP with all the arguments and then the methods.

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And the only thing that we have to change now is this one and this one, and we will go through all

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these methods.

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So I've prepared this already.

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So let me just copy this so that we don't lose too much time.

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If you want, you can type along and basically just have to copy this five times enough to change these

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names and these names.

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So it isn't that much of an effort.

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And then we can go ahead and there's a problem here.

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Oh, this is because I didn't define the Tara.

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Tara is basically this one.

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So I write to pray like this.

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Yeah, now it works.

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And now we want to plotz and for the plots.

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I will.

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Yeah.

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How should we do it?

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I would say we will compare all these solutions in comparison to the R.K. 45 methods.

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So for example, for this solution here of our old methods, I would write solution and then or maybe

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let me do it like this.

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Sorry, I used the archive 45, and then I will just subtract here the solution

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from our old method.

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And then you see, this is the difference between the two curves.

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I'm not quite sure if this is correct now, but it should be.

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Yeah.

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So it looks a bit maybe exaggerated at first, but it is really a large, large difference at points.

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This is because here, for example, it looks as the difference is pretty small because it only appears

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to be shifted.

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But for example, this value here corresponds to some value here, for example.

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So the difference is really larger than two.

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So it's really a gigantic difference.

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And now, of course, we can continue and compare this also to the other functions.

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And this I have prepared as well.

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So basically using the same syntax, we used a solution of the R.K. 45 and subtracted solutions of all

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the other methods here.

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Going run this.

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We get a curve like this where you can see basically only the black curve has a very large differences,

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and then the others have pretty small differences.

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For example, the blue one seems to have only know of almost no difference.

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So this is OK.

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23.

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So it means that our whole our method that we have programmed ourselves to are the method seems to be

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the first method of all because it is the only one that really is different to all the other methods.

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So, yeah, bad news.

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Our our own method is the worst, unfortunately, but this was pretty clear and was expected because

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it's also the most simple one.

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So in the following, I will just copy these all of these actually, and I will just change the plot

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range a bit.

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So Peel T Dot X.

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And we could go from zero to 22 of this range here.

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And then for the Y range we go from, yeah, we have to zoom in here a bit, maybe minus zero point

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one to zero point one.

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And also, we know that the black curve is gigantic.

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So let's just look plotted here.

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Yeah, so that's it.

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So now we can pretty much see how the different methods compare to each other.

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So overall, I can't really see what's the correct method to use, and probably the correct solution

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is even difference to these that are suggested here.

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But the point is that all of these methods are pretty close to each other.

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And the only difference you see is that the R.K. 23 is pretty close to the arcade.

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Forty five, it's has the slowest deviation.

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So that's of course, because they are they belong to the same family of methods.

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So of course, the results will be pretty similar than we have here.

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The green and the red one, which art is the top eight, five, three and around all methods, and they

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are pretty similar.

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And then we have the purple and the yellow one or orange, which are the PDAF and also the which are

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themselves pretty similar, but they have the largest deviation from the R.K. methods.

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So as I said, it doesn't mean that one of these methods is wrong and the other one is correct.

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It just means that you see that overall, they give pretty similar results and we can check this, by

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the way, by just.

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Plotting this one more time and just plotting here like this so that we don't show difference, but

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the actual value and you will see that they are very much the same.

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So you see there is almost no difference that can be can be seen here.

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Especially important is, of course, the ending.

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So we could zoom in here a bit more at the ending parts of 90 to 100 and in the range of the law.

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Like this?

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Maybe.

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And still, you cannot really see that much of a difference.

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So overall, it doesn't matter so much which of the methods you use.

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But if you want to have the correct result, you should probably use one of these methods that belonged

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to the of IBP module, and you should probably not use to oil the method that we have programmed ourselves.

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Still, I think it was revolting and useful that we have programmed the oil method ourself because only

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in this way we know how to solve such problems and how to program such algorithms and to really understand

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what's happening in the background.

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And also, we have seen to some extent, especially for the analytical examples like in the beginning

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here, for example, this one was at a time of amplified decay.

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We could really restore the analytical solution, and the old method wasn't so bad, after all.