1
00:00:00 --> 00:00:01
 
 

2
00:00:01 --> 00:00:02
The following content is
provided under a Creative

3
00:00:02 --> 00:00:03
Commons license.
 

4
00:00:03 --> 00:00:06
Your support will help MIT
OpenCourseWare continue to

5
00:00:06 --> 00:00:09
offer high-quality educational
resources for free.

6
00:00:09 --> 00:00:12
To make a donation, or to view
additional materials from

7
00:00:12 --> 00:00:16
hundreds of MIT courses, if
visit MIT OpenCourseWare

8
00:00:16 --> 00:00:20
at ocw.mit.edu.
 

9
00:00:20 --> 00:00:23
PROFESSOR STRANG: So this
is review session number

10
00:00:23 --> 00:00:26
five, I guess it is.
 

11
00:00:26 --> 00:00:31
And it comes before an exam
next Tuesday, and actually

12
00:00:31 --> 00:00:38
there'll be a further review
session number six on Monday,

13
00:00:38 --> 00:00:40
right before the exam.
 

14
00:00:40 --> 00:00:48
So maybe today we would,
there's a homework problem set

15
00:00:48 --> 00:00:56
on Chapter 2, mostly the
oscillating masses and springs

16
00:00:56 --> 00:01:08
and today's lecture that you
see traces of, networks.

17
00:01:08 --> 00:01:12
And masses and springs
also this Saturday case.

18
00:01:12 --> 00:01:15
So I'm open as always
to questions.

19
00:01:15 --> 00:01:17
Yes please, thank you?
 

20
00:01:17 --> 00:01:19
AUDIENCE: 2.2 number 6.
 

21
00:01:19 --> 00:01:24
PROFESSOR STRANG:
2.2, number six.

22
00:01:24 --> 00:01:24
OK.
 

23
00:01:24 --> 00:01:25
Yeah.
 

24
00:01:25 --> 00:01:31
So this is, and of course
you understand that, so

25
00:01:31 --> 00:01:34
I'm happy, that's a good
question to discuss.

26
00:01:34 --> 00:01:37
And maybe number seven
people well have

27
00:01:37 --> 00:01:40
something to say about.
 

28
00:01:40 --> 00:01:42
Good.
 

29
00:01:42 --> 00:01:47
So that just fine, so let me
start right in on those.

30
00:01:47 --> 00:01:53
So, number six is the fact, I
mean everybody understands that

31
00:01:53 --> 00:01:57
when energy is conserved,
that's an important thing.

32
00:01:57 --> 00:02:02
And so the question is first
when is energy conserved in

33
00:02:02 --> 00:02:03
the differential equations?
 

34
00:02:03 --> 00:02:05
In the equation we're
trying to solve?

35
00:02:05 --> 00:02:11
And if it is, then we want to
know, we would like to choose

36
00:02:11 --> 00:02:15
difference methods that
also conserve energy.

37
00:02:15 --> 00:02:20
They may not be exactly right,
they may not be exactly at a

38
00:02:20 --> 00:02:25
right point on this circle, if
we're in that model problem,

39
00:02:25 --> 00:02:28
but still on the circle.
 

40
00:02:28 --> 00:02:32
So, and the point is that the
trapezoidal method does stay on

41
00:02:32 --> 00:02:34
the circle, and of course the
differential equation

42
00:02:34 --> 00:02:36
stays on the circle.
 

43
00:02:36 --> 00:02:42
Can I, so, and I put quite a
bit into this problem six.

44
00:02:42 --> 00:02:51
So this is 2.2.6, and and
let me try say something

45
00:02:51 --> 00:02:53
about that, OK.
 

46
00:02:53 --> 00:02:57
So, first of all, there's the
continuous problem. du/dt=Au.

47
00:02:59 --> 00:03:01
When does that conserve energy?
 

48
00:03:01 --> 00:03:06
And then there's the discrete
problem, which we know that

49
00:03:06 --> 00:03:11
Euler's doesn't conserve
energy, because we've seen it,

50
00:03:11 --> 00:03:13
the computer shows
you right away.

51
00:03:13 --> 00:03:19
It spirals up from the circle,
it spirals in, or forward

52
00:03:19 --> 00:03:20
and backward, or whatever.
 

