1 00:00:00,360 --> 00:00:06,240 So in the previous lecture, we have found out that the central differences method is much better than 2 00:00:06,240 --> 00:00:08,640 the forward and to back what differences methods. 3 00:00:09,780 --> 00:00:15,570 So this is maybe a bit surprising because when we scroll up our notebook and go back to the very beginning, 4 00:00:16,050 --> 00:00:21,300 we found that mathematically speaking, the three methods are exactly the same. 5 00:00:22,050 --> 00:00:29,610 But this is only true if we take the limit of age to zero, which is something that we cannot precisely 6 00:00:29,610 --> 00:00:30,630 do in the Marek's. 7 00:00:31,710 --> 00:00:38,280 So in the metrics, we always have this approximation and the derivative is never really perfect. 8 00:00:38,700 --> 00:00:39,930 So of course it can happen. 9 00:00:39,930 --> 00:00:43,830 Then one of the methods works better than than one of the other methods. 10 00:00:44,880 --> 00:00:53,190 And in this lecture, I want to show you mathematically why this is the case and to figure this out, 11 00:00:53,190 --> 00:01:01,860 we used to so-called big own notation, and the big o notation tells us which order are the following 12 00:01:01,890 --> 00:01:03,660 terms that we are leaving out here. 13 00:01:04,410 --> 00:01:12,090 So previously, our roads that this derivative here is only approximately the same as this term, and 14 00:01:12,600 --> 00:01:20,430 the error that leads to the approximate instead of the equals sign is given by this big arrow here. 15 00:01:21,090 --> 00:01:28,680 And it turns out that the forward and backward differences is has an error of order h and the central 16 00:01:28,680 --> 00:01:29,070 difference. 17 00:01:29,070 --> 00:01:32,070 This method has an error of the Order H Square. 18 00:01:32,730 --> 00:01:39,780 So this means the remaining terms here that are missing due to for for these terms to be equal, the 19 00:01:39,780 --> 00:01:42,990 missing terms of order a square and higher. 20 00:01:43,350 --> 00:01:45,750 So there is no term of order H. 21 00:01:46,680 --> 00:01:54,630 And when H is a very small number like zero point zero one or in our case, also 0.1 and we squared 22 00:01:54,630 --> 00:01:56,430 this number, then it becomes even smaller. 23 00:01:57,210 --> 00:02:00,540 So a zero point one square is zero point zero one. 24 00:02:01,290 --> 00:02:07,290 So here the error is some constant times zero point zero one. 25 00:02:07,290 --> 00:02:11,270 And here the error is some constant times zero point one. 26 00:02:11,760 --> 00:02:15,750 So, of course, the error is much smaller for the central differences method. 27 00:02:17,430 --> 00:02:22,320 So why is the error of Order H Square? 28 00:02:22,410 --> 00:02:26,040 This is something that we can really show analytically. 29 00:02:26,760 --> 00:02:33,600 And for this, we need to rely on our Taylor expansion methods, and we need to assume that we can Taylor 30 00:02:33,600 --> 00:02:38,880 expand our function f at some point, which is, of course, true if it's a continuous function. 31 00:02:40,080 --> 00:02:47,370 So we can say we start from our Point X. So we want to expand the function if we want to know the value 32 00:02:47,370 --> 00:02:51,090 of the function at the point X plus h instead. 33 00:02:51,930 --> 00:02:56,940 And if we expand our function around the Point X, this works as follows. 34 00:02:56,940 --> 00:03:05,850 We write F of x plus first derivative of X Times H plus one of a two second derivative of X times a 35 00:03:05,850 --> 00:03:12,990 square, then one of two times three times three derivative of F times H two. 36 00:03:12,990 --> 00:03:13,740 The power of three. 37 00:03:14,070 --> 00:03:15,620 And then so on and so on. 38 00:03:15,630 --> 00:03:20,040 So here then there come the fourth or the terms five fifth, all the terms and so on. 39 00:03:20,790 --> 00:03:23,610 But we only need these terms for the derivation. 40 00:03:24,030 --> 00:03:30,150 But of course, we can also do the same thing for finding the the value of the function at the point 41 00:03:30,150 --> 00:03:31,360 X minus h. 42 00:03:31,380 --> 00:03:39,690 And again, we expend around to point X. And this time we get the result, which is pretty similar. 43 00:03:40,110 --> 00:03:44,130 But instead of having H and this function, we have minus h. 44 00:03:44,790 --> 00:03:50,430 So this will mean this term, if you will change the sign, this one will stay because it's H Square 45 00:03:50,820 --> 00:03:52,890 and this one will also change the sign. 46 00:03:53,160 --> 00:03:56,010 And so every second term will have a different sign. 47 00:03:56,490 --> 00:04:00,270 But besides that, these two functions are the same. 48 00:04:01,620 --> 00:04:08,610 Now, from the first line, individually, we can immediately see that the four what difference this 49 00:04:08,610 --> 00:04:10,950 method is of order H. 50 00:04:11,610 --> 00:04:16,230 And from the second line, we can see that the backwards differences method is of Order H. 51 00:04:16,920 --> 00:04:18,690 Let me show you how this works. 52 00:04:19,290 --> 00:04:21,790 We basically solve this equation. 