1
00:00:00,720 --> 00:00:03,840
Last term, we talked about linear algebra and optimisation.
2
00:00:03,840 --> 00:00:10,050
And this term, the other big half, if you like, of scientific computing is differential equations,
3
00:00:10,350 --> 00:00:13,710
in particular ordinary and partial differential equations.
4
00:00:14,040 --> 00:00:20,790
As always, we're going to go at a great rate, not very theoretically, but with a lot of hands on material.
5
00:00:21,210 --> 00:00:24,390
There's a new URL for the course. That's how it works around here.
6
00:00:24,390 --> 00:00:29,430
So all the lecture notes will be deposited there along with the assignments and so on.
7
00:00:29,820 --> 00:00:35,610
The first assignment is due next Tuesday. As usual, the first one is smaller than the other three.
8
00:00:35,850 --> 00:00:41,790
So the same pattern as last term, four assignments, the first one a little bit smaller, and then a mark at the end.
9
00:00:43,080 --> 00:00:48,420
There is a pink handout, which is a variation on last term's blue handout.
10
00:00:49,410 --> 00:00:52,830
Let's just quickly go through it to remind you of how things work.
11
00:00:54,210 --> 00:00:57,690
So the course outline, as I say, is ODS and E's.
12
00:00:57,690 --> 00:01:04,860
And in particular, I'm interested in easy methods for high accuracy and simple geometries called spectral methods.
13
00:01:06,000 --> 00:01:11,040
The lectures me and then the two Tas are Adrienne Mont Anelli and Mikael Savitsky.
14
00:01:11,220 --> 00:01:17,340
Are either of you here? I guess not. Okay. 12 lectures Tuesdays.
15
00:01:17,340 --> 00:01:27,300
Fridays at 9:00. I've reminded you here under software tools of the collection of stuff we have online linking to many different things.
16
00:01:27,510 --> 00:01:30,570
We won't talk about that again explicitly, but it's always there.
17
00:01:30,780 --> 00:01:37,380
And remember, there are a couple of hundred pointers in that Web page, too, all sorts of interesting things.
18
00:01:37,890 --> 00:01:43,200
So if you're new to the course, I definitely recommend you go to that Software Tools page and explore.
19
00:01:45,390 --> 00:01:48,750
Regarding the assignments, the same pattern as last term as usual.
20
00:01:48,760 --> 00:01:54,720
It's nice if you publish the things in MATLAB published, even though that's not the nicest system on earth.
21
00:01:55,050 --> 00:02:01,050
But that's not required. What is required is that you turn in good solutions, including the programs you used,
22
00:02:01,050 --> 00:02:07,770
of course, and that you don't just include programs and print outs.
23
00:02:08,040 --> 00:02:11,370
It's crucial that you say a few words to explain what you've done.
24
00:02:12,780 --> 00:02:19,410
One person left, turn by mistake, turned in assignment for it was 310 pages long.
25
00:02:19,950 --> 00:02:27,090
He did this online somehow, so be sure to take a look at your assignments before you accidentally turned in 310 pages.
26
00:02:29,550 --> 00:02:34,590
Okay. The last thing on the page is to remind you that the lectures are available online.
27
00:02:34,890 --> 00:02:39,420
I realise most of you by definition don't need that, but they're there if you need them.
28
00:02:39,720 --> 00:02:46,920
We haven't yet gotten around to putting last term's lectures in a publicly available place, but that's going to happen one of these days.
29
00:02:48,150 --> 00:02:51,500
Okay. Any questions? Okay.
30
00:02:51,740 --> 00:02:57,680
So let's begin with talking about. Ordinary differential equations.
31
00:03:00,560 --> 00:03:08,090
So we call this part four of the course and I call that ody ease and nonlinear dynamics.
32
00:03:20,300 --> 00:03:25,610
You know, those of you who've been here last term that I'm very interested in history always.
33
00:03:25,610 --> 00:03:33,440
And I think historically speaking, not linearity was the most important development in mathematics in the last 50 years.
34
00:03:33,680 --> 00:03:40,160
Essentially, computers got invented, and that made it possible to explore things that really couldn't be explored before.
35
00:03:40,580 --> 00:03:47,210
So, for example, chaos is one of the many outgrowth of that new way of working with mathematical problems.
36
00:03:49,160 --> 00:03:59,149
So with computers, we can explore all sorts of things, and the basic ideas we use are an ordinary differential equation means that you
37
00:03:59,150 --> 00:04:04,880
have one independent variable as opposed to a partial differential equation,
38
00:04:05,030 --> 00:04:16,040
which has more than one independent variable. And when I speak of an IVP, I mean an initial value problem as opposed to a boundary value problem.
39
00:04:19,580 --> 00:04:25,060
So the starting point of ODI is a discussion of ODI vs.
40
00:04:25,910 --> 00:04:32,070
And the basic thing to be said about those is that they can all be put in the same form.
41
00:04:32,120 --> 00:04:41,380
We can always write. You Prime equals F of t u.
42
00:04:41,960 --> 00:04:51,890
That's our universal standard form for an O.D. and we might as well suppose that it's posed on the interval to bigger than zero,
43
00:04:52,520 --> 00:05:01,730
either the whole semi infinite interval or a finite part of it. And we have an initial condition for an initial value problem.
44
00:05:02,090 --> 00:05:07,100
You not equal some prescribed data, you sub zero.
45
00:05:07,580 --> 00:05:10,910
So there's the initial data for our IVP.
46
00:05:18,070 --> 00:05:22,330
So that looks like a very specialised form. For one thing, it's a first order problem.
47
00:05:22,720 --> 00:05:26,080
For another thing, it looks like a scalar, but it doesn't have to be a scalar.
48
00:05:26,080 --> 00:05:32,200
The same notation works for systems of equations, so you could be either a scalar or a vector.
49
00:05:33,280 --> 00:05:43,210
So this is scalar or vector, and I'll often call that an end vector, i.e. a system of equations.
50
00:05:47,100 --> 00:05:55,080
And it's the usual story in scientific computing that sometimes you get systems of equations because you have many different things you want to track.
51
00:05:55,710 --> 00:05:59,880
But other times you get systems because your disk criticising something continues.
52
00:06:00,180 --> 00:06:09,480
So we saw last term that matrices arise of dimensions in the millions because we're doing finite difference or finite element approximations to eve.