53
00:03:20 --> 00:03:22
But trapezoidal method,
is that the one that

54
00:03:22 --> 00:03:24
turns out to be well?
 

55
00:03:24 --> 00:03:29
OK, so it refers to equation 24
as the trapezoidal method, and

56
00:03:29 --> 00:03:32
let me try to follow
that notation.

57
00:03:32 --> 00:03:45
Yep. the trapezoidal
method is this one.

58
00:03:45 --> 00:03:48
Did we get music there for
the trapezoidal method?

59
00:03:48 --> 00:03:49
OK.
 

60
00:03:49 --> 00:03:50
U_(n+1)=(I+A*delta t/2)U_n.
 

61
00:03:50 --> 00:03:58
 
 

62
00:03:58 --> 00:04:01
OK.
 

63
00:04:01 --> 00:04:02
Right.
 

64
00:04:02 --> 00:04:03
OK.
 

65
00:04:03 --> 00:04:09
So that, and actually problem
seven, if you maybe want to

66
00:04:09 --> 00:04:15
discuss too, is the question of
how accurate this is compared

67
00:04:15 --> 00:04:16
to the differential equation.
 

68
00:04:16 --> 00:04:22
Everybody should see that this
really came from the original

69
00:04:22 --> 00:04:28
way to look at this was
(U_(n+1)-U_n)/delta t, that

70
00:04:28 --> 00:04:33
approximated the derivative
equals A and then I'm taking

71
00:04:33 --> 00:04:41
half at U, at the new time
and half at the old time.

72
00:04:41 --> 00:04:43
So it's got that centering.
 

73
00:04:43 --> 00:04:48
That we suspect will give us
a little extra accuracy.

74
00:04:48 --> 00:04:55
OK. so two questions then, one
was the stability, so problem

75
00:04:55 --> 00:05:07
six was the energy conserved,
and problem 2.2.7, if I

76
00:05:07 --> 00:05:12
anticipated, is the
order of accuracy.

77
00:05:12 --> 00:05:18
So these are both topics that
are extremely important in

78
00:05:18 --> 00:05:23
choosing a difference method.
 

79
00:05:23 --> 00:05:28
We would like to know first
when is energy conserved there?

80
00:05:28 --> 00:05:31
What differential equations
have conserved energy?

81
00:05:31 --> 00:05:35
Physically we kind of
can see them coming.

82
00:05:35 --> 00:05:42
If a physical universe is not
being somehow, lots of physical

83
00:05:42 --> 00:05:47
problems we see those masses
and springs oscillating, say

84
00:05:47 --> 00:05:51
OK, nothing's coming in from
outside, how could energy there

85
00:05:51 --> 00:05:57
would be the sum of the kinetic
energy of the masses, and the

86
00:05:57 --> 00:05:59
potential energy
in the springs.

87
00:05:59 --> 00:06:05
So energy passes between
kinetic when the mass is

88
00:06:05 --> 00:06:11
zooming past equilibrium, and
potential energy when the mass

89
00:06:11 --> 00:06:16
is stretching the spring
so we've got two cases.

90
00:06:16 --> 00:06:19
And we would hope, and this
trapezoidal method comes

91
00:06:19 --> 00:06:22
through, that energy
is conserved.

92
00:06:22 --> 00:06:27
So can I just begin
with this one?

93
00:06:27 --> 00:06:30
Maybe I always ought to say,
because you guys are also

94
00:06:30 --> 00:06:37
thinking about the quiz.
 

95
00:06:37 --> 00:06:41
So, for example, this question
about how to find the order of

96
00:06:41 --> 00:06:48
accuracy, I'll speak about
that. but let me just say

97
00:06:48 --> 00:06:52
that's not something that we've
done in enough detail that I

98
00:06:52 --> 00:06:55
would expect you to be quick on
the quiz and just be

99
00:06:55 --> 00:06:59
able to do it out.
 

100
00:06:59 --> 00:07:05
I'll try to choose questions
on the exam that you really

101
00:07:05 --> 00:07:07
have had more practice with.
 

102
00:07:07 --> 00:07:10
But this is certainly
important, so it was that

103
00:07:10 --> 00:07:13
definitely right to
put on the homework.

