1 00:00:00,180 --> 00:00:06,270 So in a previous lecture, we have expanded the sound function into position zero is equal to zero. 2 00:00:06,990 --> 00:00:13,080 And the reason why we were only able to do it at this particular position is that we wanted to calculate 3 00:00:13,080 --> 00:00:15,120 the derivatives analytically. 4 00:00:15,270 --> 00:00:20,430 And here it is particularly easy because it's just zero minus one and plus one. 5 00:00:21,450 --> 00:00:28,410 However, if we want to do it in a more general way, then we also must be able to define or to expand 6 00:00:28,410 --> 00:00:32,400 the sine function or any other function at any given position. 7 00:00:33,000 --> 00:00:39,630 And for this to to be able to do it in general, we must define the derivative numerically and this 8 00:00:39,630 --> 00:00:41,220 is what we are going to do next. 9 00:00:42,120 --> 00:00:50,100 So first of all, we have to define the derivative, that first derivative, but also the higher derivatives. 10 00:00:50,700 --> 00:00:57,840 And I have dedicated in this course a whole section on derivatives so that we will explore this in much 11 00:00:57,840 --> 00:01:04,530 more detail and we will we will explore how you can calculate derivatives with a much higher accuracy. 12 00:01:04,920 --> 00:01:11,630 But in very simple terms, you can define a derivative by this argument here or by the way, this term, 13 00:01:11,640 --> 00:01:12,780 mathematically speaking. 14 00:01:13,200 --> 00:01:20,370 So you introduce a small number each, which will be a distance in x direction and then you take the 15 00:01:20,370 --> 00:01:25,430 difference for the Y values of this function at these two positions. 16 00:01:25,440 --> 00:01:32,070 So you take it at X plus h and an X, and then you just divide by the distance and X direction. 17 00:01:32,490 --> 00:01:38,910 And this gives you the slope of the triangle and the slope of the tangent of the function, and this 18 00:01:38,910 --> 00:01:40,260 is the first derivative. 19 00:01:41,040 --> 00:01:49,140 And we can the find this quite easily, so we can define here a function called the derivative and for 20 00:01:49,140 --> 00:01:49,770 the arguments. 21 00:01:49,770 --> 00:01:55,090 We need, of course, to function and the position, but also we must define the step size. 22 00:01:55,140 --> 00:01:57,000 So basically the value of H. 23 00:01:58,170 --> 00:02:11,520 So as I said after B, the function X is the argument of the function and H is the step size. 24 00:02:15,150 --> 00:02:18,330 And the value that we return is pretty easy. 25 00:02:18,360 --> 00:02:22,890 It's, of course, just this expression that is given here. 26 00:02:23,250 --> 00:02:30,710 So if the Position X plus each line is at the position X and then we divide the whole thing by H. 27 00:02:31,560 --> 00:02:35,790 So here in terms of mathematics, we have to do a limit of age goes to zero. 28 00:02:36,090 --> 00:02:38,340 Only then the derivative is really exact. 29 00:02:38,730 --> 00:02:40,980 But of course, in America, we cannot really do this. 30 00:02:40,980 --> 00:02:46,230 So we must say, OK, we just take a small value for h, and that's all we can do. 31 00:02:46,950 --> 00:02:52,950 Of course, as I said, we will also explore in the separate lecture how we can improve this version 32 00:02:52,950 --> 00:02:56,580 and how we can make the accuracy much, much better than what is written here. 33 00:02:56,850 --> 00:03:00,720 Now, for the higher derivative terms, it is not so easy. 34 00:03:00,990 --> 00:03:06,390 I mean, in terms of mathematics, it is pretty easy to first calculate the first derivative of the 35 00:03:06,390 --> 00:03:09,480 function F and to calculate the second derivative. 36 00:03:09,990 --> 00:03:13,770 You just calculate the same thing again, but here with the first derivative. 37 00:03:14,580 --> 00:03:20,670 So basically, you do this and iterations to calculate, for example, the third derivative, you first 38 00:03:20,670 --> 00:03:22,440 must calculate the first derivative. 39 00:03:22,920 --> 00:03:28,980 Then from here, you take this equation again with the first derivative if you calculate the second 40 00:03:28,980 --> 00:03:29,640 derivative. 