53
00:06:09,990 --> 00:06:16,980
Well, similarly, if you desc or ties at time dependent partial differential equation by sampling it in a lot of points,
54
00:06:17,520 --> 00:06:24,060
you get a system of ordinary differential equations and that can easily have dimensions in the thousands or more.
55
00:06:24,750 --> 00:06:29,460
So people really do care about systems of equations with very large dimension.
56
00:06:31,670 --> 00:06:37,550
So it can be a scalar or it can be a system and it can be first order or higher order because.
57
00:06:40,520 --> 00:06:48,050
Whenever you have a higher order problem, you can introduce additional variables to make it into a first order problem.
58
00:06:48,200 --> 00:06:51,260
So, for example, if I had well, let's write one down.
59
00:06:53,090 --> 00:07:02,710
Here's an example of the simple harmonic oscillator. The simplest ODI of all.
60
00:07:03,190 --> 00:07:09,670
Let's write it with a w I could say w double prime equals minus w.
61
00:07:11,370 --> 00:07:21,930
And I could give it some initial data. So for example, suppose I had w not equals one and w prime not equals zero.
62
00:07:23,160 --> 00:07:27,870
So there is a simple example of a second order linear differential equation,
63
00:07:28,110 --> 00:07:33,810
but I could turn it into a first order problem by defining a vector u with two components.
64
00:07:34,080 --> 00:07:44,850
So I could define the first component u one to be W and the second component you to to BW Prime.
65
00:07:46,740 --> 00:07:52,800
And then I got a system of two ordinary differential equations which I could even write in matrix form.
66
00:07:53,370 --> 00:07:59,220
You one, you two, prime equals.
67
00:08:00,930 --> 00:08:04,020
Well, because it's linear. I could write it as a two by two matrix.
68
00:08:04,230 --> 00:08:11,370
It would be zero one minus one zero times you one, you two.
69
00:08:14,540 --> 00:08:23,450
And there would be initial data. You need a vector of initial data so that the vector would be you one, you two.
70
00:08:24,650 --> 00:08:29,180
At the time, zero is equal to one zero.
71
00:08:33,580 --> 00:08:39,940
So any system of higher order odds can be written as a system of first order oddities.
72
00:08:40,120 --> 00:08:46,870
That's amazingly convenient for the theory that most basic theorem of all about existence and uniqueness is always
73
00:08:46,870 --> 00:08:52,240
proved in this first order context because it immediately translates into something for higher order problems.
74
00:08:52,840 --> 00:09:02,830
At the software level, it's a subtler question. You can, in principle solve any O.D on a computer by translating it into a first order problem.
75
00:09:03,280 --> 00:09:08,200
And that's certainly the way people do it 90% of the time, but it's not necessarily the best.
76
00:09:08,210 --> 00:09:16,510
So if you really want optimal computations with a second order opening, maybe you don't want to reformulate it as a first order system.
77
00:09:18,810 --> 00:09:25,470
Okay. So there's our simplest example. I emphasise that if it were a nonlinear problem, we couldn't write it in this matrix form.
78
00:09:25,590 --> 00:09:31,230
We would need a function F, which would be a nonlinear function written in some more arbitrary way.
79
00:09:31,620 --> 00:09:36,000
Let me also emphasise the word autonomous.
80
00:09:42,210 --> 00:09:49,350
So when you say that it is autonomous, if it's independent of time now, what do I mean by independent of time?
81
00:09:49,890 --> 00:09:52,110
Of course, the solution depends on time.
82
00:09:54,210 --> 00:10:01,770
But I mean that the equation is independent of time, so there aren't any coefficients in the equation that depend on t.
83
00:10:05,410 --> 00:10:09,260
But let me also make explicit what's implicit.
84
00:10:09,260 --> 00:10:17,720
I guess that in applications, a lot of problems involve time and a lot of problems involve space.
85
00:10:18,560 --> 00:10:26,660
Of course, mathematically it doesn't matter, but almost anything you do with odds, the independent variable is going to be one or the other of these.
86
00:10:26,840 --> 00:10:30,110
So you could, for example, use a T here and an X here.
87
00:10:30,890 --> 00:10:34,310
I won't be very systematic about making a distinction like that.
88
00:10:35,300 --> 00:10:42,650
Another difference is that problems dependent on time usually not always tend to be IVP.
89
00:10:42,710 --> 00:10:49,760
The initial value problems and problems dependent on space fairly often are boundary value problems.
90
00:10:51,050 --> 00:10:58,760
But that distinction isn't absolute. Of course, there are boundary value problems in time, and there are initial value problems in space.
91
00:11:01,580 --> 00:11:05,510
Here's another example. A nonlinear one. Consider the solar system.
92
00:11:11,490 --> 00:11:18,420
The solar system, of course, has many bodies in it, but you could simplify it and say that it's got the sun.
93
00:11:19,800 --> 00:11:23,280
Plus nine planets. Maybe it's eight planets. Maybe ten planets.
94
00:11:23,310 --> 00:11:30,360
Nobody seems to be sure how many planets there are. And we could privilege our particular moon because that's a very nice moon.
95
00:11:31,710 --> 00:11:37,170
So you could say that the solar system has 11 bodies. So 11 bodies.
96
00:11:37,170 --> 00:11:41,400
But each one, you could say, has three coordinates each.
97
00:11:45,420 --> 00:11:49,440
But each one also has three velocities because.
98
00:11:51,770 --> 00:11:59,630
Newton's laws for the solar system are written in terms of force equals mass times acceleration, and acceleration is the second derivative.
99
00:11:59,900 --> 00:12:09,650
So if we wanted to write the solar system as an oddity in the simplest fashion, we would have 11 bodies, each with six coordinates.
100
00:12:09,920 --> 00:12:13,580
So there would be 66 dependent variables.
101
00:12:18,580 --> 00:12:24,430
Assuming we're treating all of the bodies as simply points orbiting each other in the obvious fashion.
102
00:12:26,670 --> 00:12:31,980
Or you could do it as a second order formulation with 33 independent variables.
103
00:12:32,220 --> 00:12:35,910
Or you could pretend it's all in a plane, and then you'd have 22 independent variables.
104
00:12:35,920 --> 00:12:42,450
But some of the planets aren't really in the plane. Okay.
105
00:12:42,450 --> 00:12:47,880
Let's do an example. We will compute with everybody's favourite.
106
00:12:48,000 --> 00:12:52,710
Simpler, nonlinear example of an everyday being is the so called Vanderpoel equation.