104
00:07:13 --> 00:07:14
And this is important.
 

105
00:07:14 --> 00:07:20
OK, so let me tackle this one.
 

106
00:07:20 --> 00:07:23
First the differential
equation.

107
00:07:23 --> 00:07:29
So by energy here I'm meaning
just the length of u squared.

108
00:07:29 --> 00:07:35
So now I'm looking at
energy conserved, OK.

109
00:07:35 --> 00:07:39
So I'm hoping energy conserved
would mean that the derivative

110
00:07:39 --> 00:07:44
of u squared was zero.
 

111
00:07:44 --> 00:07:48
That what conserving
energy would mean.

112
00:07:48 --> 00:07:52
And my question is which
differential equations,

113
00:07:52 --> 00:07:57
what's the condition
on A, in other words?

114
00:07:57 --> 00:08:02
How would I recognize from this
matrix A that I have this

115
00:08:02 --> 00:08:04
interesting property?
 

116
00:08:04 --> 00:08:04
OK.
 

117
00:08:04 --> 00:08:09
So, let me just show
you what I would do.

118
00:08:09 --> 00:08:15
Another way to write u
squared is u transposed u.

119
00:08:15 --> 00:08:19
These are vectors, of course.
 

120
00:08:19 --> 00:08:23
So what's the derivative
of u transpose u?

121
00:08:23 --> 00:08:26
My equation is telling me what
the derivative of u is here,

122
00:08:26 --> 00:08:28
I've got the thing squared.
 

123
00:08:28 --> 00:08:31
OK.
 

124
00:08:31 --> 00:08:34
I have a product here, right?
 

125
00:08:34 --> 00:08:36
I've u's times u's.
 

126
00:08:36 --> 00:08:40
So I'm going to use the
standard, they happen to be

127
00:08:40 --> 00:08:44
vectors, so if I want to use
like freshman calculus I'd have

128
00:08:44 --> 00:08:48
to get down to the scalar to
get down to the numbers, but I

129
00:08:48 --> 00:08:51
absolutely could do that and
just follow them along,

130
00:08:51 --> 00:08:55
component by component, or I
could try to do it a vector,

131
00:08:55 --> 00:08:57
with a whole column at a time.
 

132
00:08:57 --> 00:09:01
And let me try that.
 

133
00:09:01 --> 00:09:04
It's going to be the
product rule, right?

134
00:09:04 --> 00:09:12
In some form I'll have this guy
times the derivative of this

135
00:09:12 --> 00:09:16
plus the derivative, I'll keep
them in order, the derivative

136
00:09:16 --> 00:09:20
of this thing, times this guy.
 

137
00:09:20 --> 00:09:22
Right?
 

138
00:09:22 --> 00:09:23
That's the product rule.
 

139
00:09:23 --> 00:09:26
OK, now what do I know here?
 

140
00:09:26 --> 00:09:29
I know that the
du/dt is Au, right?

141
00:09:29 --> 00:09:34
So this is u transposed Au.
 

142
00:09:34 --> 00:09:41
And I know du, is that
dt meant to be dt?

143
00:09:41 --> 00:09:46
So du/dt Au, again.
 

144
00:09:46 --> 00:09:49
Look, this isn't difficult.
 

145
00:09:49 --> 00:09:54
It's Au transpose u.
 

146
00:09:54 --> 00:09:59
And the question is,
when is this zero, OK?

147
00:09:59 --> 00:10:03
So those were the two terms
from the product rule, and

148
00:10:03 --> 00:10:05
notice they're not
exactly the same.

149
00:10:05 --> 00:10:11
This is u transpose Au,
and what's this guy? u

150
00:10:11 --> 00:10:14
transpose A transpose u.
 

151
00:10:14 --> 00:10:20
So if I put them together, I
have u transpose times the A

152
00:10:20 --> 00:10:24
and the A transpose times u.
 

153
00:10:24 --> 00:10:29
And I'm hoping that this will
be zero, for all the solutions

154
00:10:29 --> 00:10:31
that I've come up with.
 

155
00:10:31 --> 00:10:38
So the condition is simply that
A plus A, we want that to be

156
00:10:38 --> 00:10:42
the zero matrix, A
plus A transpose.