41 00:03:30,090 --> 00:03:32,400 And from there, you calculate the third derivative. 42 00:03:33,420 --> 00:03:40,740 But in our case, we want to have a function that is able to instantly calculate the and derivative. 43 00:03:41,520 --> 00:03:48,540 And so you can imagine that what you have to do then if you think about iterations when you do this 44 00:03:48,570 --> 00:03:55,320 over and over again, is that for the second derivative, for example, you must calculate here the 45 00:03:55,320 --> 00:04:01,290 first derivative not only at Position X, but also at the Position X plus h. 46 00:04:02,010 --> 00:04:03,090 And this is the problem. 47 00:04:03,480 --> 00:04:09,300 To calculate the derivative at the position X plus h, you must know the. 48 00:04:11,720 --> 00:04:17,269 The the value of the function itself at the position X plus h and X plus two h. 49 00:04:18,290 --> 00:04:24,020 And so if you go to the fourth derivative for higher, then you must know the function at the Position 50 00:04:24,020 --> 00:04:27,890 X plus order of derivative times h. 51 00:04:29,030 --> 00:04:33,560 And so this is really, yeah, what's what, what we will do next year. 52 00:04:34,010 --> 00:04:41,150 And you can also imagine that when you do this, when you have several terms here and some of these, 53 00:04:41,930 --> 00:04:45,170 you can count how often they occur. 54 00:04:45,380 --> 00:04:47,570 So some of them will occur more than once. 55 00:04:47,870 --> 00:04:50,240 And this is where this term here comes from. 56 00:04:50,630 --> 00:04:56,300 And yeah, if you think about it very carefully, I think you could come up with the solution. 57 00:04:56,690 --> 00:05:01,250 But for the moment, let's just take this equation and that's try to program it. 58 00:05:01,280 --> 00:05:04,670 So this is just something that you can find, for example, on Wikipedia. 59 00:05:05,180 --> 00:05:07,220 Well, as I said, it totally makes sense. 60 00:05:07,220 --> 00:05:12,710 And if you look at it carefully, for example, if you try to first calculate the second derivative, 61 00:05:12,710 --> 00:05:16,400 then you will see how you end up with this expression. 62 00:05:17,480 --> 00:05:19,920 So it looks more difficult than it actually is. 63 00:05:19,940 --> 00:05:21,080 This is what I want to say. 64 00:05:22,100 --> 00:05:26,690 So let us program this and then we will see that it's not so bad, actually. 65 00:05:27,320 --> 00:05:33,980 So we can just go ahead and calculate to find a function called and derivative. 66 00:05:35,240 --> 00:05:42,380 And this time we need the function to position the step, size and of course, also the order. 67 00:05:42,530 --> 00:05:50,120 And so has previously the first three arguments are the same as for the first derivative, and then 68 00:05:50,120 --> 00:05:55,100 the end will be the basically the end derivative. 69 00:05:56,990 --> 00:06:01,340 And then we have to add up terms because we have the same here. 70 00:06:01,970 --> 00:06:08,120 So once again, we do it similarly to our Taylor expansion and we create such a loop where we can, 71 00:06:08,120 --> 00:06:17,060 for example, take a for loop and we have a running index case starting at zero and going to the value 72 00:06:17,060 --> 00:06:17,480 end. 73 00:06:18,110 --> 00:06:21,440 And we include and so therefore we write and plus one here. 74 00:06:22,790 --> 00:06:28,400 And then in the end, we return for value of T. 75 00:06:28,670 --> 00:06:30,530 So T will be updated here. 76 00:06:30,530 --> 00:06:33,380 T is equal to T plus some terms. 77 00:06:33,980 --> 00:06:35,990 And we will return the value of T. 78 00:06:37,100 --> 00:06:40,700 And you can see here that this term is outside of the sum. 79 00:06:41,030 --> 00:06:44,180 So of course, I could put it inside of the sum and write it here. 80 00:06:44,450 --> 00:06:50,810 But it saves a bit of calculation time too, just after summing up the terms to divide by what is written 81 00:06:50,810 --> 00:06:51,130 here. 82 00:06:51,140 --> 00:06:53,570 So this will be h to the power of end. 83 00:06:54,590 --> 00:07:01,060 So no, we have to just program this term, so it is actually not so difficult. 