107
00:12:58,480 --> 00:13:03,580
So the Vanderpoel equation is an oscillator like the simple harmonic oscillator,
108
00:13:03,760 --> 00:13:09,940
except that it's damped and it's also nonlinear because the damping depends on the amplitude.
109
00:13:10,210 --> 00:13:16,180
So whenever you have a physical process that depends on amplitude, then that probably means it's a nonlinear problem.
110
00:13:17,500 --> 00:13:28,620
So it's a nonlinear oscillator. And even with without driving down the equation, I can describe it in words.
111
00:13:29,580 --> 00:13:35,219
So it likes to oscillate. If the amplitude is large, then it is adapt system.
112
00:13:35,220 --> 00:13:39,810
And so the amplitude decreases. If the amplitude is small, it's negatively damped.
113
00:13:40,110 --> 00:13:47,190
So the amplitude increases. And so there's a preferred amplitude of the steady state solutions of the Vanderpoel equation.
114
00:13:47,430 --> 00:13:56,159
So let's write it down w double prime plus a positive constant times w squared
115
00:13:56,160 --> 00:14:05,970
minus one double w prime plus w equals zero and c is a parameter that is fixed.
116
00:14:10,000 --> 00:14:13,870
So if we didn't have this term, it would simply be the harmonic oscillator.
117
00:14:14,680 --> 00:14:20,440
But we do have this term. As long as this part of it is positive, you have damping.
118
00:14:20,770 --> 00:14:28,540
Whenever this is negative, you have negative down. So if W is bigger than absolute one, an absolute value, it's damned if it's less than one.
119
00:14:28,540 --> 00:14:32,950
It's negatively back. And we're going to compete with that in a moment.
120
00:14:33,610 --> 00:14:40,990
What we find is that as time goes to infinity, the solutions converge.
121
00:14:44,030 --> 00:14:47,150
To something of amplitude, approximately one.
122
00:14:47,540 --> 00:15:07,339
And that's called our limit soccer. Now, of course, we can always write a higher order problem as a system of first order problems.
123
00:15:07,340 --> 00:15:11,150
So let's do that one more time. Maybe the last time I'll do it.
124
00:15:11,540 --> 00:15:29,960
If we write it in first order form, then the Vanderpoel equation becomes U one prime equals U of two, so w prime equals w prime and you two prime.
125
00:15:32,530 --> 00:15:43,240
Is equal to minus u one minus the constant times u one squared.
126
00:15:45,170 --> 00:15:48,610
Minus one. You too.
127
00:15:52,620 --> 00:16:03,140
I hope I've written that correctly. Now, if I talk rather quickly and breezily about OBE as if it's a pretty easy subject.
128
00:16:03,410 --> 00:16:07,879
The reason for that is it actually is a pretty easy subject. The odds are straightforward.
129
00:16:07,880 --> 00:16:13,430
They're one of the most important things in mathematics, but they're not one of the difficult things in mathematics.
130
00:16:13,610 --> 00:16:16,430
Let's play around a bit now on the computer.
131
00:16:27,980 --> 00:16:38,330
Now there's one of the handouts, has some code on it, and the first bit of code at the top of that page is just a reminder of anonymous functions.
132
00:16:38,540 --> 00:16:43,490
Such a convenient trick in MATLAB. So let me just remind you how anonymous functions work.
133
00:16:43,760 --> 00:16:51,410
The sort of thing you can do is say F equals out of x x squared.
134
00:16:52,520 --> 00:16:59,190
So that's an anonymous function. And I can now do things like F of three or F of three, four or five.
135
00:17:00,470 --> 00:17:12,440
And MATLAB does the obvious thing. The standard way that the MATLAB guys work is to use anonymous functions to do various things.
136
00:17:12,440 --> 00:17:20,630
So, for example, if you want to integrate an anonymous function or an M file, there's a code called Quad.
137
00:17:21,620 --> 00:17:26,719
So Quad implements a numerical method which takes this anonymous function as
138
00:17:26,720 --> 00:17:31,010
input and does the right thing to integrate it over the interval from 0 to 1.
139
00:17:32,120 --> 00:17:38,450
As you know, I prefer the fun way of thinking where all functions are in the same setting of funds.
140
00:17:39,260 --> 00:17:45,680
This more traditional way of thinking is that different operations like Quadrature Odds or whatever all have different interfaces,
141
00:17:46,040 --> 00:17:52,070
but the net result is similar. So here's another example of a different interface.
142
00:17:53,570 --> 00:17:57,710
Suppose I set a equals three. I'm just following the thing here.
143
00:17:57,920 --> 00:18:02,780
Easy plot. I could now make another anonymous function.
144
00:18:02,780 --> 00:18:09,790
So I'll say easy plot of A2 of x square root of abs, of f of x minus eight.
145
00:18:10,760 --> 00:18:15,160
So you see, I'm piling up anonymous functions on anonymous functions.
146
00:18:15,170 --> 00:18:21,670
Let's plot that. On the interval from -10 to 10.
147
00:18:27,520 --> 00:18:35,320
What's happening? Yeah. So MATLAB has this command called plot or A-Z plot, which confuses people easy plot.
148
00:18:35,920 --> 00:18:39,270
And it does the right thing to sample your function and draw a picture.
149
00:18:39,280 --> 00:18:43,660
So that's an example of another interface to an anonymous function built into MATLAB.
150
00:18:45,730 --> 00:18:51,640
Finally gets to emphasise the vector side of things. Suppose I made an anonymous function g.
151
00:18:53,890 --> 00:19:02,920
Of several variables, so I could say g equals anonymous function of ABC and then I could say A two times B three times C.
152
00:19:04,510 --> 00:19:16,390
So if I now say G of 111 with three arguments, I get one, two, three, or I could say F of g of one, one, one, and I get the squares of those.
153
00:19:16,840 --> 00:19:21,580
So it's just a convenient way of doing a lot of coding in a single line.
154
00:19:25,160 --> 00:19:30,100
Okay. So let's now assume we're good at anonymous functions and start playing with words.
155
00:19:30,260 --> 00:19:37,520
The next code called Vendor Call. I'll just say a word or two about it and then we'll run it to see what the vendor policy later looks like.
156
00:19:38,330 --> 00:19:48,020
So. M 25 Vanderpoel You see, first of all, we declare the initial condition, which is .01 for the function value and zero for the derivative.
157
00:19:48,350 --> 00:19:53,540
So that's a vector of two components because we're going to convert it to first order form.
158
00:19:54,800 --> 00:19:57,620
Then we give it a time span which goes from 0 to 50.
