1 00:00:00,000 --> 00:00:00,030 2 00:00:00,030 --> 00:00:02,330 The following content is provided under a Creative 3 00:00:02,330 --> 00:00:03,610 Commons license. 4 00:00:03,610 --> 00:00:05,960 Your support will help MIT OpenCourseWare 5 00:00:05,960 --> 00:00:09,960 continue to offer high quality educational resources for free. 6 00:00:09,960 --> 00:00:12,590 To make a donation, or to view additional materials 7 00:00:12,590 --> 00:00:16,150 from hundreds of MIT courses, if visit MIT OpenCourseWare 8 00:00:16,150 --> 00:00:20,410 at ocw.mit.edu. 9 00:00:20,410 --> 00:00:23,940 PROFESSOR STRANG: So this is review session number five, 10 00:00:23,940 --> 00:00:26,530 I guess it is. 11 00:00:26,530 --> 00:00:31,310 And it comes before an exam next Tuesday, 12 00:00:31,310 --> 00:00:34,110 and actually there'll be a further review session number 13 00:00:34,110 --> 00:00:40,250 six on Monday, right before the exam. 14 00:00:40,250 --> 00:00:48,360 So maybe today we would, there's a homework problem set 15 00:00:48,360 --> 00:00:55,490 on Chapter 2, mostly the oscillating masses 16 00:00:55,490 --> 00:01:01,680 and springs and today's lecture that you 17 00:01:01,680 --> 00:01:08,870 see traces of, networks. 18 00:01:08,870 --> 00:01:12,730 And masses and springs also the static case. 19 00:01:12,730 --> 00:01:15,840 So I'm open as always to questions. 20 00:01:15,840 --> 00:01:17,200 Yes please, thank you? 21 00:01:17,200 --> 00:01:19,670 AUDIENCE: 2.2 number six. 22 00:01:19,670 --> 00:01:24,080 PROFESSOR STRANG: 2.2, number six. 23 00:01:24,080 --> 00:01:24,731 OK. 24 00:01:24,731 --> 00:01:25,230 Yeah. 25 00:01:25,230 --> 00:01:32,270 So this is, and of course you understand that, so I'm happy, 26 00:01:32,270 --> 00:01:34,420 that's a good question to discuss. 27 00:01:34,420 --> 00:01:36,720 And maybe number seven people well 28 00:01:36,720 --> 00:01:40,000 have something to say about. 29 00:01:40,000 --> 00:01:42,100 Good. 30 00:01:42,100 --> 00:01:47,310 So that just fine, so let me start right in on those. 31 00:01:47,310 --> 00:01:52,470 So, number six is the fact, I mean 32 00:01:52,470 --> 00:01:54,870 everybody understands that when energy is conserved, 33 00:01:54,870 --> 00:01:57,590 that's an important thing. 34 00:01:57,590 --> 00:02:01,950 And so the question is first when is energy conserved 35 00:02:01,950 --> 00:02:03,400 in the differential equations? 36 00:02:03,400 --> 00:02:05,980 In the equation we're trying to solve? 37 00:02:05,980 --> 00:02:09,540 And if it is, then we want to know, 38 00:02:09,540 --> 00:02:12,900 we would like to choose difference methods that 39 00:02:12,900 --> 00:02:15,710 also conserve energy. 40 00:02:15,710 --> 00:02:20,060 They may not be exactly right, they may not be exactly 41 00:02:20,060 --> 00:02:25,760 at a right point on this circle, if we're in that model problem, 42 00:02:25,760 --> 00:02:28,710 but still on the circle. 43 00:02:28,710 --> 00:02:31,850 So, and the point is that the trapezoidal method does 44 00:02:31,850 --> 00:02:34,350 stay on the circle, and of course the differential 45 00:02:34,350 --> 00:02:36,110 equation stays on the circle. 46 00:02:36,110 --> 00:02:42,660 Can I, so, and I put quite a bit into this problem six. 47 00:02:42,660 --> 00:02:52,440 So this is 2.2.6, and and let me try say something about that, 48 00:02:52,440 --> 00:02:53,610 OK. 49 00:02:53,610 --> 00:02:59,430 So, first of all, there's the continuous problem. du/dt=Au. 50 00:02:59,430 --> 00:03:01,810 When does that conserve energy? 51 00:03:01,810 --> 00:03:03,780 And then there's the discrete problem, 52 00:03:03,780 --> 00:03:08,880 which we know that Euler's doesn't conserve energy, 53 00:03:08,880 --> 00:03:13,660 because we've seen it, the computer shows you right away. 54 00:03:13,660 --> 00:03:17,200 It spirals up from the circle, it 55 00:03:17,200 --> 00:03:20,140 spirals in, or forward and backward, or whatever. 56 00:03:20,140 --> 00:03:22,960 But trapezoidal method, is that the one that turns out 57 00:03:22,960 --> 00:03:24,010 to be well? 58 00:03:24,010 --> 00:03:29,390 OK, so it refers to equation 24 as the trapezoidal method, 59 00:03:29,390 --> 00:03:32,000 and let me try to follow that notation. 60 00:03:32,000 --> 00:03:45,480 Yep. the trapezoidal method is this one. 61 00:03:45,480 --> 00:03:48,220 Did we get music there for the trapezoidal method? 62 00:03:48,220 --> 00:03:54,820 OK. u_(n+1) equal (I+A*delta t/2)u_n. 63 00:03:54,820 --> 00:03:58,760 64 00:03:58,760 --> 00:04:01,230 OK. 65 00:04:01,230 --> 00:04:02,530 Right. 66 00:04:02,530 --> 00:04:03,470 OK. 67 00:04:03,470 --> 00:04:07,560 So that, and actually problem seven, 68 00:04:07,560 --> 00:04:11,110 that you maybe want to discuss too, 69 00:04:11,110 --> 00:04:15,060 is the question of how accurate this is compared 70 00:04:15,060 --> 00:04:16,600 to the differential equation. 71 00:04:16,600 --> 00:04:22,370 Everybody should see that this really came from, 72 00:04:22,370 --> 00:04:28,100 the original way to look at this was (u_(n+1)-u_n)/delta t, 73 00:04:28,100 --> 00:04:30,440 that approximated the derivative, 74 00:04:30,440 --> 00:04:37,000 equals A and then I'm taking half at u-- 75 00:04:37,000 --> 00:04:41,060 at the new time and half at the old time. 76 00:04:41,060 --> 00:04:43,520 So it's got that centering. 77 00:04:43,520 --> 00:04:48,040 That we suspect will give us a little extra accuracy. 78 00:04:48,040 --> 00:04:49,370 OK. 79 00:04:49,370 --> 00:04:50,770 So two questions then. 80 00:04:50,770 --> 00:04:55,980 One was the stability, so problem six 81 00:04:55,980 --> 00:05:06,930 was the energy conserved, and problem 2.2.7, 82 00:05:06,930 --> 00:05:12,050 if I anticipate it, is the order of accuracy. 83 00:05:12,050 --> 00:05:17,290 So these are both topics that are 84 00:05:17,290 --> 00:05:23,960 extremely important in choosing a difference method. 85 00:05:23,960 --> 00:05:28,290 We would like to know first when is energy conserved there? 86 00:05:28,290 --> 00:05:31,570 What differential equations have conserved energy? 87 00:05:31,570 --> 00:05:35,160 Physically we kind of can see them coming. 88 00:05:35,160 --> 00:05:41,730 If a physical universe is not being-- Somehow, 89 00:05:41,730 --> 00:05:44,970 lots of physical problems, we see those masses and springs 90 00:05:44,970 --> 00:05:48,550 oscillating, and we say OK, nothing's 91 00:05:48,550 --> 00:05:51,270 coming in from outside, how could-- Energy 92 00:05:51,270 --> 00:05:56,950 there would be the sum of the kinetic energy of the masses, 93 00:05:56,950 --> 00:05:59,890 and the potential energy in the springs. 94 00:05:59,890 --> 00:06:04,040 So energy passes between kinetic, 95 00:06:04,040 --> 00:06:07,160 when the mass is zooming past equilibrium, 96 00:06:07,160 --> 00:06:13,260 and potential energy, when the mass is stretching the spring. 97 00:06:13,260 --> 00:06:16,030 So we've got two cases. 98 00:06:16,030 --> 00:06:19,540 And we would hope, and this trapezoidal method 99 00:06:19,540 --> 00:06:22,280 comes through, that energy is conserved. 100 00:06:22,280 --> 00:06:27,390 So can I just begin with this one? 101 00:06:27,390 --> 00:06:30,010 Maybe I always ought to say, because you guys are also 102 00:06:30,010 --> 00:06:37,150 thinking about the quiz. 103 00:06:37,150 --> 00:06:40,010 So, for example, this question about how 104 00:06:40,010 --> 00:06:45,870 to find the order of accuracy, I'll speak about that. 105 00:06:45,870 --> 00:06:49,460 but let me just say that's not something 106 00:06:49,460 --> 00:06:52,270 that we've done in enough detail that I would 107 00:06:52,270 --> 00:06:55,220 expect you to be quick on the quiz 108 00:06:55,220 --> 00:06:59,700 and just be able to do it out. 109 00:06:59,700 --> 00:07:04,260 I'll try to choose questions on the exam 110 00:07:04,260 --> 00:07:07,180 that you really have had more practice with. 111 00:07:07,180 --> 00:07:11,700 But this is certainly important, so it was that definitely right 112 00:07:11,700 --> 00:07:13,060 to put on the homework. 