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So welcome back.

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This is the solution of the exercise, we want to calculate the velocity and the acceleration of some

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object.

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So you see, I have just loaded the same file.

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You also have the template and you see here's again the plot and the task.

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And I just started to type along with my solution.

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And here you see the first solution where I program the forward differences methods.

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And first of all, I think it's very important to understand how we can access the individual columns.

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So this means the time points T and D coordinates X, and we can do this by writing data, all the points

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and then zero full time and one for coordinate.

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So you see here, for example, the time goes from zero to 100 in steps of zero point one.

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And then I already gave you the hint that we cannot specify here any step with age because this is already

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determined by the data.

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So the step with will always be the step size from one point to another.

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So in this point, since it in this case, since it's equidistant, it will be always zero point one.

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And this means if we want to program this whole thing by a function, we only have a single inputs,

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which will be our data file.

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We do not need each anymore.

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OK.

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And just another hint I programmed this using a function which I called forward for for differences

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methods.

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But you don't have to do this.

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You could also just do it with a loop, with a while loop, for example, and you don't need to program

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this with a function.

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So as long as in the end, you get the correct result.

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Everything is good.

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So this is just my proposal of a solution.

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OK, so the first task was to calculate VE and DNA using for what differences methods?

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Here's how I did it.

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So you see, I returned here again, an array which consists out of data all points zero, which are

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two points of time.

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And then here I do not return to coordinates, but instead the derivative, and I start by defining

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this variable as an array, which consists only of zeros.

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So this is basically like the initialization of the array with a zero values, and it will have the

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length, which is the same as this one.

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And also of this one.

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So this is what's happened here and now we can use different ways on how to calculate this.

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So since we have accurate lists and data, we can also do it like this.

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We could say first with the final step with age, which is always the same.

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So for example, we can say calculate the difference of this element and this element.

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So it's zero point one, and then the velocity will basically be take all the points, except the last

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one.

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This is how you can write it.

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So it means start from the very first entry and then go until minus one, which is actually the last

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one.

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But the way the Python syntax works is that this point here is actually excluded.

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So it means all the points except for the last one.

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And you calculate this by taking the data file or the data entry at the same points and then you subtract.

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So here I see actually, this is incorrect, which I didn't notice because I commented, So actually,

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it has to be the other way around.

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So it means we have to calculate here the difference of the whole dataset for the next point.

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So this means basically take these indices and shift them by one.

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So this means take all the indices except for the first one, because here zero is not included.

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And then subtract the previous data point always and divide it by the step size.

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So this works for equidistant data.

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But if you want to have a general where the data is not equidistant, so this means here you have and

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or a list of points that, for example, would be zero or 0.2 0.3, then you couldn't do it like this.

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You explicitly put here everywhere.

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The step size, which can be variable.

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OK.

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And you see here, it's the same thing as in this column.

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It's just a different index.

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So here is to coordinate index.

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Here's the time index, OK?

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And then the only problem is that we cannot easily do this for the very last data point, because then

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you cannot do format differences, methods.

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So you could say now, OK, I just leave it at zero.

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I don't care about it.

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You could say I just set it to the second last value because it's probably very similar.

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Or you could say, OK, for this last point, I used it backwards differences, methods.

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So this is what I did here.

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But don't worry, I did not specify that you should handle the last data point in a specific way.

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So if you didn't do it, it's no big deal.

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It's just a single data point.

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It's different then.

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OK, so now we have a function forward of data, and now we can use it to our dataset to calculate the

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velocity forwards.

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So this will now be a dataset which has the same shape as this one.

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But here we will not have to coordinate the velocity.

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And then we can actually apply this function twice.

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And so this means we get the second derivative.

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And so we have now two two data sets velocity forward acceleration.

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And then we can do the same thing using the central difference as methods and the Richardson methods,

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which I think I don't have to explain here.

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Very much so you see here, for example, for the central difference is methods.

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We take all the points except for the first and the last.

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And then we shift here to indices upwards and downwards and tier the same upwards downwards.

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And here we take the coordinate in next time index.

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And so this calculates the derivative and then for the first and for the last value, we cannot use

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central differences.

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So I decided to use for differences for the first value and backwards differences for the last value.

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But to be honest, you could just say these are zero or you could take the previous value.