159
00:19:58,670 --> 00:20:11,360
Then we input our damping parameter C and then we call one of the standard codes in MATLAB called ODI E four or five to solve this.
160
00:20:12,560 --> 00:20:17,750
So it solves the equation defined by a function F which and you see there's an anonymous
161
00:20:17,750 --> 00:20:22,850
function at the top that defines it on the given time span with the given initial condition.
162
00:20:24,610 --> 00:20:31,419
And then it plots the result and it also makes a phase portrait. So let's run that to get a sense of things, I'll say.
163
00:20:31,420 --> 00:20:38,290
And 25 Vanderpoel and it asks me for the damping constant.
164
00:20:38,300 --> 00:20:43,550
So let's start in the middle and say one. So it's now chugging away.
165
00:20:43,760 --> 00:20:48,100
And this is the result. Plenty of physics in that picture.
166
00:20:48,110 --> 00:20:56,280
What you see is that. The initial condition is, of course, much smaller than a size 1.0.
167
00:20:56,950 --> 00:21:00,820
So we have negative nine. So the negative damping makes the thing get much bigger.
168
00:21:01,030 --> 00:21:04,060
And obviously, it's approaching a steady oscillation.
169
00:21:04,330 --> 00:21:10,720
That's the limit factor whose average amplitude is about one that we see at maximum amplitude is about two.
170
00:21:13,550 --> 00:21:19,490
If we press return, there's a pause. Now, if I press return, it will give me a face portrait,
171
00:21:20,630 --> 00:21:28,190
which means we've now plotted this trajectory on the plane with you in that direction and you prime in that direction.
172
00:21:28,520 --> 00:21:34,340
So as with most oscillations, you have an interplay between the kinetic and the potential energy, if you like.
173
00:21:34,580 --> 00:21:38,790
Sometimes it's got a large amplitude and a small velocity.
174
00:21:38,810 --> 00:21:44,000
Other times it's got a small amplitude and a large velocity, and it's moving around in that phase.
175
00:21:48,810 --> 00:21:52,230
Let's change. See? To get a sense of dependence on that damn thing.
176
00:21:52,770 --> 00:21:58,380
So if I change it to ten. So now we have much stronger damping parameter.
177
00:21:59,280 --> 00:22:04,200
Then you can see the oscillation becomes much more visibly nonlinear there.
178
00:22:04,530 --> 00:22:11,670
People have studied this a great deal. It's a fascinating case of slow motion and then a very sudden flip to the other state.
179
00:22:13,830 --> 00:22:18,120
The face portrait you see also looks interesting.
180
00:22:19,650 --> 00:22:24,330
Now at the other extreme. Suppose I run it with damping parameters zero.
181
00:22:24,900 --> 00:22:33,270
Now I really do have the simple harmonic oscillator. The initial condition is point one and it just keeps going at .01 for at.
182
00:22:35,560 --> 00:22:39,370
And the face portrait is a circle of radius point at one.
183
00:22:42,270 --> 00:22:48,060
Finally, let's run it with a small but non-zero damping parameter.
184
00:22:49,200 --> 00:22:57,510
So now we have C equals point three and what you see there is again, the negative damping makes it increase but more slowly.
185
00:22:58,020 --> 00:23:04,860
Eventually it's reaching some kind of a steady oscillation and the phase portrait similarly starts
186
00:23:04,860 --> 00:23:09,810
near the middle and it takes a lot of rotations around before it finally gets out to the limit cycle.
187
00:23:14,560 --> 00:23:20,530
You see in the curve, it looks jagged. That's because MATLAB is by default plotting.
188
00:23:20,530 --> 00:23:25,719
Piecewise linear is between data points. It doesn't mean that the data are that inaccurate.
189
00:23:25,720 --> 00:23:30,470
They're probably quite accurate. But the plot doesn't happen to sample enough to show the accuracy.
190
00:23:32,790 --> 00:23:40,170
Okay. So that's our simplest example of an oldie. And now let's say a bit about how these things are done.
191
00:23:53,070 --> 00:24:41,070
That's a reasonable. Okay.
192
00:24:44,320 --> 00:24:52,930
So the idea of solving Obis numerically is very, very old and it's one of the most concrete things in mathematics.
193
00:24:53,440 --> 00:24:58,750
The basic idea, the starting point of an idea is that you're following some trajectory.
194
00:25:00,010 --> 00:25:03,160
The body tells you the slope at a point.
195
00:25:04,000 --> 00:25:08,469
So it's naturally enough to think of going a little bit in that distance, and then we get a new slope.
196
00:25:08,470 --> 00:25:16,540
And it's the most obvious thing in the world to numerically do some kind of thing like that.
197
00:25:16,870 --> 00:25:20,770
To use the Audi E as the starting point of a numerical approximation.
198
00:25:21,010 --> 00:25:30,450
And this way of thinking was made famous by Euler, along with so much else that's going back to 1768, I guess.
199
00:25:33,350 --> 00:25:40,750
I wouldn't say that Euler did a lot of really numerical mathematics, but a century later people began to do more.
200
00:25:40,760 --> 00:25:46,700
And I want to mention the two big categories of numerical methods for obedience.
201
00:25:46,970 --> 00:25:54,670
They're called wrong cutter methods. And multistep formulas.
202
00:25:55,240 --> 00:26:04,389
These are the two big games in town. And they've been around for a long time.
203
00:26:04,390 --> 00:26:09,010
So the multistep idea is really from the middle of the 19th century.
204
00:26:09,220 --> 00:26:12,880
And the key guy there was John Adams.
205
00:26:14,650 --> 00:26:18,970
In the 1850s. So about well, in the 1850s, that's when he did the work.
206
00:26:19,420 --> 00:26:25,690
He didn't really publish it then. Adams was a great genius of numerical computing at Cambridge.
207
00:26:25,720 --> 00:26:35,590
He was very young when he started this work, and he basically took Euler's low accuracy idea and looked at higher order accuracy analogues of that.
208
00:26:35,830 --> 00:26:37,480
And these are the multi-step methods.
209
00:26:38,020 --> 00:26:45,820
So this is the same guy who predicted the existence of the planet Neptune based on perturbations of the planet Earth.
210
00:26:46,360 --> 00:26:51,460
In 1845, he was 26 years old at the time.
211
00:26:52,390 --> 00:26:57,480
Sadly, the telescopes were not trained where he said they should be trained.