113 00:07:13,060 --> 00:07:14,630 And this is important. 114 00:07:14,630 --> 00:07:20,930 OK, so let me tackle this one. 115 00:07:20,930 --> 00:07:23,490 First the differential equation. 116 00:07:23,490 --> 00:07:29,820 So by energy here I'm meaning just the length of u squared. 117 00:07:29,820 --> 00:07:35,700 So now I'm looking at energy conserved, OK. 118 00:07:35,700 --> 00:07:38,310 So I'm hoping energy conserved would 119 00:07:38,310 --> 00:07:44,220 mean that the derivative of u squared was zero. 120 00:07:44,220 --> 00:07:48,280 That's what conserving energy would mean. 121 00:07:48,280 --> 00:07:51,040 And my question is which differential 122 00:07:51,040 --> 00:07:57,490 equations-- What's the condition on A, in other words? 123 00:07:57,490 --> 00:08:00,400 How would I recognize from this matrix A 124 00:08:00,400 --> 00:08:04,390 that I have this interesting property? 125 00:08:04,390 --> 00:08:04,960 OK. 126 00:08:04,960 --> 00:08:09,170 So, let me just show you what I would do. 127 00:08:09,170 --> 00:08:15,340 Another way to write u squared is u transposed u. 128 00:08:15,340 --> 00:08:19,950 These are vectors, of course. 129 00:08:19,950 --> 00:08:23,270 So what's the derivative of u transpose u? 130 00:08:23,270 --> 00:08:26,960 My equation is telling me what the derivative of u is here, 131 00:08:26,960 --> 00:08:28,890 I've got the thing squared. 132 00:08:28,890 --> 00:08:31,720 OK. 133 00:08:31,720 --> 00:08:34,440 I have a product here, right? 134 00:08:34,440 --> 00:08:36,760 I've u's times u's. 135 00:08:36,760 --> 00:08:39,280 So I'm going to use the standard-- 136 00:08:39,280 --> 00:08:43,200 They happen to be vectors, so if I want to use like freshman 137 00:08:43,200 --> 00:08:46,810 calculus, I'd have to get down to the scalar 138 00:08:46,810 --> 00:08:48,680 to get down to the numbers, but I absolutely 139 00:08:48,680 --> 00:08:52,070 could do that and just follow them along, component 140 00:08:52,070 --> 00:08:52,920 by component. 141 00:08:52,920 --> 00:08:57,630 Or I could try to do it a whole column at a time. 142 00:08:57,630 --> 00:09:01,920 And let me try that. 143 00:09:01,920 --> 00:09:04,730 It's going to be the product rule, right? 144 00:09:04,730 --> 00:09:12,130 In some form I'll have this guy times the derivative of this 145 00:09:12,130 --> 00:09:15,100 plus the derivative, I'll keep them 146 00:09:15,100 --> 00:09:20,370 in order, the derivative of this thing, times this guy. 147 00:09:20,370 --> 00:09:22,110 Right? 148 00:09:22,110 --> 00:09:23,500 That's the product rule. 149 00:09:23,500 --> 00:09:26,300 OK, now what do I know here? 150 00:09:26,300 --> 00:09:29,410 I know that the du/dt is Au, right? 151 00:09:29,410 --> 00:09:34,820 So this is u transposed Au. 152 00:09:34,820 --> 00:09:41,440 And I know du, is that dt meant to be dt? 153 00:09:41,440 --> 00:09:46,240 So du/dt is Au, again. 154 00:09:46,240 --> 00:09:54,110 Look, this isn't difficult. It's (Au) transpose u. 155 00:09:54,110 --> 00:09:59,050 And the question is, when is this zero, OK? 156 00:09:59,050 --> 00:10:03,090 So those were the two terms from the product rule, 157 00:10:03,090 --> 00:10:05,500 and notice they're not exactly the same. 158 00:10:05,500 --> 00:10:11,640 This is u transpose Au, and what's this guy? 159 00:10:11,640 --> 00:10:14,710 u transpose A transpose u. 160 00:10:14,710 --> 00:10:16,940 So if I put them together, I have 161 00:10:16,940 --> 00:10:24,660 u transpose times the A and the A transpose times u. 162 00:10:24,660 --> 00:10:29,910 And I'm hoping that this will be zero, for all the solutions 163 00:10:29,910 --> 00:10:31,500 that I've come up with. 164 00:10:31,500 --> 00:10:37,470 So the condition is simply that A plus A-- 165 00:10:37,470 --> 00:10:42,290 we want that to be the zero matrix, A plus A transpose. 166 00:10:42,290 --> 00:10:46,860 In other words, if A transpose is minus A, 167 00:10:46,860 --> 00:10:49,030 that's the good one. 168 00:10:49,030 --> 00:10:50,920 If A transpose is minus A, this is 169 00:10:50,920 --> 00:10:53,580 zero, that's zero, that's zero, that's 170 00:10:53,580 --> 00:10:55,460 zero, energy's conserved. 171 00:10:55,460 --> 00:10:59,340 So the energy is conserved when A transpose is minus 172 00:10:59,340 --> 00:11:08,070 A. It's for the anti-symmetric A's that energy is conserved. 173 00:11:08,070 --> 00:11:12,340 And of course this all makes sense. 174 00:11:12,340 --> 00:11:17,140 What are the special solutions to that differential equation? 175 00:11:17,140 --> 00:11:22,800 The special solutions to this equation are e to the, just, 176 00:11:22,800 --> 00:11:29,120 this is connecting now with things that really are basic. 177 00:11:29,120 --> 00:11:34,270 The special solutions, the pure eigensolutions, 178 00:11:34,270 --> 00:11:37,840 the ones that follow their own paths, 179 00:11:37,840 --> 00:11:41,310 are the e^(lambda*t)x's, right? 180 00:11:41,310 --> 00:11:46,300 Where x is an eigenvector of A, and lambda's an eigenvalue. 181 00:11:46,300 --> 00:11:50,190 Those are the guys, and we expect to have n of them, 182 00:11:50,190 --> 00:11:52,340 and we expect a combination of those 183 00:11:52,340 --> 00:11:55,920 to give us the general solution and to match the boundary 184 00:11:55,920 --> 00:11:56,500 conditions. 185 00:11:56,500 --> 00:11:59,055 So these are the, these n of these guys 186 00:11:59,055 --> 00:12:01,180 with n different eigenvectors and their eigenvalues 187 00:12:01,180 --> 00:12:08,370 are the heart of problems like this. 188 00:12:08,370 --> 00:12:13,430 And of course that's the additional homework problem 189 00:12:13,430 --> 00:12:19,530 that wasn't in the book but I added as an additional problem 190 00:12:19,530 --> 00:12:22,120 was exactly that, to get you to practice 191 00:12:22,120 --> 00:12:24,280 with these eigenvectors and eigenvalues. 192 00:12:24,280 --> 00:12:28,590 OK, now, what's the deal, I want to connect 193 00:12:28,590 --> 00:12:32,860 this energy conserving with this picture of solutions. 194 00:12:32,860 --> 00:12:38,080 When would this keep the same energy? 195 00:12:38,080 --> 00:12:46,110 When would this have constant energy, constant length? 196 00:12:46,110 --> 00:12:50,130 The length would be constant, since x is certainly, 197 00:12:50,130 --> 00:12:52,770 that's an eigenvector, whatever it is. 198 00:12:52,770 --> 00:12:54,670 This is what's changing. 199 00:12:54,670 --> 00:12:58,260 And now I want to know, when does the length not change? 200 00:12:58,260 --> 00:13:02,630 Well, the test would be that this number should 201 00:13:02,630 --> 00:13:07,570 have absolute value one, right? 202 00:13:07,570 --> 00:13:12,030 If this keeps absolute value one, then 203 00:13:12,030 --> 00:13:16,710 in the eigenvalue picture I have energy staying the same. 204 00:13:16,710 --> 00:13:17,420 OK? 205 00:13:17,420 --> 00:13:22,130 Now, when will this have magnitude one? 206 00:13:22,130 --> 00:13:26,700 Time is running along, this is e to the lambda t, so which 207 00:13:26,700 --> 00:13:28,680 lambdas? 208 00:13:28,680 --> 00:13:31,850 Zero, certainly, but now there's more, 209 00:13:31,850 --> 00:13:33,660 you gotta know the others. 210 00:13:33,660 --> 00:13:37,210 What other lambdas will have, what 211 00:13:37,210 --> 00:13:40,990 other lambdas give me this thing stays on the unit 212 00:13:40,990 --> 00:13:43,360 circle, absolute value one? 213 00:13:43,360 --> 00:13:48,800 Key question you must know. lambda could be? 214 00:13:48,800 --> 00:13:51,550 Imaginary. lambda could be imaginary; 215 00:13:51,550 --> 00:13:53,767 right, lambda could be imaginary. 216 00:13:53,767 --> 00:13:54,350 e^(i*omega*t). 217 00:13:54,350 --> 00:13:59,150 218 00:13:59,150 --> 00:14:02,110 That's just like basic fact about complex numbers, 219 00:14:02,110 --> 00:14:07,000 that if lambda's imaginary we would have cosine of something 220 00:14:07,000 --> 00:14:09,980 t plus i times the sine of something t, 221 00:14:09,980 --> 00:14:12,350 cos squared plus sine squared being one, 222 00:14:12,350 --> 00:14:13,920 we'd be on the unit circle. 