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Welcome back to the course, but we have so far discussed how to calculate first derivatives of functions.

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And now we want to discuss how we can calculate higher derivatives.

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So for example, here is a second derivative and mathematically speaking, this is totally easy.

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You just take the function, calculate the first derivative and then from the first derivative, you

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calculate the second derivative in exactly the same way.

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So once again, you take the limit to zero of the first derivative and you introduce such a small value,

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which then goes to zero.

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So you see when we now use our definition for the first derivative and plug it in here.

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This gives us many possibilities for the definition of the second derivative.

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So, for example, we could use here the forward method here, the Beckworth method we could use here

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the central method to forward methods so we can combine all of these methods and we have to figure out

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which one is the best way to do it.

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So as an example, we could, for example, use double forwhat.

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So this means we start from this equation and we use for the f first derivative and this one also the

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forward method.

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So this means for this term, we will get the function and x plus two h minus the function of X plus

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one h divided by h divided by h.

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So this means we have this term here divided by h squared.

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And then for the other term we get function and x plus h minus function and x divided by h divided by

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h.

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So we get this divided by square.

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And so if we solve these brackets here, we get this expression with the three terms.

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If we to the double backwards, it's pretty similar.

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We get a similar results.

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We get here just different arguments where we'll use the previous point and the second nabor point in

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the previous direction or in the backward direction.

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What we can also do is, of course, we can combine forward and backward methods.

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So this means we start from here.

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But do you use the backward method for this one and for this one?

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So here we will get f off x plus h minus f of X, and here we will get f off x minus f of x minus h.

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And so if we solve then the brackets, we get this one.

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So we get f at the next point minus two times F at the current point, + f at x minus the previous point.

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And then we divide all cases by age square and take the limit h to zero.

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This is exactly the same result as we get for using the double central methods.

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Here we would just have then x plus two h minus two times F and X plus f x minus two h divided by four

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h square.

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But since h, uh, goes to zero, we can also substitute it with a new variable to Asia's equal to G.

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And then we have this expression and this is exactly the same as this one.

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So don't worry if this is too mathematical for you, it's not a big deal.

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The main message here is that in the end, we have once again a forward method, a backward method and

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a central point methods.

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And these ones we are going to implement now and we will like previously for the first derivative,

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we will compare them.

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And since it's so similar, let me copy the code from previously.

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So this one?

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And let me just paste it here at the bottom and updated for the second derivative.

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So we have here the analytical de-list, which we don't need anymore.

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And instead, we need the analytical D to list for the second derivative.

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And so here we need to think a bit.

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Maybe let's first do the last term because this is most easy.

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It's just this one.

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And then we calculate here the derivative, which is

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cosine.

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Then from here we get to terms.

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The first term will be cosine times one.

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So we get this term second time and the other term will be like this with a minus sign because a derivative

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of cosine is minus sine.

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So we update this two analytical D2 lists.

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And here we update the label to the second derivative.

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And now we must update these lists here to the to the to the two tier as well.

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And of course, now we must.

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Update the values themselves, so in all cases, we have here to square.

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And here we now also have an h square and here we have X list plus two times h minus two times f x list

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plus each.

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And then we have plus f x list.

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And the other terms are pretty similar.

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So I will just copy them.

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And here the difference is that we have had like this minus two times h.

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And here we have it like this.

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Yes, so I have not implemented the double forward, double backwards and the double central, which

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is equivalent to the forward and backward methods.

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And when I run this, we see first of all, it seems to be correct because all four curves are roughly

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the same.

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And we know that the analytical result is, of course, always correct.

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And you see also that there are some differences you can really see here yellow, red and also green.

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So once again, let us compare the differences and we do this, of course, in the same way as previously

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we right here Arrow.

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And we right here.

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That we calculate the difference between these two lists, and we always take the analytical list as

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a reference point.

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And then we also change the color of the plot.

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So here we use red.

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Here we use blue.

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And here we use green.

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So as previously, the red and blue curves double forward and the double back watch are not as good

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as the double central or combined forward and backward mechanism.

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And the reason is, of course, it's pretty similar to previously when we would do now the Taylor expansion,

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we would see that the double central methods would have pearls that cancel out.

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And so the accuracy would be much, much better.

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Now, this is not really surprising.

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And also, it's not surprising that we can now also take into account further nearest neighboring points

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to achieve a higher accuracy.

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And this time, the Richardson formula is given by this expression.

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So once again, we take the nearest and to second nearest neighboring points.

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And I will just use the code from previously to update to update our function.

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So I will use this one and also I will use the following cell.

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Let me copy this as well.

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Until this time, of course, we will now have to detour Richardson, and we must update now this equation,

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which will be minus F and X minus two h plus 16, that we have an additional term that cancels out previously,

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which is minus three times f of X than we have plus 16 and minus those one is OK is correct.

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OK, now we just change what we have written down here.

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D2, Richardson, Richardson, d to list and then we just update here everywhere.

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D two and here as well.

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And now I run both cells, and we see that apparently I made a mistake, which is OK, but let's see

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if we can find the mistake.

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So it seems as if the Richardson function gives a large arrow, but we know that this is not what's

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supposed to happen.

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And of course, here I forgot something here.

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It's one of a 12h square.

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So here we missed the square.

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And this time it looks much better.

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Yeah.

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This time we have an error for the double of central methods of approximately six times 10 to the minus

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three and four the Richardson method.

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We have an error of 10 to the minus five.

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So it is, I don't know, maybe, yeah, like 500 times better.

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So the logic behind this is really, really similar to the first derivatives.

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You can calculate the second derivatives pretty similarly.

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You just have to use the first derivative equations and plug them into each other so that you end up

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with some double forward, double backward or double the central methods.

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And then you will also find once again that some of these equations are better than others, and you

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could understand this analytically if you would write down the tether expansion.

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But yeah, that's a bit too much of a hassle here.

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We just believe these results and just analyze it numerically.

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And what we found is that the double central method works much, much better than the double forwhat

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and the double backward methods, and that there exist even better methods like the Richardson equation

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that is given here, which is based on taking into account also second nearest neighboring points.

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So this is all not really surprising because it is absolutely identical to the first derivative procedure

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that we have discussed in the previous lectures.

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Still is, of course, very important to keep in mind that this also works for the higher derivatives.