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Before we can really get started with interpolating our data in terms of functions, I want to talk

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about a concept called Taylor expansion and Taylor expansion means that you can basically take any function

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and you can expand.

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It is a series or as a sum that goes to infinity over these terms that you can see here.

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So you can expand it as a polynomial.

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And the equation is a sum of these terms, which are basically here.

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The end the derivative of the function f at the position x zero, which is the position or the argument

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around which we expand the function and that we have here are X dependence.

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So here the distance from the zero will then be squared and then taken to the power three and four and

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so on.

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So of course, in numerics, you cannot go to infinity here.

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So at some point this series has to terminate and you have to define some and max value at which you

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stop expanding the function.

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And then hopefully in the range where you are interested in the data, then the function is approximated

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to a good amount and we want to start and we want to test this concept, the Taylor expansion at the

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example of the exponential function.

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But first of all, I will run the cell here where I will import our two most important modules non pi

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and multiplied lib.

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So here I've written down already that we are investigating the exponential function, which is f of

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X is equal to exponential function of X.

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And as you can see from this equation here, we need the first derivative, the second to third and

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so on.

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We need all the derivatives and the special feature about the exponential function.

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This is how it is defined, is that the derivatives are all the same.

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So every derivative of the function is the function itself.

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It's still exponential function of X.

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So what we can do then very easily is we can define a function called EEGs P Taylor, which will be

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our Taylor expanded series of the function of X is equal to exponential function of X.

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So of course, we need an argument x.

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Then you can see we need to define some position at which we expand our series.

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This would be x zero and then we must define the upper boundary here because, as I said, we cannot

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go to infinity.

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So we have to define some and maximal u.

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And just to make sure that we remember later on, I will explain here all of these arguments.

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So X is the argument of the function that zero is the argument at which the derivatives will be calculated

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and then the and maximal you will be the end at which the series will terminate or at which it will

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break.

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You could also say and now what we will do is we will sum up several parts here and you can do this

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in several methods.

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You could, for example, define the individual terms as an array and then you can use the N pitot sum

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equation.

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We will do this also later in this notebook.

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But here I will just use a simple loop.

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So I will define here some value T, which will in the end, be the value of our Taylor expansion.

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And initially it is zero.

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And then we make a loop here for n in range.

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You can, of course, also take another type of loop.

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So there are many possibilities how you can do this, and we are summing up starting from zero and going

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to an max.

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And we include the value of N Max, which is why I wrote, which is why I wrote and max plus one here.

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So now what we do is we add up all of these values.

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So we write T is equal to the old T plus some value.

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And now, of course, it is a bit stupid to use the exponential function to define the Taylor expansion

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here.

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But yeah, let's let's do it for the moment is just an exercise here.

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So we will calculate them the and P X value at the position x zero.

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So for example, if we would select zero equal to zero, then we could just write here one, because

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we know the exponential function of zero is always one.

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So then it would maybe be a bit more reason.

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But then we would be restricted to just to one argument, zero.

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And I want to show you that this series can also look different when we choose different values of zero.

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So this is why I take your export of cereal.

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And then we multiply.

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So we take the the argument x minus x zero to the power of end.

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And so we have these two terms and now we are just missing this one here, which is two factorial.

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So for example, four factorial four exclamation mark means four times, three times, two times one.

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And we could, of course, program this here.

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But there is already a function that's implemented in nonpaying math module, which is called factorial.

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So we can just use this one here.

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And then don't forget, we have to return some value and of course, we want to return to.

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So as I've said, you can also write all of this in just one line when you use the and p dot some command

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and just sum up over an array where the elements of the array on these terms here.

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But yeah, let's not make it too complicated.

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And I think this one is even easier to understand.

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And now we can write the P Taylor.

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So now we run this function and we can choose, for example, the argument one for the evolution point,

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we take zero and then for the terms, let's use ten for the beginning.

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And so we get this number here.

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Two point seven one.

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So maybe this reminds you of the Euler constant and that check.

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So and p dot s p of one is almost this value.

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So you see, by taking into account only 10 terms of this infinitely large series here allows us to

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reproduce the Euler a number by a very large accuracy.

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So it works pretty well for this particular function.

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And also the reason why this works well is because our position one is quite close to our ex zero.

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And I want to show you what happens when you select a different position for the evolution.

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So I want to make a plot here, so I write X list is equal to NP total in space.

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And I want to plot in the range of four minus five to five.

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And then first of all, I want to do a sketch of plot and I want to plot this list of coordinates that

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I've just created.

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And then we just plot the exponential function.

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So this will be our reference point.

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This will be the correct exponential function as it is implemented in non pi.

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So this will be the one that we are trying to recreate here, and I will do this now.

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Maybe first let me add labels here on appeal to you told X Label X, of course.

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Then we need a y label.

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Oops, sorry y label, which is why and we need some limits.

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So I write pl t y limits x limit as OK, we have already defined this by this command and the Y limit

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will be minus five to 100.

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All right.

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And now I can go ahead and now I can plot the Taylor expansion that we have just find.

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So here I will do a line plot, all right, to appeal to the plot and then axe lists come out XP Taylor.

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And for the argument, we do not only take a single value, but we take our whole list.

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And of course, this works.

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And then 40 zero.

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Here I want to take zero and then we to find some value and max, which we have previously chosen to

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be 10.

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So that's right.

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And Max.

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And let's create these points or this line in a color blue.

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So you see, if we take 10 points into account, then it works quite well.

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I mean, you see maybe a bit here, some change, some deviation, but overall it works quite well.

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For example, if we take five only, then you see in the vicinity of X is equal to zero, which is zero.

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In all case, it works well.

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But then when we go way beyond plus minus two and we see some large deviation of the plot, and this

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changes if we take the different reference point for the evolution, for example, we could take here

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minus three and we can plot it in red or we could take two and we plotted in green.

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And then I see for the to function, which corresponds to minus three.

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It works pretty well here for small values, but not for high values and for the green function, which

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is expanded near zero equal to two.

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It is exactly the opposite way.

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So here and works well in this area, but it doesn't work at all for small values.

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So you see Taylor expansions work and we can tune the accuracy by this parameter and max.

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So of course, the higher end max will be the more terms we take into account and the longer our calculation

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will run.

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But you see, if we make it very large, then we can basically fit any function that's continuous in

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any range for the X value.

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So I will go back to the initial value because here you can see nicely that the position of zero matters

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and in the next lecture, we will expand our next function and I want to show you something else.