53 00:04:21,870 --> 00:04:29,370 And also this this equation for the first derivative of F at the Position X, because this is what we 54 00:04:29,370 --> 00:04:30,540 want, after all. 55 00:04:31,560 --> 00:04:34,740 So what we do is we bring this to the other side. 56 00:04:34,950 --> 00:04:40,380 Then we bring this to the other side and then we account for the negative sign here. 57 00:04:41,070 --> 00:04:53,880 And so what we get is then f derivative of x times h is equal to minus f of x plus f of x plus h minus 58 00:04:53,880 --> 00:04:58,620 one half this term, minus one of a sixth this term and so on and so on. 59 00:05:00,000 --> 00:05:06,300 And then we have here also the factor h and we divide by age, and this gives us what is written down 60 00:05:06,300 --> 00:05:06,630 here. 61 00:05:07,110 --> 00:05:14,070 We have the first derivative of F at the position X is equal to one over H and then all of these terms 62 00:05:14,070 --> 00:05:15,210 that I have just mentioned. 63 00:05:15,510 --> 00:05:21,270 So this will be this term than negative this term, negative this term, negative this term. 64 00:05:23,350 --> 00:05:29,440 And when you look at these two terms and divide by age, this is exactly what we have here for the forward's 65 00:05:29,440 --> 00:05:30,070 differences. 66 00:05:30,880 --> 00:05:35,710 So we have here our forward differences solution and then we have additional terms. 67 00:05:36,070 --> 00:05:41,590 And the first term will be minus one over two times second derivative Times Square. 68 00:05:41,860 --> 00:05:45,040 But we divide by age, so it's only an age here. 69 00:05:45,940 --> 00:05:48,570 And then the other terms are of higher order. 70 00:05:48,580 --> 00:05:52,120 We have to each square a short par three and so on and so on. 71 00:05:53,170 --> 00:05:59,710 So we can summarize and say that all of these terms here are the error of our forwards differences methods. 72 00:06:00,100 --> 00:06:03,430 And they are of older age and higher. 73 00:06:04,090 --> 00:06:05,320 So this is why, right? 74 00:06:05,710 --> 00:06:06,580 Order of age. 75 00:06:06,880 --> 00:06:09,640 And this will be the main term that determines the error. 76 00:06:09,650 --> 00:06:12,160 And these will, of course, also determine the arrow. 77 00:06:12,160 --> 00:06:13,450 But they will be much smaller. 78 00:06:14,830 --> 00:06:19,090 And for the backwards differences methods, we take the second equation. 79 00:06:19,540 --> 00:06:21,140 And of course, we do the same thing. 80 00:06:21,160 --> 00:06:22,960 We bring this one to the left side. 81 00:06:23,200 --> 00:06:24,460 This one to the right side. 82 00:06:24,460 --> 00:06:25,720 And we divide by age. 83 00:06:26,020 --> 00:06:31,400 And so this time we have here different signs or most of these terms will change their sign. 84 00:06:31,420 --> 00:06:33,760 So we have such an expression here. 85 00:06:35,020 --> 00:06:37,780 And so this will here be a lot back. 86 00:06:37,870 --> 00:06:39,070 What's difference this term? 87 00:06:39,460 --> 00:06:43,120 And this one divided by age will be our arrow. 88 00:06:43,420 --> 00:06:49,390 So once again, it's of older age to the power of one, and then we have also higher terms. 89 00:06:50,920 --> 00:06:58,090 However, here I wrote down that the central differences term has an arrow of older age square. 90 00:06:58,600 --> 00:07:01,660 So there is no term proportional to age. 91 00:07:02,380 --> 00:07:04,300 And here is why this is the case. 92 00:07:06,230 --> 00:07:10,400 So I wrote down to find this older H2 dependence. 93 00:07:10,880 --> 00:07:16,040 We have to subtract the two terms, so we have to subtract these two equations from each other. 94 00:07:16,940 --> 00:07:25,910 So the left side will become f of x plus h minus f of x minus h is equal to these two terms will cancel. 95 00:07:26,780 --> 00:07:31,850 These two terms will then add up because we have this term minus minus this term. 96 00:07:31,880 --> 00:07:39,470 So we have two times first derivative of F times h and then these two terms will cancel again and these 97 00:07:39,470 --> 00:07:40,970 two terms will add up again. 98 00:07:41,630 --> 00:07:47,390 So here we have one of the three third derivative and then h to the power three. 99 00:07:48,890 --> 00:07:55,460 And now you see, this is here the denominator of this function here of the central difference, this 100 00:07:55,460 --> 00:07:55,880 method. 101 00:07:56,780 --> 00:07:59,900 So we divide by two h and then we have this term. 102 00:08:00,350 --> 00:08:04,820 So we have one over two h times. 103 00:08:04,820 --> 00:08:12,980 This one is equal to the first derivative of F at the position X and then the next term will be this 104 00:08:12,980 --> 00:08:16,250 term third derivative and h two the power of two. 105 00:08:17,330 --> 00:08:23,870 So this means when we solve for this one, that this is equal to the expression that we had above this 106 00:08:23,870 --> 00:08:25,940 was the central differences method. 107 00:08:26,540 --> 00:08:34,610 And then the first term that appears will be one of a six third derivative of F and an H to the power 108 00:08:34,640 --> 00:08:35,240 of two. 109 00:08:36,380 --> 00:08:40,190 And so this is an arrow of the order h square. 110 00:08:41,270 --> 00:08:48,890 And the very reason for this not to be power one is that these two terms cancel out when we subtract 111 00:08:48,890 --> 00:08:49,760 these two terms. 112 00:08:50,870 --> 00:08:58,140 And this is the whole trick for finding new methods to calculate derivatives with a lot. 113 00:08:58,160 --> 00:09:06,740 With a smaller arrow, we have to combine several different terms so that our rules of Law Order will 114 00:09:06,740 --> 00:09:07,550 cancel out. 115 00:09:08,510 --> 00:09:13,340 And this is really a trick that is used very often, and we will explore this also in the next lecture 116 00:09:13,790 --> 00:09:20,840 that we will use a function or methods that allows us to calculate the derivative of a function with 117 00:09:20,840 --> 00:09:22,610 a much higher accuracy.