1 00:00:00,150 --> 00:00:02,310 Now, do we know what a derivative is? 2 00:00:02,580 --> 00:00:05,730 Let us discuss how we can define this in Python. 3 00:00:06,870 --> 00:00:11,610 So we basically have to implement one of these three equations. 4 00:00:11,910 --> 00:00:15,960 But the problem is that we have this limit h goes to zero. 5 00:00:16,440 --> 00:00:21,150 And of course, in numerics, the H can never really go to zero. 6 00:00:21,570 --> 00:00:27,810 And also, it could happen that we have some great some data points, some X coordinate, some Y coordinates, 7 00:00:28,170 --> 00:00:31,620 and we do not really know what is in between these data points. 8 00:00:32,430 --> 00:00:38,040 So of course, in this case, we could first to add interpolation and we could interpolate the function 9 00:00:38,040 --> 00:00:40,260 to have access to the points in between. 10 00:00:40,800 --> 00:00:46,950 But also, it's quite convenient to just be able to calculate the derivative based on the points that 11 00:00:46,950 --> 00:00:47,430 we have. 12 00:00:48,120 --> 00:00:52,410 So what this means, basically, is that h will have some fixed value. 13 00:00:52,410 --> 00:00:58,530 It will be the difference between two points or the difference of the X coordinates of the two points. 14 00:00:58,950 --> 00:01:02,550 And then here we will have the difference of the Y coordinates of the two points. 15 00:01:03,420 --> 00:01:05,400 And this is exactly what I have written down here. 16 00:01:05,850 --> 00:01:13,860 In a good approximation, we can write down that the derivative of F at some point and is given by the 17 00:01:13,860 --> 00:01:17,940 difference of the Y coordinates divided by the difference of the X coordinates. 18 00:01:18,840 --> 00:01:24,360 And as discussed previously, we can do this in terms of forward differences, backward differences 19 00:01:24,360 --> 00:01:26,130 and also central differences. 20 00:01:27,210 --> 00:01:34,590 And in this case, since we don't have the limit h goes to zero, all three methods can give a different 21 00:01:34,590 --> 00:01:35,100 result. 22 00:01:35,790 --> 00:01:37,620 And this is what we will explore first. 23 00:01:37,650 --> 00:01:45,810 We will implement these three methods and we will investigate how the derivatives will look different 24 00:01:46,440 --> 00:01:49,170 for the three methods for this example, function. 25 00:01:49,890 --> 00:01:55,960 We have chosen a bit of a more difficult function so that we can really see some maxima and minima and 26 00:01:55,960 --> 00:01:56,490 the function. 27 00:01:56,490 --> 00:02:01,980 It's sine function multiplied by X and then some third order polynomial function. 28 00:02:03,180 --> 00:02:07,920 So as always, we will first import Nampai and much plot lib. 29 00:02:08,400 --> 00:02:13,560 So let me run the cell and then I will first define our function. 30 00:02:14,280 --> 00:02:21,600 So I write def half of X, and of course, the output will be a return function and it will just return 31 00:02:21,600 --> 00:02:22,740 to value of the function. 32 00:02:22,750 --> 00:02:32,400 So I write and pitot sine of X and then times X, and then the other turn will be minus one over 100 33 00:02:33,090 --> 00:02:35,580 times X to the power of three. 34 00:02:36,930 --> 00:02:45,060 So this is our function, which is now defined and now we can continue and we can, yeah, have a look 35 00:02:45,060 --> 00:02:45,780 at the function. 36 00:02:46,500 --> 00:02:52,710 So for this, I will write X list over the final list of X coordinates flyer and P total in space. 37 00:02:53,940 --> 00:03:03,750 And for the plot range, I've chosen minus 10 to 10 and I will use 201 points because by doing so, 38 00:03:04,650 --> 00:03:08,250 the difference between each of these points will be zero point one. 39 00:03:09,630 --> 00:03:11,690 So we have now our X list. 40 00:03:12,030 --> 00:03:20,400 I can define Y list, so this will be, you know, just f our function. 41 00:03:20,400 --> 00:03:22,470 And then the argument will be D X list. 42 00:03:23,190 --> 00:03:28,740 And since this is here and every since I wrote and put model in space, we can apply the function to 43 00:03:28,740 --> 00:03:30,900 every element by writing it down like this. 44 00:03:32,040 --> 00:03:33,690 And so now we are almost finished. 