212
00:26:57,490 --> 00:27:01,900
So a year later Neptune was actually discovered by La Vecchia in France.
213
00:27:03,880 --> 00:27:08,860
Adams was an incredible genius. He was the senior wrangler, the best student at Cambridge in his year.
214
00:27:09,160 --> 00:27:12,310
And in the examinations, the second best student.
215
00:27:12,340 --> 00:27:17,380
This is back fourth got exam marks that were half those of Adams.
216
00:27:17,650 --> 00:27:24,420
So he was really simply an extraordinary person who was much loved and certainly widely admired.
217
00:27:24,430 --> 00:27:27,969
He was offered the post of Astronomer Royal and turned it down.
218
00:27:27,970 --> 00:27:35,470
He was offered a knighthood and turned it down. Amazing guy. This guy best for his roommate.
219
00:27:35,470 --> 00:27:40,120
I like to think they weren't actually roommates, I suppose, but they were pretty much college roommates.
220
00:27:40,120 --> 00:27:47,080
Translated into the modern era, actually wrote down a lot of this stuff 30 years later in 1883.
221
00:27:48,580 --> 00:27:52,750
So these this is the origin of the multistep methods.
222
00:27:54,740 --> 00:27:59,510
And then it was a generation or so later that the wrong a cut of message came in.
223
00:27:59,780 --> 00:28:06,110
So that was Carl Reiner, who is one of my heroes of numerical computation, did many, many things.
224
00:28:07,280 --> 00:28:14,780
That was in 1895. He invented the fast Fourier transform, though nobody noticed, and many other things.
225
00:28:15,470 --> 00:28:21,920
Another key name was Heine in 1900, and cut to himself was, I think, in 1901.
226
00:28:26,590 --> 00:28:28,900
So basically that was the Brits and then the Germans.
227
00:28:29,050 --> 00:28:36,340
And incidentally, regarding pronunciation in German and in English, the right thing to say is wrong to cut off or run a cut or something.
228
00:28:36,970 --> 00:28:42,730
If you say runs, that's a complete error. However, in French they do say runes or something.
229
00:28:42,760 --> 00:28:46,960
So if you're speaking French, you're allowed to say runes. Otherwise, I don't want to hear anyone say.
230
00:28:47,200 --> 00:28:52,810
Right. Let me mention another interesting name.
231
00:28:56,800 --> 00:29:00,550
Forest, Ray Moulton.
232
00:29:02,130 --> 00:29:09,270
So there are formulas called atoms, best forth formulas, and there are formulas called atoms, molten formulas.
233
00:29:09,540 --> 00:29:14,130
This guy, Molten, was an American, which was very unusual in those days.
234
00:29:14,550 --> 00:29:17,590
America wasn't doing much academically in 1926.
235
00:29:18,870 --> 00:29:23,430
And I think he may be related to Derek Moulton, who's a faculty member in this department.
236
00:29:23,640 --> 00:29:31,800
So Derek isn't sure. But he says he had a great, great uncle who was supposedly very good at mathematics, and he thinks that may be the very good one.
237
00:29:34,110 --> 00:29:41,909
One more name to mention, an early contributor is the very famous physicist Richard Farmhouses, who did many things.
238
00:29:41,910 --> 00:29:46,620
And one of those was to get quite involved in these formulas in the 1930s.
239
00:29:47,520 --> 00:29:51,180
Okay. So as I say, there are two big classes and.
240
00:29:54,200 --> 00:30:01,760
The distinction between them is that the multi-step methods mar using formulas that coupled together many different steps.
241
00:30:02,090 --> 00:30:05,610
Their only academy methods are, in some sense, one step method.
242
00:30:05,690 --> 00:30:14,470
So now let me say what I mean by that. So what we always do in all of these methods is take the time, access and disparate it.
243
00:30:15,680 --> 00:30:23,720
In the simplest setting. You imagine a uniform is for possession, though in practice, software will make that non-uniform.
244
00:30:24,620 --> 00:30:29,660
So you imagine at time zero. You have t not.
245
00:30:29,930 --> 00:30:38,870
And then t one we could call K. So K is our timestamp t two is two k and so so k is the timestamp.
246
00:30:44,490 --> 00:30:51,330
Now you can do what Euler did and just can predict the future by little straight lines of length k.
247
00:30:51,780 --> 00:30:56,790
Of course that works in some sense, but it's not an accurate method if the only converges linearly.
248
00:30:57,090 --> 00:31:01,650
If you cut the time, step in half. The accuracy only doubles, which is not so great.
249
00:31:02,340 --> 00:31:06,480
So the point is to get higher order second, third, fourth, or higher order methods.
250
00:31:09,670 --> 00:31:17,810
So our goal in any of these formulas is to approximate the true solution.
251
00:31:17,830 --> 00:31:30,790
You at the time, let's say t seven. Buy some number, which I could call this seven.
252
00:31:32,230 --> 00:31:36,970
Which we're going to compute by some algebraic approximation to the differential equation.
253
00:31:37,450 --> 00:31:40,870
So this will be computed somehow algebraically.
254
00:31:48,080 --> 00:31:51,400
I tend to talk about numbers, but they don't have to be numbers in general.
255
00:31:51,410 --> 00:31:54,440
You as a vector and V as a vector of the same size.
256
00:31:54,890 --> 00:32:00,140
So we're approximating the vectors at these time values by.
257
00:32:02,130 --> 00:32:07,690
Other factors that are approximations. T seven of course refers to ten times the times.
258
00:32:10,410 --> 00:32:14,220
Let me just tell you the first four atoms formulas.
259
00:32:17,380 --> 00:32:24,170
Or more precisely, the first for what are called atoms dash forth formulas.
260
00:32:28,730 --> 00:32:36,410
So we're going back now 130 years. And the first one is Euler's formula.
261
00:32:36,680 --> 00:32:43,880
So we write it like this. We say V and plus one equals the N.
262
00:32:45,120 --> 00:32:50,850
Plus Hey Times fan and I better say what FM means.
263
00:32:51,460 --> 00:33:02,970
F1 is an abbreviation for the function F evaluated at time t, n, and the current estimate of u.
264
00:33:03,240 --> 00:33:08,810
V. So this formula is exactly this picture.
265
00:33:09,560 --> 00:33:16,610
You're going along estimating your solution. You're at some point in order to do the next one, you take the current slope,
266
00:33:16,910 --> 00:33:22,010
you apply F to your current estimate and march then a distant K in the time direction.