223 00:14:13,920 --> 00:14:18,020 So from this picture we would want the lambdas 224 00:14:18,020 --> 00:14:20,920 to be pure imaginary. 225 00:14:20,920 --> 00:14:24,640 And now a little next step, what we'd 226 00:14:24,640 --> 00:14:32,320 like for eigenvectors, because the real solution will not 227 00:14:32,320 --> 00:14:37,180 be just one of these guys but a combination. 228 00:14:37,180 --> 00:14:41,620 So when we have a combination each one is doing its thing, 229 00:14:41,620 --> 00:14:44,540 each one better have lambda imaginary 230 00:14:44,540 --> 00:14:50,770 but more than that, we would want the x's to be 231 00:14:50,770 --> 00:14:51,650 perpendicular. 232 00:14:51,650 --> 00:14:57,890 Because if the x's interact, then this guy, one of these, 233 00:14:57,890 --> 00:14:59,610 you will say there's only one there, 234 00:14:59,610 --> 00:15:02,330 but I'm thinking of n of them there. 235 00:15:02,330 --> 00:15:04,930 A combination of say, two of them. 236 00:15:04,930 --> 00:15:07,970 Suppose I have an e^(lambda_1 t)*x_1, 237 00:15:07,970 --> 00:15:13,520 and an e^(lambda_2*t)*x_2, when does that conserve energy? 238 00:15:13,520 --> 00:15:20,650 Well, each one will, but the combination will be fine 239 00:15:20,650 --> 00:15:22,570 if the x's are perpendicular. 240 00:15:22,570 --> 00:15:24,920 Because if I have perpendicular vectors, 241 00:15:24,920 --> 00:15:29,080 then the length of the whole combination by Pythagoras 242 00:15:29,080 --> 00:15:30,960 is just one squared and the other squared 243 00:15:30,960 --> 00:15:34,430 and each of those pieces is constant. 244 00:15:34,430 --> 00:15:36,910 Let me say what I'm trying to say. 245 00:15:36,910 --> 00:15:40,270 That the eigenvalue, eigenfunction picture 246 00:15:40,270 --> 00:15:45,290 also tells us that we would like imaginary eigenvalues 247 00:15:45,290 --> 00:15:48,070 and perpendicular eigenvectors. 248 00:15:48,070 --> 00:15:53,010 And that is exactly what you get from A transpose equal minus A. 249 00:15:53,010 --> 00:15:57,630 So A transpose equal minus A is exactly, 250 00:15:57,630 --> 00:16:01,130 those matrices, anti-symmetric matrices, 251 00:16:01,130 --> 00:16:06,050 have perpendicular eigenvectors, just like symmetric, 252 00:16:06,050 --> 00:16:09,290 but the eigenvalues are pure imaginary. 253 00:16:09,290 --> 00:16:11,910 Instead of all being real, they're all pure imaginary. 254 00:16:11,910 --> 00:16:15,820 In other words, that answer and the discussion 255 00:16:15,820 --> 00:16:18,840 here came to the same conclusion, 256 00:16:18,840 --> 00:16:21,290 that A should be anti-symmetric. 257 00:16:21,290 --> 00:16:22,560 OK. 258 00:16:22,560 --> 00:16:27,460 Now let me look at-- Is that OK, so that's 259 00:16:27,460 --> 00:16:31,340 a discussion which is worth knowing 260 00:16:31,340 --> 00:16:33,960 about differential equations. 261 00:16:33,960 --> 00:16:35,610 When is energy conserved. 262 00:16:35,610 --> 00:16:38,250 Now I want to do, or the problem asks 263 00:16:38,250 --> 00:16:40,950 me to do, what about this difference equation? 264 00:16:40,950 --> 00:16:43,290 When is energy conserved there? 265 00:16:43,290 --> 00:16:50,690 And I believe it will be, this is 266 00:16:50,690 --> 00:16:52,830 the requirement for the differential equation 267 00:16:52,830 --> 00:16:57,340 to be OK, to conserve energy and so I'm going to expect, 268 00:16:57,340 --> 00:16:59,780 I'm going to need that in this one. 269 00:16:59,780 --> 00:17:01,340 And is that enough? 270 00:17:01,340 --> 00:17:06,070 If I have this A transpose equal minus A, anti-symmetric, 271 00:17:06,070 --> 00:17:10,460 it was good for this, does it also do the job here? 272 00:17:10,460 --> 00:17:14,240 Is this trapezoidal method just cool 273 00:17:14,240 --> 00:17:18,140 so that it will conserve energy, too. 274 00:17:18,140 --> 00:17:20,920 And the answer, I think, is yes, and the problem 275 00:17:20,920 --> 00:17:24,850 was to prove it, or to see why. 276 00:17:24,850 --> 00:17:28,230 So what do I now want? 277 00:17:28,230 --> 00:17:29,910 If you don't mind my erasing, I'm 278 00:17:29,910 --> 00:17:35,190 now going to look at the discrete guy. 279 00:17:35,190 --> 00:17:40,390 So now I'm looking at when is u_(n+1) squared equal u_n 280 00:17:40,390 --> 00:17:43,450 squared? 281 00:17:43,450 --> 00:17:47,620 That's what I mean by conserving energy in the discrete case. 282 00:17:47,620 --> 00:17:50,290 At every step, same energy. 283 00:17:50,290 --> 00:17:53,780 So now I want to look at the energy in u_(n+1) compared 284 00:17:53,780 --> 00:17:59,590 to the energy in u_n, and I want to see that this holds. 285 00:17:59,590 --> 00:18:08,240 Probably, there's some smart way to do that. 286 00:18:08,240 --> 00:18:12,200 Now we're down to just the math questions. 287 00:18:12,200 --> 00:18:16,650 Math is always looking for some, you just sort of do 288 00:18:16,650 --> 00:18:21,530 the right thing, you stand back and poof it works. 289 00:18:21,530 --> 00:18:23,110 OK, so what's the right thing? 290 00:18:23,110 --> 00:18:25,860 Hopefully I have helped you and me 291 00:18:25,860 --> 00:18:30,480 by saying what would be a good idea to do here. 292 00:18:30,480 --> 00:18:35,160 Can I just look? 293 00:18:35,160 --> 00:18:41,660 OK, it say, oh, does it say what to do? 294 00:18:41,660 --> 00:18:42,190 Yeah. 295 00:18:42,190 --> 00:18:45,640 It says multiply by u_(n+1)+u_n. 296 00:18:45,640 --> 00:18:48,480 297 00:18:48,480 --> 00:18:53,010 Take the dot product, that's interesting. 298 00:18:53,010 --> 00:18:55,920 Take the dot product, so why did that work? 299 00:18:55,920 --> 00:18:59,480 Take the dot product of both sides with u_(n+1)-u_n. 300 00:18:59,480 --> 00:19:02,630 301 00:19:02,630 --> 00:19:06,060 Did anybody succeed with this idea? 302 00:19:06,060 --> 00:19:07,700 But that's the idea. 303 00:19:07,700 --> 00:19:11,550 And hopefully we'll get it to work. 304 00:19:11,550 --> 00:19:16,890 That if I multiply both sides by u_(n+1), oh, no, plus u_n. 305 00:19:16,890 --> 00:19:20,260 Maybe better if I look at it this way. 306 00:19:20,260 --> 00:19:23,690 I'm sort of OK to do it that way. 307 00:19:23,690 --> 00:19:25,440 Suppose I multiply both sides. 308 00:19:25,440 --> 00:19:30,240 So now I'm following on this idea, That equation 309 00:19:30,240 --> 00:19:34,660 I've rewritten here and without practice I don't know which one 310 00:19:34,660 --> 00:19:36,780 is the good one to start with. 311 00:19:36,780 --> 00:19:41,240 But I'm pretty OK with starting with this one. 312 00:19:41,240 --> 00:19:43,300 So what's my idea? 313 00:19:43,300 --> 00:19:47,010 That's my equation, now I'm going to multiply both sides 314 00:19:47,010 --> 00:19:51,900 by (u_(n+1)+u_n) transpose. 315 00:19:51,900 --> 00:19:54,400 Now I don't have room to do it, unfortunately. 316 00:19:54,400 --> 00:19:59,110 I want to stick in here u_(n+...) with a plus sign 317 00:19:59,110 --> 00:20:03,170 in there, and of course I have to do the same thing here. 318 00:20:03,170 --> 00:20:04,440 OK. 319 00:20:04,440 --> 00:20:08,000 Are you OK, do you see what I'm doing? 320 00:20:08,000 --> 00:20:10,400 I want to show that this equation, which 321 00:20:10,400 --> 00:20:15,510 is the same as this equation, has this property which 322 00:20:15,510 --> 00:20:18,320 is a copy of this property. 323 00:20:18,320 --> 00:20:21,550 Here would be another way to do it. 324 00:20:21,550 --> 00:20:25,420 We could do it, the way we're going 325 00:20:25,420 --> 00:20:29,240 to do it now sort of compares with the way we started 326 00:20:29,240 --> 00:20:31,780 with the derivative of the norm squared. 327 00:20:31,780 --> 00:20:34,810 I could also ask the same question 328 00:20:34,810 --> 00:20:37,960 by following eigenvectors. 329 00:20:37,960 --> 00:20:39,560 I could also ask the same question 330 00:20:39,560 --> 00:20:41,900 by following eigenvectors. 