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So in the previous lecture, we have explored the SAM method and did trapezoidal methods.

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And also, I mentioned that using the trapezoidal method gives us quite a good result, and it corresponds

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to integrating a Linnaeus line through two data points.

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So when we look here again, we have linear functions connecting to neighboring points and then we just

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integrate over these linear functions, which is quite easy.

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However, we could also interpolate these data points not with linear functions, but with polynomial

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functions.

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And this works especially well in our case because our function is in fact a polynomial.

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And this is this is done by using the so-called Simpson rule or also the Newton Courtes equations,

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which are general generalized versions of the Simpson rule.

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So they are even better, and this method corresponds to integrating a polynomial interpolation function

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through two data points.

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And you may wonder where all of these coefficients from this equation here come from, and you can really

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derive this analytically.

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And the coefficients have, of course, been optimized so that this formula, where we add up individual

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terms corresponds to having first interpolated the data with a third order polynomial and then integrating

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over this polynomial.

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So this is where all of these coefficients come from.

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So here we just take our individual data points X1 to X2 13 in this case, and we add them up with these

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coefficients.

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So it's a bit similar to this one, but even better, and you see basically here we would write one

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of a two f of x one and then one times f of x two, one times f of x three and so on.

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And then the last one will be one half.

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But here we have the same thing just different coefficients.

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One of a three four three two over three four three two over three and so on.

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And then the last one will again be one over three.

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So the idea behind it is quite similar.

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So here just one thing I have to mention you have to be careful because this method only works if you

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have an odd number of data points.

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And if you know the function itself, then this is absolutely no problem because you could always just

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say, OK, I just create my data points with an odd number.

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But if you are given the data, then it can be a bit tricky.

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However, luckily in our case, we have 13 data points, which is a lot numbers, so we can just go

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ahead and implement the function and check if it works or not.

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So I write define integral Simpson of data.

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And then I return.

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The individual, yeah, individual terms, I would say.

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So what I want to do is I want to start with the brackets here, so I just type along.

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One over three times data, one from a zero, which means y coordinate of the first data point.

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Then I will go ahead and take the second one.

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This would be for over three times data.

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One comma one.

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And then the next one.

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So don't worry, I will not Typekit of the data points because now we are a bit of a trouble because

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we in generally we don't know how many data points we have, so I cannot write down explicitly.

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So I have to say, OK, the second, the fourth, the sixth and so on data point, they will all be

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multiplied here.

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So what I can do is writing and p dot some over several data points and you can write here one two minus

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one in steps of two.

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And this means take the indices one, three, five, seven and so on.

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And this corresponds to taking the second, fourth, sixth eight data points.

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And here we can do the same thing.

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We just write two to minus one in steps of two.

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So this just excludes the very last data point because for this, um, we will once again have to take

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one of three.

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So, yeah, I forgot to write and peddled some.

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So let's just edit.

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Like this and then the rides plus the last one will be one over three times data.

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One comma minus one.

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All right.

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So I close the bracket, and now we only have to consider this one here, and we have this before already.

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It was just this expression.

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So you just multiply it here and we are good to go.

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So here now, the command is a bit long, so the trick you could use is to use such a backslash and

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then enter and then some tabs.

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So this means you continue the line in the next line and I think how it should work.

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Try to find this and I'll write integral.

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I can just copy this so I don't make a typo here.

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And the result is six point six, exactly.

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So maybe you read this already?

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I wrote it down before we get to the perfect result here.

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And the reason is because our data has been generated using a third order polynomial.

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And the Simpson rule considers also third all the polynomials for the interpolation of the data.

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So it's clear that it must get the perfect result.

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So, of course, it will not always get the perfect result.

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For example, if we would have taken an exponential function, then the result would not be perfect.

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But it would still be much better than the linear interpolation method, which corresponds to the trapezoidal

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method.

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So you see, when you cannot tune the number of data points that you take into account that you must

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really make the best out of your data.

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And a good idea is to interpolate the data first and then integrate over this function.

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And this is exactly done by the Simpson rule and the Newton Carter's equations.

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They basically consider a higher order polynomials, which work even better for more difficult functions.

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But yeah, I think since we already get the perfect result, it's not really necessary to implement

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these.

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But just as an example, for example, you can take the boules rule, which has even more complicated

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coefficients, which arise from the fact that you take a higher order polynomial to interpolate the

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data first and then you integrate over it if you want to know more of these methods.

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Here they are all listed on this Wikipedia page.