157
00:10:42 --> 00:10:46
In other words, if A
transpose is minus A,

158
00:10:46 --> 00:10:49
that's the good one.
 

159
00:10:49 --> 00:10:52
If A transpose is minus A,
this is zero, that's zero,

160
00:10:52 --> 00:10:55
that's zero, that's zero,
energy's conserved.

161
00:10:55 --> 00:11:01
So the energy is conserved
when A transpose is minus A.

162
00:11:01 --> 00:11:08
It's for the anti-symmetric
A's that energy is conserved.

163
00:11:08 --> 00:11:12
And of course this
all makes sense.

164
00:11:12 --> 00:11:17
What are the special solutions
to that differential equation?

165
00:11:17 --> 00:11:22
The special solutions to this
equation are e to the, just,

166
00:11:22 --> 00:11:29
this is connecting now with
things that really are basic.

167
00:11:29 --> 00:11:35
The special solutions, the pure
eigen solutions, the ones that

168
00:11:35 --> 00:11:41
follow their own paths, are
the e^(lambda*t)x's, right?

169
00:11:41 --> 00:11:46
Where x is an eigenvector of A,
and lambda's an eigenvalue.

170
00:11:46 --> 00:11:50
Those are the guys and we
expect to have n of them, and

171
00:11:50 --> 00:11:53
we expect a combination of
those to give us the general

172
00:11:53 --> 00:11:56
solution and to match the
boundary conditions.

173
00:11:56 --> 00:12:00
So these are the, these n of
these guys with n different

174
00:12:00 --> 00:12:04
eigenvectors and their
eigenvalues are the heart

175
00:12:04 --> 00:12:08
of problems like this.
 

176
00:12:08 --> 00:12:13
And of course that's the
additional homework problem

177
00:12:13 --> 00:12:19
that wasn't in the book but I
added as an additional problem

178
00:12:19 --> 00:12:22
was exactly that, to get
you to practice with these

179
00:12:22 --> 00:12:24
eigenvectors and eigenvalues.
 

180
00:12:24 --> 00:12:29
OK, now, what's the deal, I
want to connect this energy

181
00:12:29 --> 00:12:32
conserving with this
picture of solutions.

182
00:12:32 --> 00:12:38
When would this keep
the same energy?

183
00:12:38 --> 00:12:46
When would this have constant
energy, constant length?

184
00:12:46 --> 00:12:50
The length would be constant,
since x is certainly, that's an

185
00:12:50 --> 00:12:52
eigenvector, whatever it is.
 

186
00:12:52 --> 00:12:54
This is what's changing.
 

187
00:12:54 --> 00:12:58
And now I want to know when
does the length not change?

188
00:12:58 --> 00:13:02
Well, the test would be that
this number should have

189
00:13:02 --> 00:13:07
absolute value one, right?
 

190
00:13:07 --> 00:13:13
If this keeps absolute value
one, then in the eigenvalue

191
00:13:13 --> 00:13:16
picture I have energy
staying the same.

192
00:13:16 --> 00:13:17
OK?
 

193
00:13:17 --> 00:13:22
Now, when will this
have magnitude one?

194
00:13:22 --> 00:13:25
Time is running along,
this is either the lambda

195
00:13:25 --> 00:13:28
t, so which lambdas?
 

196
00:13:28 --> 00:13:32
Zero, certainly, but
now there's more, you

197
00:13:32 --> 00:13:33
gotta know the others.
 

198
00:13:33 --> 00:13:38
What other lambdas will have,
what other lambdas give me

199
00:13:38 --> 00:13:43
this thing stays on the unit
circle, absolute value one?

200
00:13:43 --> 00:13:48
Key question you must
know. lambda could be?

201
00:13:48 --> 00:13:52
Imaginary. lambda could be
imaginary; right, lambda could

202
00:13:52 --> 00:13:54
be imaginary. e^(i*omega*t).
 

203
00:13:54 --> 00:13:59
 
 

204
00:13:59 --> 00:14:02
That's just like basic fact
about complex numbers, that if

205
00:14:02 --> 00:14:07
lambda's imaginary we would
have cosine of something t plus

206
00:14:07 --> 00:14:10
i times the sign of something
t, cos squared plus sine

207
00:14:10 --> 00:14:13
squared being one, we'd
be on the unit circle.