45 00:03:33,690 --> 00:03:43,290 We can just write pal T dot plot and X List Comma Y list, and then we can add some labels for the XS 46 00:03:43,290 --> 00:03:50,730 Plus Tot X label, which will be X and Y label will be F of X. 47 00:03:54,760 --> 00:03:57,940 And we run the cell and have a look at our plot. 48 00:03:59,440 --> 00:04:05,410 So now you can see here, it's really a pretty complicated plot with several maxima and minima. 49 00:04:05,830 --> 00:04:12,970 And in the next step, we are going to calculate the derivative according to these three equations here. 50 00:04:14,650 --> 00:04:22,600 So what we do next is we, of course, take our X list, so it's not really necessary to define it once 51 00:04:22,600 --> 00:04:22,930 again. 52 00:04:22,930 --> 00:04:26,290 But here we'll do it anyway, just in case we change something later on. 53 00:04:26,800 --> 00:04:32,200 And then I will, first of all, calculate the derivative analytically. 54 00:04:32,890 --> 00:04:35,420 So analytical the list. 55 00:04:35,950 --> 00:04:37,920 So for derivative. 56 00:04:40,000 --> 00:04:44,390 And so our function is has two parts. 57 00:04:44,410 --> 00:04:50,020 So in X times x and then this polynomial, the derivative of the polynomial is, of course, very simple. 58 00:04:50,020 --> 00:04:55,330 It's just three times one over 100 times x squared. 59 00:04:55,780 --> 00:05:00,550 And then here for this function, since it's a product, we must use the product rule and we get two 60 00:05:00,550 --> 00:05:01,780 terms for the derivative. 61 00:05:02,230 --> 00:05:08,200 The first one will be cosine times X because the derivative of sine as cosine. 62 00:05:08,680 --> 00:05:16,330 And then the other term will be just sine x times one because the derivative of X is one. 63 00:05:17,470 --> 00:05:19,780 So we will have three terms in total. 64 00:05:20,200 --> 00:05:27,040 This will be an impedance cosine of X lists times x list. 65 00:05:29,170 --> 00:05:34,750 Then the other term will be A. Dots sine of X list times of one. 66 00:05:35,710 --> 00:05:44,740 And then the last term will be minus three times one of 100 times X lists square. 67 00:05:46,090 --> 00:05:48,250 And of course, now we can plot this. 68 00:05:48,520 --> 00:05:53,200 So let's just copy the plot command and change this list here. 69 00:05:54,010 --> 00:05:56,140 And let's right here derivative. 70 00:05:58,300 --> 00:05:59,680 And I did the mistake. 71 00:05:59,860 --> 00:06:01,840 This is oh no, not really a mistake. 72 00:06:01,840 --> 00:06:09,010 But this is, of course, because for the dash here, for the derivative, we cannot use the same symbol. 73 00:06:09,010 --> 00:06:13,670 So let me replace two symbols that let me write it down like this. 74 00:06:13,930 --> 00:06:14,710 Then it should work. 75 00:06:16,000 --> 00:06:16,420 OK. 76 00:06:17,410 --> 00:06:21,760 So the resolution is not really that good in that plot, but I think I think it's sufficient to just 77 00:06:21,760 --> 00:06:22,540 do it like that. 78 00:06:23,290 --> 00:06:28,360 So you see, we have a derivative function, which also has several maxima and minima. 79 00:06:28,720 --> 00:06:34,090 And for example, when the function has a maximum, it goes through zero here. 80 00:06:35,140 --> 00:06:35,560 All right. 81 00:06:35,570 --> 00:06:36,880 So it seems to be correct. 82 00:06:37,300 --> 00:06:41,140 And now we can also add the numerical derivatives. 83 00:06:41,920 --> 00:06:50,980 And for this, I will first of all, define in h value, which will be zero point one so that it corresponds 84 00:06:50,980 --> 00:06:53,880 basically to the data points in our list. 85 00:06:53,950 --> 00:07:00,760 So this h two step size of zero point one is the same difference between two points in this and p total 86 00:07:00,760 --> 00:07:02,380 in space of our X list. 87 00:07:03,340 --> 00:07:08,140 And now I can define a list, for example, for what D list? 88 00:07:09,400 --> 00:07:11,590 And now I must 89 00:07:14,260 --> 00:07:15,820 define it in this way. 90 00:07:15,830 --> 00:07:19,480 So we take the value of the function at the next point. 91 00:07:19,900 --> 00:07:23,260 Minus the values value of the function at the current point. 92 00:07:23,560 --> 00:07:26,200 Divide it by H, basically. 93 00:07:27,070 --> 00:07:31,150 So I write down the value of the function. 