331 00:20:41,900 --> 00:20:51,110 I'm guessing that here the eigenvalues-- u_(n+1) is, 332 00:20:51,110 --> 00:20:53,570 you see I could do it both ways. 333 00:20:53,570 --> 00:20:55,870 Maybe having just done eigenvectors 334 00:20:55,870 --> 00:20:57,680 let me do this one by eigenvectors. 335 00:20:57,680 --> 00:21:04,010 So an eigenvector of A, when it's x itself, 336 00:21:04,010 --> 00:21:06,320 is, what happens to an eigenvector. 337 00:21:06,320 --> 00:21:12,280 Suppose u_0 is an eigenvector x of A, What's u_1? 338 00:21:12,280 --> 00:21:15,570 Yeah, you really should see this question. 339 00:21:15,570 --> 00:21:26,580 So u_0 is the eigenvector x, then what is u_1? 340 00:21:26,580 --> 00:21:28,730 Let me just write it here. 341 00:21:28,730 --> 00:21:31,530 Ax equaling lambda*x. 342 00:21:31,530 --> 00:21:33,720 So these are the eigenvalues of A, 343 00:21:33,720 --> 00:21:40,680 and we've learned that they're pure imaginary in this case 344 00:21:40,680 --> 00:21:42,920 when we're ready to go, and now I'd 345 00:21:42,920 --> 00:21:48,240 like to know that we get the good thing here. 346 00:21:48,240 --> 00:21:55,380 OK, so if u_n is an eigenvector, what is u_(n+1)? 347 00:21:55,380 --> 00:22:00,520 OK, so can I just do that? u_(n+1) is, 348 00:22:00,520 --> 00:22:10,990 so what do I have on that right hand side? x and what is Ax? 349 00:22:10,990 --> 00:22:12,590 It's lambda, right? 350 00:22:12,590 --> 00:22:14,130 It's lambda*x. 351 00:22:14,130 --> 00:22:22,860 So all this is one plus lambda delta t on two x 352 00:22:22,860 --> 00:22:27,560 but now I've also got to bring this guy 353 00:22:27,560 --> 00:22:30,720 over here, its inverse. 354 00:22:30,720 --> 00:22:33,810 And see what that does. 355 00:22:33,810 --> 00:22:36,410 Now it's the inverse, so it's going 356 00:22:36,410 --> 00:22:40,330 to have the same eigenvector and the eigenvalue's 357 00:22:40,330 --> 00:22:42,290 going to go in the denominator and it'll 358 00:22:42,290 --> 00:22:50,090 be one minus lambda delta t over two. 359 00:22:50,090 --> 00:22:53,280 OK, so that's u_(n+1). 360 00:22:53,280 --> 00:22:56,840 Do you see what's happening here? 361 00:22:56,840 --> 00:23:02,020 The eigenvector x, if we start with that eigenvector x, 362 00:23:02,020 --> 00:23:04,190 we come out with a multiple of x. 363 00:23:04,190 --> 00:23:06,800 And this is the multiple. 364 00:23:06,800 --> 00:23:11,810 So each finite difference step multiplies by a number 365 00:23:11,810 --> 00:23:15,440 just the way each, in the continuous case 366 00:23:15,440 --> 00:23:17,940 we were multiplying by e to the lambda t 367 00:23:17,940 --> 00:23:21,760 and in the discrete step by step case 368 00:23:21,760 --> 00:23:24,270 we're multiplying by that number. 369 00:23:24,270 --> 00:23:28,670 Actually, this is why problem seven is important, 370 00:23:28,670 --> 00:23:33,980 because if we want to know how accurate the comparison is I 371 00:23:33,980 --> 00:23:39,800 want to compare e to the lambda t with that number. 372 00:23:39,800 --> 00:23:47,150 So problem six is asking a question about that ratio. 373 00:23:47,150 --> 00:23:49,790 And problem seven is asking another question 374 00:23:49,790 --> 00:23:51,770 about that very same ratio. 375 00:23:51,770 --> 00:23:54,430 Now what's the question for problem six? 376 00:23:54,430 --> 00:24:01,630 When will this vector have the same length as-- This 377 00:24:01,630 --> 00:24:08,180 x was u_n. 378 00:24:08,180 --> 00:24:14,330 So I started with the u_n, I multiplied by this number 379 00:24:14,330 --> 00:24:19,750 to get u_(n+1), when do they have the same length? 380 00:24:19,750 --> 00:24:27,210 When that number has absolute value one. 381 00:24:27,210 --> 00:24:31,680 So if I'm watching eigenvectors, this guy 382 00:24:31,680 --> 00:24:35,550 had absolute value one because lambda was imaginary. 383 00:24:35,550 --> 00:24:37,570 Now, what about this guy? lambda's 384 00:24:37,570 --> 00:24:40,110 still that same lambda, imaginary. 385 00:24:40,110 --> 00:24:43,130 What can you tell me about one plus, 386 00:24:43,130 --> 00:24:48,850 so lambda is some i*omega, delta t over two and down here I have 387 00:24:48,850 --> 00:24:54,200 one minus i*omega, that's the lambda, delta t over two. 388 00:24:54,200 --> 00:25:00,460 I believe that that does have absolute value one. 389 00:25:00,460 --> 00:25:02,620 Anybody tell me why? 390 00:25:02,620 --> 00:25:07,980 So this is checking that energy is 391 00:25:07,980 --> 00:25:11,580 conserved for each eigenvector. 392 00:25:11,580 --> 00:25:15,090 The energy-- Because the eigenvector is multiplied 393 00:25:15,090 --> 00:25:18,290 by that number and that's some number, 394 00:25:18,290 --> 00:25:20,810 it's some complex number, but I believe 395 00:25:20,810 --> 00:25:24,080 it has absolute value one and I believe you can tell me why. 396 00:25:24,080 --> 00:25:25,740 Yep. 397 00:25:25,740 --> 00:25:28,450 Because they're complex conjugates. 398 00:25:28,450 --> 00:25:31,480 This numerator and the denominator are complex 399 00:25:31,480 --> 00:25:38,260 conjugates, in the complex plane here's the one, 400 00:25:38,260 --> 00:25:43,380 and I go up by i*omega*delta t over two, 401 00:25:43,380 --> 00:25:48,780 or on this one I go down by-- Bu those lengths are the same. 402 00:25:48,780 --> 00:25:51,110 That numerator, the length of the numerator 403 00:25:51,110 --> 00:25:55,740 is that guy, the length of the denominator is this guy, 404 00:25:55,740 --> 00:25:58,450 and their ratio is one. 405 00:25:58,450 --> 00:26:01,970 So I think that this gives us the point 406 00:26:01,970 --> 00:26:04,310 about complex numbers. 407 00:26:04,310 --> 00:26:09,010 That a complex number and its conjugate 408 00:26:09,010 --> 00:26:14,270 automatically have ratio of magnitude one. 409 00:26:14,270 --> 00:26:17,550 You see the difference between Euler's method. 410 00:26:17,550 --> 00:26:24,450 So Euler's method, so forward Euler-- Forward Euler 411 00:26:24,450 --> 00:26:31,560 would not have had this stuff on the left side. 412 00:26:31,560 --> 00:26:33,790 It would all have been on the right-hand side. 413 00:26:33,790 --> 00:26:38,790 Forward Euler would have been about i plus A*delta t. 414 00:26:38,790 --> 00:26:42,670 Delta t A. And what are its eigenvalues? 415 00:26:42,670 --> 00:26:48,410 One plus i omega delta t, right? 416 00:26:48,410 --> 00:26:54,600 With no, we're not dividing by anybody. 417 00:26:54,600 --> 00:26:59,030 This part is up top too, so it's one plus i*omega*delta t. 418 00:26:59,030 --> 00:27:02,990 Now, does that have absolute value one? 419 00:27:02,990 --> 00:27:05,480 Well, you know from the way I'm asking the question, what 420 00:27:05,480 --> 00:27:07,280 can you tell me about the absolute value 421 00:27:07,280 --> 00:27:11,310 of the forward Euler growth factor? 422 00:27:11,310 --> 00:27:13,400 Greater than one. 423 00:27:13,400 --> 00:27:17,310 Because this is the one, and this is the i*omega*delta t, 424 00:27:17,310 --> 00:27:19,550 maybe went up twice as far. 425 00:27:19,550 --> 00:27:21,980 And there was nobody to divide by. 426 00:27:21,980 --> 00:27:24,290 It's bigger than one, so it blows up. 427 00:27:24,290 --> 00:27:29,850 And the backward Euler had only the one over one minus 428 00:27:29,850 --> 00:27:37,050 i*omega*delta t, so the backward was like this, one over it. 429 00:27:37,050 --> 00:27:37,980 And less than one. 430 00:27:37,980 --> 00:27:41,960 But this balance has absolute value equal one. 431 00:27:41,960 --> 00:27:47,230 So, OK, that's the sort of heart of what's going on. 432 00:27:47,230 --> 00:27:54,270 Can I, before I tackle the question using the hint there, 433 00:27:54,270 --> 00:27:57,180 which would take me on another blackboard, 434 00:27:57,180 --> 00:28:00,180 can I discuss question seven? 435 00:28:00,180 --> 00:28:02,570 Were you going to ask me about number seven? 436 00:28:02,570 --> 00:28:02,750 AUDIENCE: Yeah, I was. 437 00:28:02,750 --> 00:28:03,000 PROFESSOR STRANG: You were? 438 00:28:03,000 --> 00:28:03,820 OK. 439 00:28:03,820 --> 00:28:04,670 Alright. 