208
00:14:13 --> 00:14:18
So from this picture we
would want the lambdas

209
00:14:18 --> 00:14:20
to be pure imaginary.
 

210
00:14:20 --> 00:14:25
And now a little next step,
what we'd like for

211
00:14:25 --> 00:14:33
eigenvectors, because the real
solution will not be just one

212
00:14:33 --> 00:14:37
of these guys but
a combination.

213
00:14:37 --> 00:14:41
So when we have a combination
each one is doing its thing,

214
00:14:41 --> 00:14:49
each one better have lambda
imaginary but more than that,

215
00:14:49 --> 00:14:51
we would want the x's
to be perpendicular.

216
00:14:51 --> 00:14:57
Because if the x's interact,
then this guy, one of these,

217
00:14:57 --> 00:15:00
you will say there's only one
there, but I'm thinking

218
00:15:00 --> 00:15:02
of n of them there.
 

219
00:15:02 --> 00:15:04
A combination of
say, two of them.

220
00:15:04 --> 00:15:08
Suppose I have an
e^(lambda*1t)*x_1, and an

221
00:15:08 --> 00:15:13
e^(lambda*2t)*x_2, when
does that conserve energy?

222
00:15:13 --> 00:15:20
Well, each one will, but the
combination will be fine if

223
00:15:20 --> 00:15:22
the x's are perpendicular.
 

224
00:15:22 --> 00:15:26
Because if I have perpendicular
vectors, then the length of the

225
00:15:26 --> 00:15:30
whole combination by Pythagoras
is just one squared and the

226
00:15:30 --> 00:15:34
other squared and each of
those pieces is constant.

227
00:15:34 --> 00:15:36
Let me say what I'm
trying to say.

228
00:15:36 --> 00:15:41
That the eigenvalue, eigen
function picture also tells

229
00:15:41 --> 00:15:45
us that we would like
imaginary eigenvalues and

230
00:15:45 --> 00:15:48
perpendicular eigenvectors.
 

231
00:15:48 --> 00:15:51
And that is exactly what
you get from A transpose

232
00:15:51 --> 00:15:53
equal minus A.
 

233
00:15:53 --> 00:15:58
So A transpose equal minus
A is exactly, matrices

234
00:15:58 --> 00:16:03
anti-symmetric matrices, have
perpendicular eigenvectors,

235
00:16:03 --> 00:16:09
just like symmetric, but the
eigenvalues are pure imaginary.

236
00:16:09 --> 00:16:11
Instead of all being real,
they're all pure imaginary.

237
00:16:11 --> 00:16:16
In other words, that answer and
the discussion here came to

238
00:16:16 --> 00:16:21
the same conclusion, that a
should be anti-symmetric.

239
00:16:21 --> 00:16:22
OK.
 

240
00:16:22 --> 00:16:30
Now let me look at, is that OK,
so that's a discussion which

241
00:16:30 --> 00:16:33
is worth knowing about
differential equations.

242
00:16:33 --> 00:16:35
When is energy conserved.
 

243
00:16:35 --> 00:16:38
Now I want to do, or the
problem asks me to do, what

244
00:16:38 --> 00:16:40
about this difference equation?
 

245
00:16:40 --> 00:16:43
When is energy conserved there?
 

246
00:16:43 --> 00:16:51
And I believe it will be, this
is the requirement for the

247
00:16:51 --> 00:16:56
differential equation to be OK,
to conserve energy and so I'm

248
00:16:56 --> 00:16:59
going to expect, I'm going
to need that in this one.

249
00:16:59 --> 00:17:04
And is that enough? if I have
this A transpose equal minus A,

250
00:17:04 --> 00:17:08
anti-symmetric, it was good for
this, does it also

251
00:17:08 --> 00:17:10
do the job here?
 

252
00:17:10 --> 00:17:16
Is this trapezoidal method
just cool so that it will

253
00:17:16 --> 00:17:18
conserve energy, too.
 

254
00:17:18 --> 00:17:21
And the answer, I think, is
yes, and the problem was to

255
00:17:21 --> 00:17:24
prove it, or to see why.
 

256
00:17:24 --> 00:17:28
So what do I now want?
 

