1 00:00:00,180 --> 00:00:05,790 The previous lecture, we have implemented the archival methods, and now we will go one step further 2 00:00:05,790 --> 00:00:11,760 and we will implement the archive for five methods, and for this, I will just copy the code of the 3 00:00:11,760 --> 00:00:18,840 Archive for Methods, and we will now update this to archive four or five and the archive for five methods 4 00:00:19,260 --> 00:00:21,960 in the general syntax is very, very similar. 5 00:00:22,260 --> 00:00:25,530 It's just that we have more of these additional points here. 6 00:00:25,830 --> 00:00:32,070 We do not only have one key one to care for, but we have even five and six and we have much more difficult 7 00:00:32,070 --> 00:00:33,090 coefficients here. 8 00:00:34,290 --> 00:00:37,860 So let's have a look at this link that I've opened in the new tap here. 9 00:00:38,280 --> 00:00:45,240 And you see, this is called also the fair back method or a new quota feedback methods, and it has 10 00:00:45,240 --> 00:00:49,980 two different version of of where the error is of order five and order four. 11 00:00:50,400 --> 00:00:57,150 So the coefficients have to be chosen a bit more in difficult manner so that certain arrows cancel out. 12 00:00:58,170 --> 00:01:04,920 And here we will use the upper description here because on Wikipedia, as written to first row of the 13 00:01:04,920 --> 00:01:11,060 coefficient, gives the fifth order accurate solution, so it has an higher accuracy than the fourth 14 00:01:11,070 --> 00:01:11,610 order one. 15 00:01:12,000 --> 00:01:20,250 So there's really no reason why we should bother using this other force or the solution, so it's always 16 00:01:20,250 --> 00:01:22,740 better to have the higher order solution in this case. 17 00:01:23,940 --> 00:01:26,010 So now we just have to read the table. 18 00:01:26,250 --> 00:01:31,830 So these here are the coefficients for our weighted average, for the key value. 19 00:01:32,400 --> 00:01:38,610 This one will be the coefficients for the update of each, and these ones will be the coefficients for 20 00:01:38,610 --> 00:01:40,650 the update of the key values. 21 00:01:41,730 --> 00:01:52,170 So we just have to add here New York, new and new terms k five and six. 22 00:01:53,640 --> 00:01:56,350 And here we have to also add new terms. 23 00:01:56,370 --> 00:02:05,100 So some coefficient times k five plus some coefficients time times six. 24 00:02:06,630 --> 00:02:09,840 So I will update now first to coefficients for K1. 25 00:02:10,229 --> 00:02:13,170 These will be, uh, these ones here. 26 00:02:14,160 --> 00:02:18,390 So it begins with sixteen over one hundred thirty five. 27 00:02:19,230 --> 00:02:19,740 So. 28 00:02:21,990 --> 00:02:28,590 Let me just update these ones, then the second one was zero that we have here two very difficult ones 29 00:02:28,590 --> 00:02:30,210 and in two shorter ones. 30 00:02:30,630 --> 00:02:39,450 So I have, of course, prepared this in a different notebook already six six five six over one to eight 31 00:02:39,450 --> 00:02:41,880 to five then for the K. 32 00:02:42,450 --> 00:02:47,490 We have two eight five six one over five six four three zero. 33 00:02:47,970 --> 00:02:52,140 Then here we have minus nine over 50. 34 00:02:52,650 --> 00:02:56,670 And here we have to over 55. 35 00:02:58,320 --> 00:03:04,890 So then we can next implement the update, which is a bit more simple. 36 00:03:04,890 --> 00:03:07,950 So if you have zero one over for three of eight and so on. 37 00:03:08,610 --> 00:03:10,320 So let's go ahead and do this. 38 00:03:10,710 --> 00:03:13,410 So for the first one, we just have one number. 39 00:03:13,520 --> 00:03:14,280 We had zero. 40 00:03:14,940 --> 00:03:16,260 Yeah, for the first one, we had zero. 41 00:03:16,260 --> 00:03:18,330 So we do not have to change anything for each. 42 00:03:18,870 --> 00:03:20,310 Then we have one over four. 43 00:03:21,000 --> 00:03:27,330 Then we had three over h them for the next one. 44 00:03:27,570 --> 00:03:29,460 We have 12 or 13. 45 00:03:31,890 --> 00:03:33,540 Then we have. 46 00:03:36,670 --> 00:03:39,760 One and we have one over two. 47 00:03:40,420 --> 00:03:41,920 Let's compare if that is correct. 48 00:03:42,190 --> 00:03:42,610 Yes. 49 00:03:43,480 --> 00:03:48,280 And now we have to implement the linear combinations of all of these terms here. 50 00:03:48,820 --> 00:03:51,040 So for the first one, we have one over four. 51 00:03:52,240 --> 00:03:53,650 So this will be here. 52 00:03:54,040 --> 00:03:54,280 OK. 53 00:03:54,320 --> 00:03:55,270 One over four. 54 00:03:55,840 --> 00:04:03,970 Then for the next one, we have three over 32 times K-1 and nine over 32 times K2. 55 00:04:04,750 --> 00:04:05,770 So. 56 00:04:09,080 --> 00:04:11,240 Nine over 32 times. 57 00:04:12,290 --> 00:04:16,190 And then we have also a K-1 term, which we didn't have before. 58 00:04:16,640 --> 00:04:25,850 So this is K one times three over thirty two and then we have also other terms here for K four, five 59 00:04:25,850 --> 00:04:27,590 and six, which are quite long. 60 00:04:27,620 --> 00:04:30,110 So I will just copy them for my separate notebook. 61 00:04:30,740 --> 00:04:38,420 If you really want to update this, you can of course compare this with Wikipedia, or you can just 62 00:04:38,420 --> 00:04:41,600 post a video here and just write along what I did. 63 00:04:42,170 --> 00:04:47,510 So you see, for the K for, for example, we have now really three different terms. 64 00:04:47,900 --> 00:04:54,170 So three additional terms linear combination of K one, two and three and four K six, we even have 65 00:04:54,170 --> 00:04:55,090 five terms. 66 00:04:55,130 --> 00:05:02,450 So yeah, this time all the coefficients are non zero for the K for the Why update, which is updated 67 00:05:02,450 --> 00:05:10,100 as a linear combination of the K values and the rest of the code can really stay as it is, the whole 68 00:05:10,100 --> 00:05:11,740 idea behind it is pretty similar. 69 00:05:11,750 --> 00:05:19,340 It's just that even higher order error terms cancel out since we have constructed this update of the 70 00:05:19,340 --> 00:05:23,300 Y value via the K in the more difficult way. 71 00:05:23,450 --> 00:05:32,330 So this is like when we have derived or when we have shown that there are higher order solutions for 72 00:05:32,330 --> 00:05:34,670 calculating a derivative that we had. 73 00:05:34,670 --> 00:05:44,510 Also, it was also necessary to consider more difficult coefficients and to superimpose several different 74 00:05:44,540 --> 00:05:47,390 terms so that certain arrows cancel out here. 75 00:05:47,390 --> 00:05:48,290 It's the same idea. 76 00:05:49,340 --> 00:05:52,610 So let me run this, so there is no error message. 77 00:05:52,970 --> 00:06:00,230 And so the next lecture, we can now compare our Euler methods with the R.K. four methods and the R.K. 78 00:06:00,230 --> 00:06:01,220 four five methods.