440 00:28:04,670 --> 00:28:07,840 We get the answer. 441 00:28:07,840 --> 00:28:12,280 So, question seven is about the accuracy. 442 00:28:12,280 --> 00:28:18,350 So here's the correct number, this is my e^(i*omega*t), 443 00:28:18,350 --> 00:28:24,000 that's the correct number that I should be multiplying by. 444 00:28:24,000 --> 00:28:30,080 And the actual number that I'm multiplying by is that much. 445 00:28:30,080 --> 00:28:34,680 Or, in the forward Euler case, it's that one. 446 00:28:34,680 --> 00:28:40,000 And so I'm comparing the one step accuracy. 447 00:28:40,000 --> 00:28:44,580 So let me compare one step accuracy. 448 00:28:44,580 --> 00:28:49,100 So this is the topic now, of order of accuracy. 449 00:28:49,100 --> 00:28:52,140 This is question seven. 450 00:28:52,140 --> 00:29:01,520 And it amounts to comparing the-- So what is one delta t 451 00:29:01,520 --> 00:29:03,820 step in the continuous case? 452 00:29:03,820 --> 00:29:06,600 So how much does the eigenvector x, 453 00:29:06,600 --> 00:29:11,330 what does it get multiplied by if I take a delta t 454 00:29:11,330 --> 00:29:14,530 step in the differential equation? 455 00:29:14,530 --> 00:29:17,660 So this is the exact delta t step, 456 00:29:17,660 --> 00:29:20,740 what the finite difference won't get exactly right. 457 00:29:20,740 --> 00:29:26,740 So the exact step delta t. 458 00:29:26,740 --> 00:29:36,660 The differential equation, and of course I'm always looking 459 00:29:36,660 --> 00:29:46,690 at Ax=lambda*x, the differential equation multiplies x by what? 460 00:29:46,690 --> 00:29:52,910 What's the exact growth factor, you could say, 461 00:29:52,910 --> 00:29:57,620 if my equation is du/dt=Au, that's the differential 462 00:29:57,620 --> 00:30:03,810 equation, and I'm supposing that I'm on an eigenvector x, 463 00:30:03,810 --> 00:30:09,630 so that the solution is e^(i*omega*t), 464 00:30:09,630 --> 00:30:12,440 or e^(i*lambda*x). 465 00:30:12,440 --> 00:30:19,010 Now, what happened over a delta t step? 466 00:30:19,010 --> 00:30:23,340 This is the answer like running along for all time, 467 00:30:23,340 --> 00:30:26,380 all I'm asking you to do is if the step is 468 00:30:26,380 --> 00:30:30,220 delta t, what's that number? 469 00:30:30,220 --> 00:30:34,310 I mean that number is telling us how much it grew in that delta 470 00:30:34,310 --> 00:30:37,380 t step, and of course it's e^(i*omega*delta t). 471 00:30:37,380 --> 00:30:41,420 472 00:30:41,420 --> 00:30:44,770 That's the exact growth factor, that's G_exact. 473 00:30:44,770 --> 00:30:48,350 474 00:30:48,350 --> 00:30:50,760 In one time step, the eigenvector 475 00:30:50,760 --> 00:30:54,310 gets multiplied by that, because that's the amount of time that 476 00:30:54,310 --> 00:30:55,330 elapsed. 477 00:30:55,330 --> 00:30:58,610 And what's the approximate growth, 478 00:30:58,610 --> 00:31:03,900 the growth factor from trapezoidal 479 00:31:03,900 --> 00:31:06,110 is just what we wrote down here. 480 00:31:06,110 --> 00:31:14,510 One plus lambda*delta t, maybe I'll stay with lambda rather 481 00:31:14,510 --> 00:31:15,110 than i*omega. 482 00:31:15,110 --> 00:31:19,320 483 00:31:19,320 --> 00:31:25,010 e^(lambda*delta t), and this was one plus delta t over two 484 00:31:25,010 --> 00:31:33,320 lambda, divided by one minus delta t over two lambda. 485 00:31:33,320 --> 00:31:42,560 So question seven just says compare that with that. 486 00:31:42,560 --> 00:31:48,380 Thinking of delta t as a small time step, if delta t is zero, 487 00:31:48,380 --> 00:31:51,200 then of course e to the zero is one, 488 00:31:51,200 --> 00:31:53,270 if delta t is zero I get one here, 489 00:31:53,270 --> 00:31:59,180 they're correct if delta t is zero, that's no big deal. 490 00:31:59,180 --> 00:32:09,240 How do I understand what happens for small delta t? 491 00:32:09,240 --> 00:32:13,690 I'm comparing this exponential for a small delta t 492 00:32:13,690 --> 00:32:15,980 with this guy for a small delta t. 493 00:32:15,980 --> 00:32:19,230 How do you make comparisons for a small delta t? 494 00:32:19,230 --> 00:32:22,730 Well, that's what Taylor series is all about. 495 00:32:22,730 --> 00:32:24,150 Let's do the Taylor series. 496 00:32:24,150 --> 00:32:27,870 What's the series for the exponential? 497 00:32:27,870 --> 00:32:32,790 If delta t is small, I have e to some little number, 498 00:32:32,790 --> 00:32:40,220 tell me, start me out on the exponential. 499 00:32:40,220 --> 00:32:49,010 One, thanks, one plus, lambda*delta t plus, 500 00:32:49,010 --> 00:32:51,700 this is the exponential series, there are only two series 501 00:32:51,700 --> 00:32:54,100 in this world that are worth knowing. 502 00:32:54,100 --> 00:32:55,840 Really, that's literally true. 503 00:32:55,840 --> 00:32:59,600 In calculus you study all these infinite series, 504 00:32:59,600 --> 00:33:01,400 there are two that are important, 505 00:33:01,400 --> 00:33:04,150 that are worth remembering long after calculus. 506 00:33:04,150 --> 00:33:09,600 And e to the x, e to the whatever, is one of them. 507 00:33:09,600 --> 00:33:13,070 OK, what's the next term? 508 00:33:13,070 --> 00:33:17,980 Over two, lambda delta t squared over two, 509 00:33:17,980 --> 00:33:21,880 and then there's a cube guy if you don't mind telling me 510 00:33:21,880 --> 00:33:25,180 what's the denominator in that one? 511 00:33:25,180 --> 00:33:26,750 It's three factorial six. 512 00:33:26,750 --> 00:33:27,250 Good. 513 00:33:27,250 --> 00:33:28,040 And onward. 514 00:33:28,040 --> 00:33:29,570 OK. 515 00:33:29,570 --> 00:33:32,570 So that's one of the series that everybody should know. 516 00:33:32,570 --> 00:33:37,540 OK, how we going to deal with this guy? 517 00:33:37,540 --> 00:33:40,930 We want to expand that, so what's my goal? 518 00:33:40,930 --> 00:33:45,200 I want you to expand that in powers of lambda*delta t 519 00:33:45,200 --> 00:33:46,960 and compare with this. 520 00:33:46,960 --> 00:33:51,980 And see where, they aren't going to be equal, right? 521 00:33:51,980 --> 00:33:54,800 At some point they're going to be different. 522 00:33:54,800 --> 00:33:57,160 But at least they should start out equal. 523 00:33:57,160 --> 00:34:03,550 So so here's the heart of problem seven. 524 00:34:03,550 --> 00:34:08,400 How do I expand this in powers of delta t? 525 00:34:08,400 --> 00:34:10,030 Do you mind if I just, this is just 526 00:34:10,030 --> 00:34:17,650 a number let me put it times one over, so this is times one 527 00:34:17,650 --> 00:34:23,410 minus delta t over two lambda, inverse right? 528 00:34:23,410 --> 00:34:25,930 I just bring that up as a number. 529 00:34:25,930 --> 00:34:32,770 So it's this guy times one over this guy. 530 00:34:32,770 --> 00:34:34,330 What do I do? 531 00:34:34,330 --> 00:34:46,480 This is, here's the moment when the math tools get used. 532 00:34:46,480 --> 00:34:51,040 And I'm well aware that it's like years 533 00:34:51,040 --> 00:34:55,480 since you did calculus or series or whatever, 534 00:34:55,480 --> 00:34:59,170 and those tools get rusty. 535 00:34:59,170 --> 00:35:02,510 And the point is that they're really genuine tools 536 00:35:02,510 --> 00:35:05,060 that we can now use. 537 00:35:05,060 --> 00:35:08,250 So what do you think? 538 00:35:08,250 --> 00:35:11,500 This is the problem one, this is the one coming from 539 00:35:11,500 --> 00:35:15,770 the denominator; this is 1/(1-x). 540 00:35:15,770 --> 00:35:19,560 So I have a 1/(1-x) deal. 541 00:35:19,560 --> 00:35:24,750 And what's the series for that? 542 00:35:24,750 --> 00:35:26,870 I said there were two series worth remembering, 543 00:35:26,870 --> 00:35:29,650 and sure enough the exponential was one of them 544 00:35:29,650 --> 00:35:32,060 and now we're ready for the other one. 545 00:35:32,060 --> 00:35:35,990 What's the series for that guy? 546 00:35:35,990 --> 00:35:39,530 1+x, good start. 547 00:35:39,530 --> 00:35:45,370 Plus x squared. 548 00:35:45,370 --> 00:35:49,820 Right, x squared plus x cubed and so on. 549 00:35:49,820 --> 00:35:51,530 Real simple. 