267
00:33:22,790 --> 00:33:28,380
So that's Euler. And everybody calls that Euler's method.
268
00:33:31,970 --> 00:33:38,450
But then Adams realised that if you play around with polynomials properly, you can get any order of accuracy you want.
269
00:33:38,690 --> 00:33:47,090
So the next one is the end. Plus one equals VM plus k over two.
270
00:33:48,690 --> 00:33:54,300
Times three F and minus F and minus one.
271
00:33:56,100 --> 00:34:01,770
Now you're already seeing the multi step aspect of this to get the new value it's coupling together.
272
00:34:01,770 --> 00:34:07,260
The current and the previous value and the order of accuracy has just improved by one order.
273
00:34:08,880 --> 00:34:18,090
If you work with this one, you'll find its first order accurate. So the errors look like, okay, this one is second order accurate, case square,
274
00:34:18,330 --> 00:34:23,220
and then I'll write down the next two, which are third order accurate and fourth order accurate.
275
00:34:26,360 --> 00:34:39,739
So the third order Adams batch fourth formula is V and plus one equals V and plus k over 1223 times F,
276
00:34:39,740 --> 00:34:47,720
n, -16 F and minus one plus five F and minus two.
277
00:34:49,190 --> 00:34:55,160
And then the fourth order one is V and plus one equals v n.
278
00:34:56,330 --> 00:35:14,190
Plus K over 24. You can see factorial hidden there times 55 F and -59 F and minus one plus 37.
279
00:35:15,730 --> 00:35:21,280
And minus two. Minus nine and minus three.
280
00:35:23,190 --> 00:35:26,250
And so on. It's an infinite family. You can take these forever.
281
00:35:26,970 --> 00:35:30,970
So there is a 1,000th order out of the passport formula.
282
00:35:30,990 --> 00:35:36,600
I'm sure nobody's ever written it down. You know, I like concreteness.
283
00:35:36,620 --> 00:35:41,750
So the reason I write down these numbers is to make sure you have a sense of how concrete it all is.
284
00:35:42,050 --> 00:35:45,350
Obviously, the numbers are not in any deeper sense. Very interesting.
285
00:35:45,890 --> 00:35:52,190
But you can imagine there's a lot of very pretty mathematics designed to automate generation of such formulas.
286
00:35:52,550 --> 00:35:57,470
And if you go back to the 1920s, 1930, 1940s, people were really good at that.
287
00:35:58,190 --> 00:36:03,140
Once computers came along, the interest moved into somehow a little bit higher level questions.
288
00:36:05,990 --> 00:36:10,940
Okay. So the good thing about add ons that's worth formulas is they're a very straightforward,
289
00:36:10,940 --> 00:36:21,530
infinite set of formulas of whatever order accuracy you could ask for were a bit of a drawback is how to start them up.
290
00:36:27,180 --> 00:36:35,070
Because in order to get the end plus one, I need not only v m but the previous three values with that form.
291
00:36:35,430 --> 00:36:40,440
So if I want to actually use this in software, it's not enough to have an initial condition.
292
00:36:40,680 --> 00:36:42,690
I need three more levels somehow,
293
00:36:42,960 --> 00:36:51,300
so I need to use lower order formulas with very small time steps or something like that in order to generate the first few data values.
294
00:36:51,840 --> 00:36:54,180
That's a little bit annoying on a uniform grade,
295
00:36:54,360 --> 00:37:00,540
and it's more annoying if you're varying the time step because every time you vary it, you have to do something clever again.
296
00:37:02,340 --> 00:37:06,780
So that's why I wrong and could have decided to construct a different set of formulas.
297
00:37:07,770 --> 00:37:15,810
And let me show you some examples there. I'm going to show you two of.
298
00:37:21,710 --> 00:37:26,150
So in the case of Adams formulas, I can say these are the first four and it's an infinite series.
299
00:37:26,150 --> 00:37:29,270
Everything is conceptually very straightforward.
300
00:37:29,510 --> 00:37:33,560
Wrong kind of formulas are not at all like that. There's no natural series.
301
00:37:33,680 --> 00:37:37,790
There's more than one of any given order. They have all sorts of complexities.
302
00:37:37,940 --> 00:37:45,380
You can't assume that the seventh order formula has seven stages in it, and you'll see what I mean by a stage.
303
00:37:45,830 --> 00:37:49,850
Much more complicated and therefore maybe interesting mathematical theory.
304
00:37:50,450 --> 00:37:56,480
The two I'm going to show you are a standard second order one and the standard fourth order.
305
00:37:57,890 --> 00:38:03,370
So. Okay. Squared and that. I love K to the fourth.
306
00:38:08,130 --> 00:38:14,190
As I say, there's more than one. But the one that I'll write down here is called Modified Euler.
307
00:38:20,190 --> 00:38:28,760
And it looks like this. We take a look and define it to be the timestamp times F evaluated at the current point.
308
00:38:28,770 --> 00:38:42,340
So that's F of key and the. And then we take B and define it to be the timestamp times f a value that t n plus a half of k.
309
00:38:45,360 --> 00:38:48,780
And then plus a half of it.
310
00:38:51,520 --> 00:38:54,759
So these are called stages. We've now done two stages.
311
00:38:54,760 --> 00:39:06,280
We've used the function F twice, and then we combine the stages in modified order like this, we simply say the end plus one equals V and plus B.
312
00:39:09,150 --> 00:39:19,530
So what's going on intuitively is that whereas Euler generates a slope from the left hand of the endpoint, thereby introducing I know of Keira,
313
00:39:20,520 --> 00:39:25,230
this little bit of cleverness gives you some approximation to the average slope over the interval,
314
00:39:25,500 --> 00:39:31,350
thereby giving you o of k squared and then you can crank up the order.
315
00:39:31,800 --> 00:39:34,710
So here is the famous fourth order and another kind of scheme.
316
00:39:36,210 --> 00:39:42,030
It's not the only fourth order running scheme, but it does have the property that it's often called the fourth order task.
317
00:39:46,900 --> 00:39:52,030
And that one looks like this. And this one is very famous. I'm sure many of you have used it.
318
00:39:52,720 --> 00:39:57,670
It has four stages. So you say AA equals K times F of ten then.
319
00:39:58,180 --> 00:40:02,410
So we're starting as if we're doing Euler, but then we improve it.
320
00:40:02,620 --> 00:40:12,340
We say B equals K times F of ten plus k over to V and plus over to.