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It would all just be fine.

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OK, and now you can.

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We can use this new function and again apply it to the dataset and get a new dataset to Velocity Central

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Acceleration Central and for the Richardson method.

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It's, of course, also very similar.

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Here we have to shift by to an.

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Sees so this means we take all the values, except for the first two and the last two, so this means

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we shift into the positive direction by two, we shift by one, we shift by minus one and we shift by

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minus two, and then we divide by 12 times the step size.

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And you see that this is pretty much the same way we have programmed this previously.

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So you see there was this equation where we had to shift by to age by one age minus one age minus two

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age and the factors minus one eight minus eight one.

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So if you don't remember, just have a look at our previous notebook.

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It's really the same thing here.

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OK, and once again, we can now calculate our new data sets velocity Richardson, acceleration Richardson.

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And finally, we can now plot all of these things.

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So you see here I have two plots.

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The first one is velocity versus time, and I plot here three curves on top of each other.

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And I plot, of course, velocity, forward velocity, central velocity, Richards and then the same

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thing with the acceleration.

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And this is what we get.

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You see, first of all that all the points I hear really on top of each other.

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So especially for the velocity you see we have here a yellow curve, a green curve and a blue curve.

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And they are all on top of each other, which means on this zoom, on this magnification, we do not

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really see a difference of the methods.

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And also we see that in the beginning, the velocity is positive, but it decreases to zero.

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Almost four, I would say, even below zero.

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And then it oscillates, and in the end it will be positive and it has a pretty large value.

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And this makes sense if you look at the data file.

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First of all, the velocity is positive, but then decreases.

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So this means here the car is maybe even going a tiny bit backwards.

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And then in the end, it will increase because the velocity is positive.

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And here we have some backwards and forwards motion because the velocity is positive and also the negative

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at points.

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So it makes sense for the acceleration.

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We take the derivative of this curve.

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So you see in the beginning, the velocity decreases, which means the acceleration is negative.

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But then at a later point of time here, the acceleration has to turn positive.

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So there is a zero here.

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And you see forward acceleration.

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We have the same phenomenon that all the three curves are on top of each other, except for the last

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two or three points.

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All right.

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Yeah, that's two points, actually.

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So you see it in here.

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We have a blue point, a green point in the yellow point.

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And this is because the acceleration calculates the second order time derivative.

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So the problem is here that we did not calculate the first and the last value in the proper way.

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We had to cheat here a bit because our method was not working because we didn't know the next value

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that would come after the last value.

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So the problem is, now that this value and this value are actually not really accurate.

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And if we continue to calculate the derivative, then it gets even worse from the last value and also

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for the second last value.

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This is where these differences come from.

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So I decided to rebuild everything and to leave out the two first values and the two last values for

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the velocity and the full first and for last value, for the exploration.

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And then you see everything fits perfectly well, maybe not perfectly.

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So now is the end of your first task if you have accomplished this really, really nice.

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And if not, please have a look again at my solution.

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I hope this helps you to find your ranch or your mistake.

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And now I want to show you a bit of an additional analysis of the error.

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So here I can tell you now how I have generated the data set to offload it.

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This was actually here's the function that they used.

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So a cosine function power two to five, power two to four and power to one, and some coefficients

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which you see here.

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And I have programmed this, actually, I have changed this one here later on.

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So the changes?

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OK.

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That's correct.

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And this is the function that I have programs and you see, it gives us the same dataset that we have

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loaded previously starts from approximately one goes to almost 16.

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This was also what we had here.

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And in fact, I have here in the data set, and this is actually how I export it, the data set that

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you have downloaded and if you look at the function this one, we can actually calculate the derivative

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and the second derivative analytically.

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So here we will get a minus sign and a derivative will be a he would get five times b t to the power

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of four four times C over three and D.

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And this is what I've written down here.

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And then what we derive again, we get minus cosine.

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Another in a derivative.

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So we have a squared.

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Here we have five times for beta two to power three four times, three times c t to the power of two

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and then zero for this term.

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So again, I have programmed these two functions and we have plotted them.

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So this is here the analytical solution.

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So if you remember, it looks a bit similar to what we had previously.

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So in fact, it should look exactly the same, and I hope it does.

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But we can figure this out by just taking this data set.