550 00:35:51,530 --> 00:35:54,610 It's all the same stuff with no factorials. 551 00:35:54,610 --> 00:35:57,360 Those are the two series to know. 552 00:35:57,360 --> 00:36:00,320 The exponential series and the geometric series. 553 00:36:00,320 --> 00:36:03,710 Right, that's the geometric series. 554 00:36:03,710 --> 00:36:07,020 OK, so that's what I've got out of this stuff. 555 00:36:07,020 --> 00:36:08,030 Can I write it below? 556 00:36:08,030 --> 00:36:12,490 I have one plus delta t over two lambda. 557 00:36:12,490 --> 00:36:15,270 Let me just call that x for the moment. 558 00:36:15,270 --> 00:36:17,920 Delta t over two lambda is my x. 559 00:36:17,920 --> 00:36:24,190 One plus x, and this is 1/(1-x), which you just told me is one 560 00:36:24,190 --> 00:36:29,730 plus x plus x squared plus x cubed and so on. 561 00:36:29,730 --> 00:36:33,370 And now I've got to do that multiplication. 562 00:36:33,370 --> 00:36:40,020 OK, x is, remember this is x, I'm just saving space. 563 00:36:40,020 --> 00:36:43,580 Can you multiply those guys? 564 00:36:43,580 --> 00:36:47,650 So that's one plus x times a lot of stuff here. 565 00:36:47,650 --> 00:36:49,150 What do I have all together? 566 00:36:49,150 --> 00:36:53,210 Well, the one, what's the next term? 567 00:36:53,210 --> 00:36:54,270 Two x's? 568 00:36:54,270 --> 00:36:57,430 Everybody spots the two x's there? 569 00:36:57,430 --> 00:37:03,040 And then the next term, you have to get these terms right 570 00:37:03,040 --> 00:37:05,380 because we plan to compare with this guy 571 00:37:05,380 --> 00:37:07,710 and see how many we get. 572 00:37:07,710 --> 00:37:10,650 How many x squareds are in there? 573 00:37:10,650 --> 00:37:12,100 Is it two? 574 00:37:12,100 --> 00:37:13,110 Looks like two. 575 00:37:13,110 --> 00:37:14,810 Two x squareds. 576 00:37:14,810 --> 00:37:16,370 And two x cubes, and so on. 577 00:37:16,370 --> 00:37:18,710 Yeah, that looks right, OK. 578 00:37:18,710 --> 00:37:22,670 Now I'm ready, what am I ready for? 579 00:37:22,670 --> 00:37:28,230 I'm ready to say what x is, x is this delta t over two lambda. 580 00:37:28,230 --> 00:37:29,850 So what have I got here, one? 581 00:37:29,850 --> 00:37:32,190 What is this guy now? 582 00:37:32,190 --> 00:37:37,540 Two x's is delta t lambda. 583 00:37:37,540 --> 00:37:39,180 Is this good? 584 00:37:39,180 --> 00:37:40,280 Yes, right? 585 00:37:40,280 --> 00:37:41,420 We're pleased. 586 00:37:41,420 --> 00:37:45,880 Because the two x is the, two of these 587 00:37:45,880 --> 00:37:49,250 is delta t lambda and that's what we wanted to match. 588 00:37:49,250 --> 00:37:50,090 Absolutely. 589 00:37:50,090 --> 00:37:52,520 Delta t lambda, lambda delta t. 590 00:37:52,520 --> 00:37:54,400 Now let's keep going. 591 00:37:54,400 --> 00:37:57,860 By the way if this first term hadn't matched 592 00:37:57,860 --> 00:38:00,770 we would be extremely surprised. 593 00:38:00,770 --> 00:38:04,730 Because that first matching is only 594 00:38:04,730 --> 00:38:08,760 saying that my difference equation is quite consistent, 595 00:38:08,760 --> 00:38:14,610 it's a reasonable creation out of the differential equation. 596 00:38:14,610 --> 00:38:16,230 And we knew that. 597 00:38:16,230 --> 00:38:19,490 The question is how much further are we going to get? 598 00:38:19,490 --> 00:38:21,900 Euler will not get any further. 599 00:38:21,900 --> 00:38:24,550 With Euler the next ones will fail. 600 00:38:24,550 --> 00:38:28,240 But I think with trapezoidal the next ones are going to work. 601 00:38:28,240 --> 00:38:31,750 Does it work? 602 00:38:31,750 --> 00:38:35,170 It's like we're holding our breath, right? 603 00:38:35,170 --> 00:38:37,380 Two now, I'm going to put in x squared 604 00:38:37,380 --> 00:38:41,490 and see about this term. x is what? 605 00:38:41,490 --> 00:38:45,640 x is this guy, delta t over two lambda. 606 00:38:45,640 --> 00:38:49,230 Delta t lambda over two, squared. 607 00:38:49,230 --> 00:38:52,490 And now you get the fun. 608 00:38:52,490 --> 00:38:56,930 Because you're going to compare this term with what? 609 00:38:56,930 --> 00:39:00,530 With this term. 610 00:39:00,530 --> 00:39:03,190 And are they the same? 611 00:39:03,190 --> 00:39:04,010 Yes. 612 00:39:04,010 --> 00:39:05,100 Yes. 613 00:39:05,100 --> 00:39:07,790 So that's the way, you see, that you 614 00:39:07,790 --> 00:39:12,380 got the extra accuracy which Euler did not give you, 615 00:39:12,380 --> 00:39:14,820 but that's why the trapezoidal rule is 616 00:39:14,820 --> 00:39:17,970 a is a second order accurate method. 617 00:39:17,970 --> 00:39:29,270 OK, you may say that I went overboard to say all that. 618 00:39:29,270 --> 00:39:31,690 You may say I didn't ask that question. 619 00:39:31,690 --> 00:39:36,120 But it's the right question to ask about order of accuracy, 620 00:39:36,120 --> 00:39:40,740 and it's what problem seven was intending to bring. 621 00:39:40,740 --> 00:39:50,140 Maybe I called it h in problem seven rather than x here. 622 00:39:50,140 --> 00:39:52,260 Well. 623 00:39:52,260 --> 00:39:55,660 Oh gosh, I realize I I'm supposed 624 00:39:55,660 --> 00:39:57,510 to come back to this one. 625 00:39:57,510 --> 00:39:59,880 But some people might have other problems 626 00:39:59,880 --> 00:40:01,330 that they're interested in. 627 00:40:01,330 --> 00:40:06,340 But let me, because time is pushing along, 628 00:40:06,340 --> 00:40:09,260 and the solution to this one we'll post, 629 00:40:09,260 --> 00:40:12,220 let me at least offer the possibility to ask me 630 00:40:12,220 --> 00:40:15,830 about something completely, not six or seven here, 631 00:40:15,830 --> 00:40:17,860 but something entirely different, 632 00:40:17,860 --> 00:40:20,920 like what's the first question on the quiz or anything. 633 00:40:20,920 --> 00:40:32,510 And that, let me say I'll hope to know by Tuesday. 634 00:40:32,510 --> 00:40:37,260 I love to teach, but making up exams is serious work. 635 00:40:37,260 --> 00:40:39,430 Anyway. 636 00:40:39,430 --> 00:40:46,100 Let me open a board and open to another question of any sort. 637 00:40:46,100 --> 00:40:50,540 Any place, Chapter 1, Chapter 2, whatever. 638 00:40:50,540 --> 00:40:52,870 Is there anything? 639 00:40:52,870 --> 00:40:56,610 So I know that you're in the middle of this homework. 640 00:40:56,610 --> 00:41:03,220 So I can say a little more here about 641 00:41:03,220 --> 00:41:07,720 that number six if you want, but I wanted to allow, yep. 642 00:41:07,720 --> 00:41:14,580 AUDIENCE: [INAUDIBLE]. 643 00:41:14,580 --> 00:41:18,240 PROFESSOR STRANG: The A, from today's lecture 644 00:41:18,240 --> 00:41:21,280 this was the incidence matrix, and this 645 00:41:21,280 --> 00:41:23,630 was the A transpose A that's probably still 646 00:41:23,630 --> 00:41:28,201 on the board somewhere. 647 00:41:28,201 --> 00:41:28,700 Yep. 648 00:41:28,700 --> 00:41:30,030 Yep. 649 00:41:30,030 --> 00:41:36,980 So this is the A, which you should take in and be 650 00:41:36,980 --> 00:41:39,610 able to create if I gave you the graph, 651 00:41:39,610 --> 00:41:41,730 and this is the A transpose A, so it's 652 00:41:41,730 --> 00:41:44,020 through today's lecture, yeah. 653 00:41:44,020 --> 00:41:47,200 Next lecture I'll be talking about the A transpose 654 00:41:47,200 --> 00:41:50,430 by itself, which involves Kirchhoff's current law, 655 00:41:50,430 --> 00:41:52,090 it's beautiful. 656 00:41:52,090 --> 00:41:55,210 A transpose w equals zero. 657 00:41:55,210 --> 00:42:00,550 But I think this part was straightforward enough 658 00:42:00,550 --> 00:42:06,410 to be able to add this to our list of problems 659 00:42:06,410 --> 00:42:08,720 which fit the framework. 660 00:42:08,720 --> 00:42:11,790 So that's what that was about. 661 00:42:11,790 --> 00:42:15,260 It doesn't mean that this will be on but it could be, right. 662 00:42:15,260 --> 00:42:17,380 OK, what else? 663 00:42:17,380 --> 00:42:20,844 You guys are patient, I come on-- Yeah, thanks. 664 00:42:20,844 --> 00:42:21,760 AUDIENCE: [INAUDIBLE]. 