321
00:40:15,650 --> 00:40:29,990
And then amazingly, we do sort of the same thing. We now say see equals K times F of ten plus k over to V and plus B over to.
322
00:40:35,040 --> 00:40:44,650
And then we do a fast one. We say D is equal to K times F of t, n plus k.
323
00:40:44,670 --> 00:40:52,200
So now we're evaluating the function at fully the new time step V and plus C.
324
00:40:54,540 --> 00:40:57,630
So we've done these four stages and then we put them together.
325
00:40:58,260 --> 00:41:07,589
V And plus one equals v n +16 times a plus two.
326
00:41:07,590 --> 00:41:10,740
B plus two C plus D.
327
00:41:14,850 --> 00:41:20,460
So you can probably guess by looking at that that the analysis of these things is a little bit complicated.
328
00:41:20,760 --> 00:41:27,210
It wasn't until the modern era that people were even able to generate a kind of formulas of order seven and eight and so on.
329
00:41:27,810 --> 00:41:31,050
And there are all sorts of open conjectures about the kind of formulas.
330
00:41:31,530 --> 00:41:35,490
The theory is actually quite beautiful. It involves trees in the sense of graphs.
331
00:41:36,450 --> 00:41:44,820
And the person who is the great man who put the theory of wrong kind of methods into mathematics is called John Butcher.
332
00:41:45,090 --> 00:41:49,740
He did this starting in the 1950s. He's still alive. He's a Kiwi from New Zealand.
333
00:41:50,610 --> 00:41:58,320
Fascinating map. If you had one formula to take to a desert island, this is the one.
334
00:41:58,860 --> 00:42:04,829
It's complicated to write down, but it's simple to use because there's no start problem.
335
00:42:04,830 --> 00:42:10,320
You've got to do it. And it's got fourth order accuracy, which for many applications is very good accuracy.
336
00:42:12,330 --> 00:42:16,570
So I'm curious. Raise your hand if you've used that form. That's amazing.
337
00:42:16,590 --> 00:42:22,680
Okay, so here's a historical note.
338
00:42:24,280 --> 00:42:29,770
Another great figure in numerical owning is Gerhard Vollmer from the University of Geneva.
339
00:42:30,100 --> 00:42:34,450
And I heard him give a talk once about multi-asset versus market.
340
00:42:35,830 --> 00:42:42,639
And he made the observation that over the years, Americans and British people tend to prefer multi-asset,
341
00:42:42,640 --> 00:42:47,290
formal and Continentals tend to prefer kind of formulas.
342
00:42:47,620 --> 00:42:56,650
And he actually argued convincingly that this difference ultimately goes back to the fight between lightness and Newton about calculus.
343
00:42:56,980 --> 00:43:02,590
I forget the details of the argument, but somehow the Newtonian approach sort of led to this through Adams,
344
00:43:02,590 --> 00:43:08,680
if you like, and this continental approach, the line, its approach through the Germans led to this.
345
00:43:09,420 --> 00:43:14,520
Maybe true. Okay.
346
00:43:14,520 --> 00:43:18,140
Let me say a word about the software available in MATLAB.
347
00:43:24,810 --> 00:43:28,320
So this is officially Section 4.3.
348
00:43:29,190 --> 00:43:38,820
Initial value problem codes in MATLAB and simulate.
349
00:43:45,430 --> 00:43:53,050
So first of all, let me say all I'm going to say about Simula. You know, MathWorks is a very successful company with two or 3000 employees.
350
00:43:53,350 --> 00:43:56,410
It all started from MATLAB, which was rooted in matrices.
351
00:43:56,740 --> 00:44:01,180
But the real money-maker is differential equations and dynamics,
352
00:44:01,450 --> 00:44:09,880
and the way most of the money is made is through this thing called simulant, which would be called a an environment for visual programming.
353
00:44:10,210 --> 00:44:18,370
So in simulation you have you draw something on your screen and you connect various boxes and suitable ways to to construct the dynamics.
354
00:44:18,850 --> 00:44:21,940
People like me, numerically, analysts never use simulated.
355
00:44:22,480 --> 00:44:25,960
So I have actually never used it in my life, except once I tried the demo.
356
00:44:27,430 --> 00:44:30,980
However, there's no denying that for many people it's the way to do it.
357
00:44:31,030 --> 00:44:36,160
So I'm curious as you sit here. All right, have any used it a lot?
358
00:44:36,820 --> 00:44:46,180
Yeah. So one day you'll give us a demo. Okay. But anyway, numerical analysts, mathematicians are a little more old fashioned.
359
00:44:46,180 --> 00:44:50,410
We like to know more of what's going on. So we like to call codes to do things.
360
00:44:52,240 --> 00:45:02,590
Let me just say 3 minutes about software. Oh, these are really the great subject in which numerical software took off 50 years ago,
361
00:45:02,890 --> 00:45:10,420
because it has just that right level of complexity where nobody is going to solve an nobody on paper usually.
362
00:45:11,140 --> 00:45:17,890
But to do it numerically is not that hard. So in the sixties, people started writing adaptive code.
363
00:45:17,890 --> 00:45:24,190
So what does adaptive mean? For 50 years they've all been adaptive.
364
00:45:24,550 --> 00:45:32,320
And there are two senses in which that word applies in the time steps or the space steps.
365
00:45:34,560 --> 00:45:44,460
So that's always true. Basically, every ODP code in existence varies the time stamp depending on how the curve seems to be evolving.
366
00:45:45,300 --> 00:45:53,940
We might say a little more about that later, but not today. It may also vary in the order, but not all of them do that.
367
00:45:56,290 --> 00:46:02,140
So some of them will stick with a single formula like Ronnie Carter for something and vary the time step.
368
00:46:02,470 --> 00:46:13,480
Others might go up and down a hierarchy of multistep formulas from order 1 to 13 or something, just to mention some classics in Fortran.
369
00:46:13,750 --> 00:46:25,360
Going really back to the sixties. The two that you can find online of the oldest, best known codes are called Pack and Arcade Suite.
370
00:46:28,130 --> 00:46:32,390
Easily found on the web, and they're also in our software tools directories.
371
00:46:32,630 --> 00:46:36,340
A lot of these originate from the national labs in the US.
372
00:46:39,140 --> 00:46:47,470
In MATLAB. There actually are eight codes that hardly any of us use, more than a couple of them.
373
00:46:48,850 --> 00:46:51,969
And let me tell you the names.
374
00:46:51,970 --> 00:47:01,840
The big three, really. r0de23034, five and O.D. 113.