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So T list a list and T list and B list and we subtract our numerically calculated values.

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For example, make this a write t list for the x axis and for the y axis I plot velocity forward minus

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the analytical v list.

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So this gives us note the arrow.

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So we have you the arrow for velocity and for the acceleration.

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And you see this is what the plot gives us.

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So the first curve, the blue one, which is the forward difference, is methods will have the largest

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arrow.

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It will be 0.0 two and four the acceleration and will be zero point zero zero one.

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And then the other two curves almost have no arrow.

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So maybe we maybe let's look at the curves again.

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What means an arrow of zero point zero zero three?

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So you see on velocity and error of zero point zero zero three is really tiny.

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So this is why we cannot see it.

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Still, we figure out that the central difference is methods, and the Richardson methods are far more

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accurate.

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And if you look carefully, you see that the Richardson method is even better.

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So to further quantify this, I calculate here to total error, and I do this by calculating the the

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the total square of the error.

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So for every data point, I calculate the error I squirts and then I added up.

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And this is what I wrote here, and this is what we get for the values.

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So for the central, the story for what difference is methods.

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We have a total error of zero point zero zero two.

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Then we have the central differences methods approximately nine times 10 to the power of minus eight.

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And Richardson 10 to the power of minus 16.

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So you really see how this scales with the step size and how to our drastically reduces for the Richardson

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methods and also for these central differences methods.

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It's way smaller than for the forward differences methods.

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So this is what is meant by saying that there is an error of the order of each of each square and even

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higher.

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And in our previous example, in previous lectures where we knew the function, this wasn't really a

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big deal because we could always just decrease the step size.

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So even for the full what difference this methods, we could achieve a very accurate result by just

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decreasing the step, size, age.

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But here, since we work with a fixed data set, we cannot do this.

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So this is where really these Richardson method and also the central difference, this method where

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they shine compared to the other methods, because working with the same data file, you get a much

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higher precision.

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And then for the acceleration, we get a similar observation, wasteful error here.

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OK, and then you had the second task.

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Let me scroll up to the tasks again, determine the maximum value of the acceleration and the corresponding

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point of time.

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So we basically look at this graph here acceleration, and we see the maximum value is here almost 0.06

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meters per square second.

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And that point of time is seventy eight seventy nine, maybe.

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So let's see if we can figure this out using our methods, and we can do this by loading one of our

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of our data sets actually here A-roads velocity, but its acceleration.

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And this is our dataset, and now we can use several methods, of course, to do this.

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And one of the most easy ones is to, first of all, look at the maximum value of this column here.

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So we write maximum of exploration Richards and all the values and then the exploration column here

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one.

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And this gives us a value which I have plotted here.

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So this is zero point zero five eight eight nine.

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This corresponds to the value on the acceleration axis.

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So, for example, here this one.

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Yeah, so it really is, you see almost zero point zero six.

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So it's correct.

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And then we can write, find out the index.

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So basically, the number of the elements, the index where it is values reached.

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So we say please find and p dot max of this dataset acceleration Richardson, all the values and one.

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So now we have the index, which is seven hundred ninety six.

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So since we know our step sizes zero point one, we already know, okay, the time has done ninety seven

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point six seconds.

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But if we wouldn't know the step size and if it would be more homogenous than we could now, right?

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Just take the acceleration of Richardson.

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So this is our data sets here.

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And then for the first index, we used the index, of course, which we have just figured out.

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So take the the seven hundred ninety 796 and then take the first values.

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So take the time.

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And so the time value is ten seventy nine point six seconds.

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All right.

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So we have now solved both of these tasks.

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We have calculated the velocity and the acceleration using different methods, and we have figured out

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the maximum of the acceleration.

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And furthermore, I have also shown you a brief analysis of the error here, and I hope you believe

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me now that the Richardson method and also other methods are sometimes way better than the.

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What differences methods.

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And it is worth it to really use these advanced weapons methods from time to time, especially if you

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have to work with the fixed dataset where you cannot use the step size.

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So there it's really good and really essential to decrease the error and to improve the result.

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So I hope you took the time and you also tried to do this yourself, and I hope you had some success.

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If not, then I hope that now, since you have seen this solution, these differences or these problems

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that you had are now resolved and you can continue with the next lecture of the course.