257
00:17:28 --> 00:17:30
If you don't mind my racing,
I'm now going to look

258
00:17:30 --> 00:17:35
at the discrete guide.
 

259
00:17:35 --> 00:17:39
So now I'm looking at
when is U_(n+1) squared

260
00:17:39 --> 00:17:43
equal U_n squared?
 

261
00:17:43 --> 00:17:46
That's what I mean by
conserving energy in

262
00:17:46 --> 00:17:47
the discrete case.
 

263
00:17:47 --> 00:17:50
At every step, same energy.
 

264
00:17:50 --> 00:17:53
So now I want to look at the
energy in U_(n+1) compared to

265
00:17:53 --> 00:17:59
the energy in U_n, and I want
to see that this holds.

266
00:17:59 --> 00:18:08
Probably, there's some
smart way to do that.

267
00:18:08 --> 00:18:12
Now we're down to just
the math questions.

268
00:18:12 --> 00:18:16
Math is always looking for
some, you just sort of do

269
00:18:16 --> 00:18:21
the right thing, you stand
back and poof it works.

270
00:18:21 --> 00:18:23
OK, so what's the right thing?
 

271
00:18:23 --> 00:18:28
Hopefully I have helped you and
me by saying what would be

272
00:18:28 --> 00:18:30
a good idea to do here.
 

273
00:18:30 --> 00:18:35
Can I just look?
 

274
00:18:35 --> 00:18:41
OK, it say, oh, does
it say what to do?

275
00:18:41 --> 00:18:42
Yeah.
 

276
00:18:42 --> 00:18:45
It says multiply
by U_(n+1)+U_n.

277
00:18:45 --> 00:18:48
 
 

278
00:18:48 --> 00:18:53
Take the dotted product,
that's interesting.

279
00:18:53 --> 00:18:55
Take the dot product,
so why did that work?

280
00:18:55 --> 00:18:59
Take the dot product of both
sides with U_(n+1)-U_n.

281
00:18:59 --> 00:19:02
 
 

282
00:19:02 --> 00:19:06
Did anybody succeed
with this idea?

283
00:19:06 --> 00:19:07
But that's the idea.
 

284
00:19:07 --> 00:19:11
And hopefully we'll
get it to work.

285
00:19:11 --> 00:19:17
That if I multiply both sides
by U_n+U_n, maybe better

286
00:19:17 --> 00:19:20
if I look at it this way.
 

287
00:19:20 --> 00:19:23
I'm sort of OK to
do it that way.

288
00:19:23 --> 00:19:25
Suppose I multiply both sides.
 

289
00:19:25 --> 00:19:30
So now I'm following on this
idea, That equation I've

290
00:19:30 --> 00:19:34
rewritten here and without
practice I don't know which one

291
00:19:34 --> 00:19:36
is the good one to start with.
 

292
00:19:36 --> 00:19:41
But I'm pretty OK with
starting with this one.

293
00:19:41 --> 00:19:43
So what's my idea?
 

294
00:19:43 --> 00:19:47
That's my equation, now I'm
going to multiply both sides

295
00:19:47 --> 00:19:51
by U_(n+1)+U_n transpose.
 

296
00:19:51 --> 00:19:54
Now I don't have room to
do it, unfortunately.

297
00:19:54 --> 00:20:00
I want to stick in here U_(n+
with a plus sign in there, and

298
00:20:00 --> 00:20:03
of course I have to do
the same thing here.

299
00:20:03 --> 00:20:08
OK. are you OK, do you
see what I'm doing?

300
00:20:08 --> 00:20:10
I want to show that this
equation, which is the same

301
00:20:10 --> 00:20:16
as this equation, has
this property which is a

302
00:20:16 --> 00:20:18
copy of this property.
 

303
00:20:18 --> 00:20:21
Here would be another
way to do it.

304
00:20:21 --> 00:20:26
We could do it, the way we're
going to do it now sort of

305
00:20:26 --> 00:20:30
compares with the way we
started with the derivative

306
00:20:30 --> 00:20:31
of the norm squared.
 

307
00:20:31 --> 00:20:35
I could also ask the same
question by following

308
00:20:35 --> 00:20:37
eigenvectors.
 

309
00:20:37 --> 00:20:40
I could also ask the same
question by following

310
00:20:40 --> 00:20:41
eigenvectors.
 