665 00:42:21,760 --> 00:42:22,676 PROFESSOR STRANG: Yep. 666 00:42:22,676 --> 00:42:25,370 AUDIENCE: This is only valid when x is less than one? 667 00:42:25,370 --> 00:42:29,740 PROFESSOR STRANG: It's only valid when x is less than one, 668 00:42:29,740 --> 00:42:34,880 so that's now the math point that this expansion for e^x 669 00:42:34,880 --> 00:42:37,070 valid for all x's. 670 00:42:37,070 --> 00:42:40,200 Because you're dividing by these bigger and bigger numbers. 671 00:42:40,200 --> 00:42:44,150 But this one is only valid up to x=1. 672 00:42:44,150 --> 00:42:47,450 At x=1 we're getting one plus one plus one, 673 00:42:47,450 --> 00:42:49,520 and we're getting one over one minus one, 674 00:42:49,520 --> 00:42:52,030 sort of infinity matches infinity, 675 00:42:52,030 --> 00:42:58,780 but then if x goes up to two, yeah what happens if x is two? 676 00:42:58,780 --> 00:43:02,090 It's sort of not good, but you know mathematics, 677 00:43:02,090 --> 00:43:05,010 it's never completely crazy, right? 678 00:43:05,010 --> 00:43:06,960 If x is two, what does this say? 679 00:43:06,960 --> 00:43:10,450 What have I got on the left hand side? 680 00:43:10,450 --> 00:43:12,640 Negative one. 681 00:43:12,640 --> 00:43:15,560 And what have I got on the right hand side? 682 00:43:15,560 --> 00:43:21,010 One plus two plus four plus eight. 683 00:43:21,010 --> 00:43:24,270 I should not allow this to be videotaped, 684 00:43:24,270 --> 00:43:30,410 but that's actually not so completely crazy. 685 00:43:30,410 --> 00:43:37,700 In some nutty way that could still make some sense. 686 00:43:37,700 --> 00:43:41,110 That's certainly will not be on the-- So you're right that x 687 00:43:41,110 --> 00:43:45,520 should be less than one, and of course it will be here 688 00:43:45,520 --> 00:43:48,550 because I'm looking at little delta t's. 689 00:43:48,550 --> 00:43:53,380 Little, so my delta t-- My x was this thing and my delta t, 690 00:43:53,380 --> 00:43:57,970 the time step was small and somehow that tells me, 691 00:43:57,970 --> 00:44:01,350 actually this is a good indication. 692 00:44:01,350 --> 00:44:03,090 It gives me the units. 693 00:44:03,090 --> 00:44:06,510 That stability and things going right 694 00:44:06,510 --> 00:44:09,710 will depend on lambda delta t. 695 00:44:09,710 --> 00:44:14,540 Will depend on lambda delta t, that's the key parameter there. 696 00:44:14,540 --> 00:44:17,640 That's like the dimensionless parameter 697 00:44:17,640 --> 00:44:21,520 that we're, or lambda delta t over two, or whatever. 698 00:44:21,520 --> 00:44:24,310 But lambda delta t is the key. 699 00:44:24,310 --> 00:44:26,860 And a highly important key. 700 00:44:26,860 --> 00:44:30,040 It tells us that as lambda gets bigger, 701 00:44:30,040 --> 00:44:32,820 as the matrix has bigger eigenvalues, 702 00:44:32,820 --> 00:44:35,670 delta t has got to get smaller. 703 00:44:35,670 --> 00:44:38,740 And I mentioned stiff equations. 704 00:44:38,740 --> 00:44:43,000 Stiff equations are equations where the eigenvalues lambda 705 00:44:43,000 --> 00:44:46,450 are out of scale. 706 00:44:46,450 --> 00:44:50,380 You know, you might have two eigenvalues, one of size one 707 00:44:50,380 --> 00:44:52,940 and the other of size ten to the fourth, 708 00:44:52,940 --> 00:44:55,330 because you've got two physical processes going on 709 00:44:55,330 --> 00:44:57,480 at the same time. 710 00:44:57,480 --> 00:45:00,350 And those equations are tough, because that ten 711 00:45:00,350 --> 00:45:04,060 to the fourth guy is forcing your delta t 712 00:45:04,060 --> 00:45:06,320 to be really small. 713 00:45:06,320 --> 00:45:08,740 Whereas the action might, the true, 714 00:45:08,740 --> 00:45:12,940 real solution might be controlled by the lambda=1 guy. 715 00:45:12,940 --> 00:45:15,620 So to follow this slow evolution, 716 00:45:15,620 --> 00:45:19,100 you're having to take very small steps because on top of that 717 00:45:19,100 --> 00:45:22,060 slow evolution with the lambda=1, 718 00:45:22,060 --> 00:45:25,590 there's some very fast evolution maybe with lambda equal minus 719 00:45:25,590 --> 00:45:27,140 10,000. 720 00:45:27,140 --> 00:45:32,630 Yeah, there's a lot happening here. 721 00:45:32,630 --> 00:45:36,450 And always you have to think OK, is there 722 00:45:36,450 --> 00:45:39,970 some way around that box. 723 00:45:39,970 --> 00:45:44,510 Because forward Euler would not get you through. 724 00:45:44,510 --> 00:45:46,611 OK, thanks for that question, you got another one? 725 00:45:46,611 --> 00:45:47,110 OK. 726 00:45:47,110 --> 00:45:50,135 AUDIENCE: So then if you weren't using small enough time 727 00:45:50,135 --> 00:45:51,420 steps, [INAUDIBLE]? 728 00:45:51,420 --> 00:45:55,390 PROFESSOR STRANG: If you weren't using small enough time steps, 729 00:45:55,390 --> 00:45:56,790 OK. 730 00:45:56,790 --> 00:45:59,120 For trapezoidal, let's say? 731 00:45:59,120 --> 00:46:01,384 AUDIENCE: I mean, that expansion wouldn't hold 732 00:46:01,384 --> 00:46:02,550 if you were using a lambda-- 733 00:46:02,550 --> 00:46:04,508 PROFESSOR STRANG: Well, the expansion is really 734 00:46:04,508 --> 00:46:05,920 intended for a small delta t. 735 00:46:05,920 --> 00:46:06,420 Yeah. 736 00:46:06,420 --> 00:46:10,730 It's not intended, I never added up the whole series. 737 00:46:10,730 --> 00:46:14,770 I just compared a couple of terms to see how am I doing, 738 00:46:14,770 --> 00:46:19,260 and I got the extra term to match from trapezoidal 739 00:46:19,260 --> 00:46:22,760 that I didn't get from Euler. 740 00:46:22,760 --> 00:46:24,600 So what's to say? 741 00:46:24,600 --> 00:46:27,060 If you took delta t too big, what 742 00:46:27,060 --> 00:46:31,050 would happen in the trapezoidal method? 743 00:46:31,050 --> 00:46:33,440 Well, you would stay on this circle 744 00:46:33,440 --> 00:46:39,410 because the absolute value of this thing is truly one. 745 00:46:39,410 --> 00:46:43,120 Even if lambda is enormous and delta t is way too big, 746 00:46:43,120 --> 00:46:49,320 we still had complex conjugates and their ratio was one. 747 00:46:49,320 --> 00:46:51,420 So we would not leave the circle, 748 00:46:51,420 --> 00:46:54,900 at least in perfect arithmetic, as everybody says. 749 00:46:54,900 --> 00:46:56,820 If we didn't make any round-off error, 750 00:46:56,820 --> 00:46:58,570 we would not leave the circle. 751 00:46:58,570 --> 00:47:02,020 But boy would we skip all over the place on that circle. 752 00:47:02,020 --> 00:47:05,080 So if we took delta t too big, we 753 00:47:05,080 --> 00:47:06,730 would be completely inaccurate. 754 00:47:06,730 --> 00:47:10,270 We wouldn't be unstable, for trapezoidal, 755 00:47:10,270 --> 00:47:11,910 because it would stay on the circle, 756 00:47:11,910 --> 00:47:16,090 but the phase would be completely wrong, yeah. 757 00:47:16,090 --> 00:47:19,120 So it would be a complex number of absolute value one, 758 00:47:19,120 --> 00:47:28,310 but it would not be close to the exact growth factor. 759 00:47:28,310 --> 00:47:30,880 Well, so many things to say. 760 00:47:30,880 --> 00:47:34,720 I realize that the course moves along pretty quickly 761 00:47:34,720 --> 00:47:40,070 but this topic of numerical methods for differential 762 00:47:40,070 --> 00:47:44,460 equations, that's a core part of 18.086. 763 00:47:44,460 --> 00:47:51,550 So I'm like anticipating here in just a couple of days what 764 00:47:51,550 --> 00:47:55,730 really takes longer is the stability and the accuracy 765 00:47:55,730 --> 00:48:05,330 and the best choices for time-dependent problems. 766 00:48:05,330 --> 00:48:07,640 OK, always good questions. 767 00:48:07,640 --> 00:48:10,010 Anything else that's on your mind of any sort? 768 00:48:10,010 --> 00:48:10,730 Yes, thanks. 769 00:48:10,730 --> 00:48:11,690 AUDIENCE: [INAUDIBLE]. 770 00:48:11,690 --> 00:48:17,940 PROFESSOR STRANG: 114? 771 00:48:17,940 --> 00:48:20,910 AUDIENCE: There is a figure 2.7. 772 00:48:20,910 --> 00:48:21,785 PROFESSOR STRANG: OK. 773 00:48:21,785 --> 00:48:22,285 OK. 774 00:48:22,285 --> 00:48:24,540 114 figure 2.7. 