84 00:07:01,070 --> 00:07:09,620 So we just have to write minus oops minus one to the power of and or no, it's not, and it's close. 85 00:07:09,620 --> 00:07:15,350 And then we have this fraction of these factorial. 86 00:07:15,650 --> 00:07:26,690 So we have then p dots, math dot factorial off and divided by and then let me copy this because it's 87 00:07:26,690 --> 00:07:28,100 quite a lot to write. 88 00:07:28,580 --> 00:07:35,570 We divide by Factorial K times factorial of and minus K. 89 00:07:37,310 --> 00:07:40,040 And then we are only left to the function. 90 00:07:40,040 --> 00:07:46,670 So we write F 50 Argument X Plus K Times page. 91 00:07:47,990 --> 00:07:50,270 And I think this is already it. 92 00:07:51,470 --> 00:07:56,990 So to test the whole thing, I will first test the derivative and then the higher derivative. 93 00:07:56,990 --> 00:08:01,730 So I will define here a function which will be a test function. 94 00:08:02,330 --> 00:08:13,160 So the function will, for example, be called two times sine of Tax Square. 95 00:08:13,400 --> 00:08:18,980 So let's just use something a bit more difficult that we cannot so easily calculate analytically. 96 00:08:19,400 --> 00:08:24,470 I mean, it would definitely be possible to calculate here easily the first, second, third derivative. 97 00:08:24,920 --> 00:08:28,610 But if you want to calculate the tenth derivative, it's pretty annoying, of course. 98 00:08:29,540 --> 00:08:30,770 So we can do it like this. 99 00:08:32,090 --> 00:08:38,679 So we have our function and now we can calculate, for example, at the position x zero is equal to 100 00:08:38,690 --> 00:08:40,820 ten point five the value that we had before. 101 00:08:41,360 --> 00:08:49,940 We can now calculate the value of the function, which was twelve point something, and we can calculate 102 00:08:49,940 --> 00:08:56,930 the first derivative of F, which we have called function func. 103 00:08:57,560 --> 00:09:00,920 And then X is zero and then we have to define some h value. 104 00:09:01,400 --> 00:09:07,220 So here I will just take, you know, I don't know, zero point one. 105 00:09:08,750 --> 00:09:09,830 So like this. 106 00:09:10,860 --> 00:09:12,670 And I get an error. 107 00:09:18,260 --> 00:09:25,310 Let me see what is the problem here, name derivative is not defined. 108 00:09:25,490 --> 00:09:26,540 Maybe I didn't run it. 109 00:09:27,440 --> 00:09:27,830 Let's see. 110 00:09:28,460 --> 00:09:28,790 OK. 111 00:09:28,860 --> 00:09:30,090 Yeah, I just forgot to run it. 112 00:09:30,110 --> 00:09:31,160 So, yeah, no problem. 113 00:09:31,580 --> 00:09:32,180 Now it works. 114 00:09:32,630 --> 00:09:35,270 So we have now our function, our derivative. 115 00:09:35,340 --> 00:09:37,610 No, we can test a higher order derivative. 116 00:09:38,150 --> 00:09:46,550 And we can basically copy this year and write and derivative of function x zero h. 117 00:09:46,550 --> 00:09:47,240 And then we need. 118 00:09:48,440 --> 00:09:53,420 And we can say and will be the and the river type. 119 00:09:53,720 --> 00:09:57,500 So if we take a zero, then we should get the actual value of the function. 120 00:09:58,010 --> 00:09:59,030 So I will write zero. 121 00:10:00,410 --> 00:10:02,960 This gives us twelve point zero four. 122 00:10:03,590 --> 00:10:05,240 Then we can write one. 123 00:10:07,130 --> 00:10:09,740 This gives us the first derivative, which is this one. 124 00:10:09,740 --> 00:10:10,490 So it worked. 125 00:10:10,970 --> 00:10:14,660 And now, of course, we could just continue and check out some values. 126 00:10:14,660 --> 00:10:16,810 But yeah, here we don't know if it's correct. 127 00:10:16,820 --> 00:10:21,170 But since the first two terms are correct, I have high hopes that this works well. 128 00:10:22,490 --> 00:10:22,960 All right. 129 00:10:22,970 --> 00:10:24,590 But to really know if it works. 130 00:10:24,770 --> 00:10:35,000 Let us define the general function for the Taylor expansion and then plot the function two times sine 131 00:10:35,000 --> 00:10:36,020 squared plus x. 132 00:10:36,350 --> 00:10:40,730 And then that is also plot the Taylor expansion and this we will do in the next lecture.