311
00:20:41 --> 00:20:51
I'm guessing that here the
eigenvalues U_(n+1) is, you

312
00:20:51 --> 00:20:53
see I could do it both ways.
 

313
00:20:53 --> 00:20:56
Maybe having just done
eigenvectors let me do

314
00:20:56 --> 00:20:57
this one by eigenvectors.
 

315
00:20:57 --> 00:21:04
So an eigenvector of A, when
it's x itself, is, what

316
00:21:04 --> 00:21:06
happens to an eigenvector.
 

317
00:21:06 --> 00:21:12
Suppose U_0 is an eigenvector
x of A, What's U_1?

318
00:21:12 --> 00:21:15
Yeah, you really should
see this question.

319
00:21:15 --> 00:21:26
So U_0 is the eigenvector
x, then what is U_1?

320
00:21:26 --> 00:21:28
Let me just write it here.
 

321
00:21:28 --> 00:21:30
Ax equaling lambda*x.
 

322
00:21:31 --> 00:21:37
So these are the eigenvalues of
A, and we've learned that

323
00:21:37 --> 00:21:42
they're pure imaginary in this
case when we're ready to go,

324
00:21:42 --> 00:21:48
and now I'd like to know that
we get the good thing here.

325
00:21:48 --> 00:21:53
OK, so if U_n is an
eigenvector, what is U_(n+1)?

326
00:21:55 --> 00:22:02
OK, so can I just do that,
U_(n+1) is, so what do I

327
00:22:02 --> 00:22:10
have on that right hand
side? x and what is Ax?

328
00:22:10 --> 00:22:12
It's lambda, right?
 

329
00:22:12 --> 00:22:13
It's lambda*x.
 

330
00:22:14 --> 00:22:24
So all this is one plus lambda
delta t on two x but now I've

331
00:22:24 --> 00:22:30
also got to bring this guy
over here, it's inverse.

332
00:22:30 --> 00:22:33
And see what that does.
 

333
00:22:33 --> 00:22:37
Now it's the inverse, so it's
going to have the same

334
00:22:37 --> 00:22:41
eigenvector and the
eigenvalue's going to go in the

335
00:22:41 --> 00:22:50
denominator and it'll be one
minus lambda delta t over two.

336
00:22:50 --> 00:22:52
OK, so that's U_(n+1).
 

337
00:22:53 --> 00:22:56
Do you see what's
happening here?

338
00:22:56 --> 00:23:02
The eigenvector x, if we start
with that eigenvector x, we

339
00:23:02 --> 00:23:04
come out with a multiple of x.
 

340
00:23:04 --> 00:23:06
And this is the multiple.
 

341
00:23:06 --> 00:23:12
So each find a different step
multiplies by a number just the

342
00:23:12 --> 00:23:16
way each, in the continuous
case we were multiplying by

343
00:23:16 --> 00:23:21
either the lambda t and in the
discrete step by step case

344
00:23:21 --> 00:23:24
we're multiplying
by that number.

345
00:23:24 --> 00:23:29
Actually, this is why problem
seven is important, because if

346
00:23:29 --> 00:23:34
we want to know how accurate
the comparison is I want to

347
00:23:34 --> 00:23:39
compare either the lambda
t with that number.

348
00:23:39 --> 00:23:47
So problem six is asking a
question about that ratio.

349
00:23:47 --> 00:23:50
And problem seven is asking
another question about

350
00:23:50 --> 00:23:51
that very same ratio.
 

351
00:23:51 --> 00:23:54
Now what's the question
for problem six?

352
00:23:54 --> 00:24:08
When will this vector have the
same length this x was U_n.

353
00:24:08 --> 00:24:14
So I started with the U_n, I
multiplied by this number to

354
00:24:14 --> 00:24:19
get U_(n+1), when do they
have the same length?

355
00:24:19 --> 00:24:27
When that number has
absolute value one.

356
00:24:27 --> 00:24:31
So if I'm watching
eigenvectors, this guy had

357
00:24:31 --> 00:24:35
absolute value one because
lambda was imaginary.

358
00:24:35 --> 00:24:38
Now, what about this guy?
lambda's still that

359
00:24:38 --> 00:24:40
same lambda, imaginary.
 