94 00:07:33,650 --> 00:07:48,590 At the Find X list plus age minus, well, you have to function at X list and then we divide the whole 95 00:07:48,590 --> 00:07:50,120 thing by H. 96 00:07:52,640 --> 00:07:59,180 So the next thing that we can do is backwards differences here we take the value of the function minus 97 00:07:59,180 --> 00:08:04,340 the value of the function at the previous point, and then we will also program the central differences, 98 00:08:04,340 --> 00:08:11,570 which will be the difference of the following point and the previous point divided by 2H this time. 99 00:08:12,980 --> 00:08:14,210 So let's program it. 100 00:08:15,770 --> 00:08:16,310 Here we are. 101 00:08:16,760 --> 00:08:24,890 So I will just copy this line twice, so I don't have to type so much and I will write backward and 102 00:08:25,010 --> 00:08:25,730 central. 103 00:08:28,290 --> 00:08:33,900 And then here we said we'd take the current point minus the previous point, divided by age. 104 00:08:34,440 --> 00:08:41,100 And here we said we take the following point minus the previous point divided by two h. 105 00:08:42,990 --> 00:08:43,409 All right. 106 00:08:43,409 --> 00:08:49,200 So now we have created these three lists characterizing derivatives calculated by three different methods. 107 00:08:49,410 --> 00:08:57,450 And now we can add these plots, and I just have to exchange the names for the lists. 108 00:08:59,080 --> 00:09:06,640 So let me do this takes a few seconds, and I hope you also program along with me, and when I run this, 109 00:09:07,900 --> 00:09:10,390 we basically see not a big difference. 110 00:09:10,570 --> 00:09:15,070 I mean, we can still see here red blow to green blood, maybe a blue plot in the background. 111 00:09:15,340 --> 00:09:21,490 So there is some small difference comparing the three plots, but the difference is quite small. 112 00:09:23,170 --> 00:09:29,140 But of course, now on the following we want to analyze this in more detail. 113 00:09:29,920 --> 00:09:34,270 And for this, I want to analyze now the arrow. 114 00:09:35,290 --> 00:09:43,240 So I will copy the plot function and I will write for the y axis out of the derivative. 115 00:09:44,230 --> 00:09:55,360 And so what we do is we take this list analytical list as a reference point and we will subtract the 116 00:09:55,360 --> 00:09:57,520 other lists from these reference points. 117 00:09:59,170 --> 00:10:07,120 So like this, and I will delete this list because this is no our reference until we will plot only 118 00:10:07,120 --> 00:10:07,720 the difference. 119 00:10:08,650 --> 00:10:16,000 This works because we choose the same X list for all of the plots, so we cannot just calculate the 120 00:10:16,000 --> 00:10:17,230 difference of the why list. 121 00:10:18,310 --> 00:10:26,140 And now I will add some color so that we know which plot corresponds to which function blue and green. 122 00:10:27,400 --> 00:10:34,260 So when I run it, we see this is all a result. 123 00:10:34,270 --> 00:10:40,420 This is the error of the derivative of X compared to the analytical result, which we know is the correct 124 00:10:40,420 --> 00:10:40,660 one. 125 00:10:41,170 --> 00:10:47,020 And we see that basically the red and the blue function have quite a large difference. 126 00:10:47,020 --> 00:10:56,320 At some points it goes even up to 0.4, which is considerable if we consider our function as a change 127 00:10:56,320 --> 00:10:58,900 from plus seven to minus 12. 128 00:10:59,590 --> 00:11:02,470 So yeah, it is some error of a few percent. 129 00:11:02,920 --> 00:11:09,070 And what's also interesting is that we can also see that the absolute error of the forward and the backwards 130 00:11:09,070 --> 00:11:10,630 method is the same. 131 00:11:10,630 --> 00:11:11,950 It's just a different sign. 132 00:11:12,760 --> 00:11:18,970 But most importantly, the green function, which is the central method, has a much, much smaller 133 00:11:18,970 --> 00:11:19,390 arrow. 134 00:11:20,050 --> 00:11:24,640 It's much better to use this method for calculating the derivative. 135 00:11:26,080 --> 00:11:31,060 And in the following lecture, we want to explore why this is the case. 136 00:11:31,060 --> 00:11:39,160 So the name is why is the central differences methods typically better than the forwhat and the backroads 137 00:11:39,220 --> 00:11:40,390 differences methods?