775 00:48:24,540 --> 00:48:26,460 Oh yes, OK. 776 00:48:26,460 --> 00:48:27,560 Oh yes. 777 00:48:27,560 --> 00:48:31,260 AUDIENCE: I figure it's about how these shapes [INAUDIBLE]. 778 00:48:31,260 --> 00:48:32,220 see. 779 00:48:32,220 --> 00:48:37,320 That has a bunch of figures, so that in order 780 00:48:37,320 --> 00:48:40,880 to say for everybody who's not looking at the book, 781 00:48:40,880 --> 00:48:44,280 those figures are about the problem we've discussed here 782 00:48:44,280 --> 00:48:48,480 with a model problem, where we're on a circle. 783 00:48:48,480 --> 00:48:52,480 So do I have space to draw a circle? 784 00:48:52,480 --> 00:48:56,690 Well, let me just make space here. 785 00:48:56,690 --> 00:49:00,610 OK, so page 114 has that model problem 786 00:49:00,610 --> 00:49:03,430 that we've drawn before. 787 00:49:03,430 --> 00:49:07,600 There's the exact solution, here's the phase plane; 788 00:49:07,600 --> 00:49:12,640 there's u and there's u', and the u was cos(t), 789 00:49:12,640 --> 00:49:16,830 so the u' was minus sin(t), and we travel around the circle. 790 00:49:16,830 --> 00:49:19,140 On the exact solution. 791 00:49:19,140 --> 00:49:22,440 Energy constant, u squared stays one-- u squared 792 00:49:22,440 --> 00:49:25,460 plus u prime squared stay one. 793 00:49:25,460 --> 00:49:30,361 Now which figure was it you wanted me to look at? 794 00:49:30,361 --> 00:49:30,860 So. 795 00:49:30,860 --> 00:49:32,476 AUDIENCE: [INAUDIBLE] 796 00:49:32,476 --> 00:49:33,850 PROFESSOR STRANG: Of any of them? 797 00:49:33,850 --> 00:49:35,330 AUDIENCE: Yeah. 798 00:49:35,330 --> 00:49:38,230 PROFESSOR STRANG: OK, that's fine. 799 00:49:38,230 --> 00:49:39,260 Let's see. 800 00:49:39,260 --> 00:49:40,720 Is trapezoidal on that one? 801 00:49:40,720 --> 00:49:41,220 Yeah. 802 00:49:41,220 --> 00:49:43,460 Trapezoidal was the first one. 803 00:49:43,460 --> 00:49:47,650 OK, so figure 2.6 shows the trapezoidal method 804 00:49:47,650 --> 00:49:50,290 moving around the circle. 805 00:49:50,290 --> 00:49:51,100 So what happens? 806 00:49:51,100 --> 00:49:55,160 Yeah, thanks, that's a very suitable question. 807 00:49:55,160 --> 00:50:02,340 OK. and I took, in that figure I took, how long does it 808 00:50:02,340 --> 00:50:05,650 take for the exact solution to get exactly back 809 00:50:05,650 --> 00:50:07,590 where it started? 810 00:50:07,590 --> 00:50:09,720 At t equal what do I come back? 811 00:50:09,720 --> 00:50:12,070 AUDIENCE: 2pi. 812 00:50:12,070 --> 00:50:14,550 PROFESSOR STRANG: t equal to 2pi, 813 00:50:14,550 --> 00:50:15,960 I'm right back where I was. 814 00:50:15,960 --> 00:50:16,460 Right? 815 00:50:16,460 --> 00:50:18,870 Cosine has period 2pi. 816 00:50:18,870 --> 00:50:24,550 OK, now a single step of size 2pi 817 00:50:24,550 --> 00:50:28,250 would be really ridiculous, right? 818 00:50:28,250 --> 00:50:31,050 I mean, I want to now delta t. 819 00:50:31,050 --> 00:50:37,930 So in that figure I took delta t to be 2pi divided by 32. 820 00:50:37,930 --> 00:50:42,750 So I'm taking delta t to be the 2pi, that 821 00:50:42,750 --> 00:50:48,390 would bring me all the way around, but I'm dividing by 32. 822 00:50:48,390 --> 00:50:49,390 So, what does that mean? 823 00:50:49,390 --> 00:50:54,420 What what does the exact solution do at those steps? 824 00:50:54,420 --> 00:50:56,120 32 steps? 825 00:50:56,120 --> 00:51:01,360 It goes on the circle, 32 equal steps, 30, 360, 826 00:51:01,360 --> 00:51:06,290 2pi divided by 32 radians every time, comes back exactly 827 00:51:06,290 --> 00:51:08,350 there, the exact solution. 828 00:51:08,350 --> 00:51:12,210 And right where I started. 829 00:51:12,210 --> 00:51:16,280 So it's like following a planet. 830 00:51:16,280 --> 00:51:20,250 Now I do it by finite differences. 831 00:51:20,250 --> 00:51:22,410 So now I'm going to follow the trapezoidal rule, 832 00:51:22,410 --> 00:51:26,280 just what we've been talking about, with that time step, 833 00:51:26,280 --> 00:51:29,060 and with the equation-- Everybody remembers 834 00:51:29,060 --> 00:51:34,040 the equation was [u, u'] equals, do you remember what the matrix 835 00:51:34,040 --> 00:51:36,950 was in that equation? 836 00:51:36,950 --> 00:51:39,950 This is the derivative of it and this is [u, u']. 837 00:51:39,950 --> 00:51:45,780 Sorry to squeeze this in, but what I'm, u' is u'. 838 00:51:45,780 --> 00:51:50,000 u'' is minus u. 839 00:51:50,000 --> 00:51:54,500 Now we know why that matrix was good, right? 840 00:51:54,500 --> 00:51:55,250 Why is that? 841 00:51:55,250 --> 00:51:59,130 That's my matrix A, why is it good? 842 00:51:59,130 --> 00:52:02,440 Because it's exactly, it fits. 843 00:52:02,440 --> 00:52:06,750 A transpose is minus A. It's anti-symmetric. 844 00:52:06,750 --> 00:52:08,670 Keeps me right on the circle. 845 00:52:08,670 --> 00:52:11,740 OK now, trapezoidal method keeps me 846 00:52:11,740 --> 00:52:14,620 right on the circle, 32 steps. 847 00:52:14,620 --> 00:52:20,170 And so the picture just shows where it goes after 32 steps. 848 00:52:20,170 --> 00:52:25,930 And 32, 32 does it come back there? 849 00:52:25,930 --> 00:52:28,070 Well, not exactly, right? 850 00:52:28,070 --> 00:52:31,420 We don't expect to find a different solution 851 00:52:31,420 --> 00:52:37,180 to be exactly in sync with cos(t), the real one. 852 00:52:37,180 --> 00:52:38,460 But it's really close. 853 00:52:38,460 --> 00:52:43,070 I think in that figure I can see that that's 854 00:52:43,070 --> 00:52:46,750 sort of a double point there, at 2pi. 855 00:52:46,750 --> 00:52:50,370 I put a little arrow indicating small phase error. 856 00:52:50,370 --> 00:52:55,350 It misses by a little bit. 857 00:52:55,350 --> 00:53:01,210 And actually, roughly what does it miss by? 858 00:53:01,210 --> 00:53:05,560 This was the point of the order of accuracy stuff. 859 00:53:05,560 --> 00:53:09,630 Roughly what size is that little error? 860 00:53:09,630 --> 00:53:12,400 That's what we did over here. 861 00:53:12,400 --> 00:53:18,290 The term that we got wrong was a delta t cubed. 862 00:53:18,290 --> 00:53:21,000 At each step. 863 00:53:21,000 --> 00:53:23,050 Can I just tell you the answer? 864 00:53:23,050 --> 00:53:27,390 The error here is of size delta t squared. 865 00:53:27,390 --> 00:53:29,720 Because over here we match those series 866 00:53:29,720 --> 00:53:33,250 and we found the error was delta t cubed. 867 00:53:33,250 --> 00:53:35,540 That's in a single step. 868 00:53:35,540 --> 00:53:38,740 But now we've got one over delta t steps, 869 00:53:38,740 --> 00:53:40,580 you see what I'm saying? 870 00:53:40,580 --> 00:53:45,970 That if the error was delta t cubed per step, 871 00:53:45,970 --> 00:53:51,090 and I have one over delta t steps, to get somewhere, 872 00:53:51,090 --> 00:53:55,861 or 2pi over delta t or whatever, then that gives me delta t 873 00:53:55,861 --> 00:53:56,360 squared. 874 00:53:56,360 --> 00:54:02,240 So that little error there is my error of size delta t squared. 875 00:54:02,240 --> 00:54:05,500 And that square tells me I've got a good method. 876 00:54:05,500 --> 00:54:07,510 At least, decent. 877 00:54:07,510 --> 00:54:09,310 Second order accurate. 878 00:54:09,310 --> 00:54:16,330 And the trapezoidal rule is sort of the natural one. 879 00:54:16,330 --> 00:54:20,390 Well, OK, so that's a full hour mostly devoted 880 00:54:20,390 --> 00:54:22,800 to two or three things. 881 00:54:22,800 --> 00:54:26,720 Actually the eigenvectors came into it. 882 00:54:26,720 --> 00:54:30,980 And the energy conservation came into it, 883 00:54:30,980 --> 00:54:34,050 the stability matching series came into it. 884 00:54:34,050 --> 00:54:36,930 And the picture. 885 00:54:36,930 --> 00:54:43,530 OK, I'll see you Friday for more about these guys, 886 00:54:43,530 --> 00:54:48,660 and then Monday evening please ask me everything 887 00:54:48,660 --> 00:54:50,190 you want to, on Monday evening. 888 00:54:50,190 --> 00:54:51,940 OK. 889 00:54:51,940 --> 00:54:53,400 Thank you.