375
00:47:03,390 --> 00:47:06,960
These are both remarkable methods.
376
00:47:09,680 --> 00:47:11,479
And the way they work, loosely speaking,
377
00:47:11,480 --> 00:47:20,330
is that they run something with a fourth order method and a fifth order method and compare the two to get an error estimate to control the time step.
378
00:47:21,140 --> 00:47:29,920
This is a multi-step one. Which actually has variable order going from order one to Order 13.
379
00:47:30,130 --> 00:47:37,330
This tends to be my preference because I like high accuracy and the high order means that if you really want ten digits, this is usually best.
380
00:47:37,510 --> 00:47:41,770
But if you only want five digits, this is fine. And if you only want two or three digits, this is hard.
381
00:47:43,240 --> 00:47:52,690
Now, in the handout, if you look on the back of the code, I've copied the first couple of pages of the paper that introduced all of these codes.
382
00:47:53,680 --> 00:47:58,510
It's by Larry Champagne and Mark Reichel Champagne.
383
00:47:59,320 --> 00:48:02,470
I think at that time was at a national lab. No, he wasn't.
384
00:48:02,480 --> 00:48:05,680
He was at Southern Methodist University, but he was connected with the labs.
385
00:48:05,980 --> 00:48:09,280
Mark recalled was at MathWorks, I guess.
386
00:48:09,550 --> 00:48:13,390
He's been at MathWorks for many years. He took some time off to found a company.
387
00:48:16,210 --> 00:48:25,870
I think this is one of the two great papers published by the MathWorks team that made their software applicable to more people.
388
00:48:25,870 --> 00:48:29,800
I've shown you last term. The other one about sparks matrices in MATLAB.
389
00:48:30,070 --> 00:48:38,050
Well, this old paper described the suite of code that is still the basis of MATLAB computing 20 years later.
390
00:48:38,740 --> 00:48:43,059
For your amusement. For my amusement. Notice the abstract.
391
00:48:43,060 --> 00:48:45,950
And also the first sentence of the paper is a little curious.
392
00:48:45,970 --> 00:48:52,150
The paper presents mathematical and software developments that are the basis for a suite of programs.
393
00:48:52,540 --> 00:48:55,630
It doesn't say the paper presents a suite of programs.
394
00:48:56,120 --> 00:49:02,020
Now, I don't know why, but I can guess it's the universal problem that publishing software papers is hard.
395
00:49:02,590 --> 00:49:04,750
Nobody respects software.
396
00:49:04,930 --> 00:49:12,220
So even when you have some software that a million people are going to use, you have to couch it in terms of mathematical development.
397
00:49:12,490 --> 00:49:15,070
This is something we on the telephone team are very aware of.
398
00:49:15,080 --> 00:49:19,750
We write a lot of papers and we're always careful to say that there's more going on in software.
399
00:49:20,020 --> 00:49:22,510
And of course, there is more going on less software.
400
00:49:23,680 --> 00:49:34,180
So let me finish guest have a couple of minutes by showing you how we can solve an initial volume problem in telephone.
401
00:49:38,970 --> 00:49:42,300
The underlying methods are the same, basically.
402
00:49:43,650 --> 00:49:47,190
In fact, our default is to call ody 113.
403
00:49:48,330 --> 00:49:52,979
But the setting is all about the way of computing, you know.
404
00:49:52,980 --> 00:50:07,190
Well. So let me just in the very little time available, just give you a sense of how this works.
405
00:50:07,550 --> 00:50:13,300
Suppose I wanted to solve the Vanderpoel equation and said, fine,
406
00:50:15,050 --> 00:50:24,380
I'm just going to do one version and I'll say C equals one, and then I'll say and equals what we call a chap.
407
00:50:24,950 --> 00:50:29,419
So it's how about this? How we define a differential operator, we give it a domain.
408
00:50:29,420 --> 00:50:39,560
So from 0 to 50 and the operator part of the differential operator is an anonymous function of time and the dependent variables.
409
00:50:39,860 --> 00:50:48,730
So for example, for the Vanderpoel, it would be the second derivative of u comma two plus the constant times use.
410
00:50:48,760 --> 00:50:51,200
Well, you don't need the dot u squared minus one.
411
00:50:53,070 --> 00:51:02,160
Times differ view plus you so notice a key difference here from the standard MATLAB way of operating is
412
00:51:02,370 --> 00:51:07,410
we're perfectly happy with higher order you don't have to reformulate it to have fun does that for you.
413
00:51:07,890 --> 00:51:14,400
So there's a second order formulation, Chad. Fun ultimately is going to do it behind the scenes, convert that to first order.
414
00:51:14,970 --> 00:51:20,190
Then, of course, you have to give it an initial condition and LBC stands for left boundary condition.
415
00:51:20,580 --> 00:51:26,820
So if we wanted to set the function value in the derivative, we can do that.
416
00:51:29,400 --> 00:51:36,360
And then to solve the problem we use backslash that we say and backslash zero.
417
00:51:36,540 --> 00:51:41,250
So what we've just done is the first time you do anything, it takes a long time.
418
00:51:42,000 --> 00:51:46,580
That's MATLAB. Let's do it again. Okay.
419
00:51:46,790 --> 00:51:53,300
So as you know, in linear algebra, backslash is matlab notation for how you solve x equals B.
420
00:51:53,510 --> 00:51:57,740
Well, in turbofan, we've overloaded that to solving a differential equation.
421
00:51:58,040 --> 00:52:11,390
So what we've just done, we have just solved an of t you equals the right hand sorry times the unknown vector you equals the right hand side zero.
422
00:52:12,260 --> 00:52:21,200
By methods I'm not talking about now. So if I played it, I get the same picture that we've seen before.
423
00:52:21,320 --> 00:52:27,129
This is now a cheddar fan, so. It's actually a polynomial.
424
00:52:27,130 --> 00:52:30,160
If I said the length of you, I'd see that it's a problem on your degree.
425
00:52:30,160 --> 00:52:40,540
1070. Or if I said flat coefficients of you, I'd see that the Chevy shaft coefficients decrease down to machine precision like that.
426
00:52:40,840 --> 00:52:50,050
And if I wanted to do the phase plane, I could say FIG. two and then plot you against the derivative here as I read my.
427
00:52:56,750 --> 00:53:00,920
So the pictures end up very similar, but the setting is a little bit different.
428
00:53:01,220 --> 00:53:02,600
Okay. See you on Friday.