360
00:24:40 --> 00:24:44
What can you tell me about one
plus, so lambda is some i,

361
00:24:44 --> 00:24:50
omega, delta t over two and
down here I have one minus i

362
00:24:50 --> 00:24:54
omega, that's the lambda
delta t over two.

363
00:24:54 --> 00:25:00
I believe that that does
have absolute value one.

364
00:25:00 --> 00:25:02
Anybody tell me why?
 

365
00:25:02 --> 00:25:09
So this is checking that
energy is conserved

366
00:25:09 --> 00:25:11
for each eigenvector.
 

367
00:25:11 --> 00:25:15
The energy, because the
eigenvector is multiplied by

368
00:25:15 --> 00:25:19
that number and that's some
number, it's some complex

369
00:25:19 --> 00:25:22
number, but I believe it has
absolute value one and I

370
00:25:22 --> 00:25:24
believe you can tell me why.
 

371
00:25:24 --> 00:25:25
Yep.
 

372
00:25:25 --> 00:25:28
Because they're
complex conjugates.

373
00:25:28 --> 00:25:31
This numerator and the
denominator are complex

374
00:25:31 --> 00:25:38
conjugates, in the complex
plane here's the one, and I go

375
00:25:38 --> 00:25:44
up and either the i omega delta
t over two, or on this one I

376
00:25:44 --> 00:25:48
go down by, but those
lengths are the same.

377
00:25:48 --> 00:25:52
That numerator, the length of
the numerator is that guy, the

378
00:25:52 --> 00:25:56
length of the denominator is
this guy, and their

379
00:25:56 --> 00:25:58
ratio is one.
 

380
00:25:58 --> 00:26:01
So I think that this
gives us the point

381
00:26:01 --> 00:26:04
about complex numbers.
 

382
00:26:04 --> 00:26:11
That a complex number and its
conjugate automatically have

383
00:26:11 --> 00:26:14
ratio of magnitude one.
 

384
00:26:14 --> 00:26:17
You see the difference
between Euler's method.

385
00:26:17 --> 00:26:25
So Euler's method, so forward
Euler Forward Euler would not

386
00:26:25 --> 00:26:31
have had this stuff
on the left side.

387
00:26:31 --> 00:26:33
It would all have been
on the right hand side.

388
00:26:33 --> 00:26:36
Forward Euler would have
been about i plus A

389
00:26:36 --> 00:26:40
delta t. delta t A.
 

390
00:26:40 --> 00:26:42
And what are its eigenvalues?
 

391
00:26:42 --> 00:26:48
One plus i omega
delta t, right?

392
00:26:48 --> 00:26:56
With no, we're not dividing by
anybody. this part is up top

393
00:26:56 --> 00:26:59
too, so it's one plus
i omega delta t.

394
00:26:59 --> 00:27:02
Now, does that have
absolute value one?

395
00:27:02 --> 00:27:05
Well, you know from the way I'm
asking the question, what can

396
00:27:05 --> 00:27:08
you tell me about the absolute
value of the forward

397
00:27:08 --> 00:27:11
Euler growth factor?
 

398
00:27:11 --> 00:27:13
Greater than one.
 

399
00:27:13 --> 00:27:17
Because this is the one, and
this is the i omega delta t,

400
00:27:17 --> 00:27:19
maybe went up twice as far.
 

401
00:27:19 --> 00:27:21
And there was nobody
to divide by.

402
00:27:21 --> 00:27:24
It's bigger than one,
so it blows up.

403
00:27:24 --> 00:27:30
And the backward Euler had only
the one over one minus i omega

404
00:27:30 --> 00:27:37
delta t, so the backward was
like this, one over it.

405
00:27:37 --> 00:27:37
And lesson one.
 

406
00:27:37 --> 00:27:41
But this balance has absolute
value of equal one.

407
00:27:41 --> 00:27:47
So, OK, that's the sort of
heart of what's going on.

408
00:27:47 --> 00:27:54
Can I, before I tackle the
question using the hint there,

409
00:27:54 --> 00:27:58
which would take me on
another blackboard, can I

410
00:27:58 --> 00:28:00
discuss question seven?
 

411
00:28:00 --> 00:28:02
Were you going to ask
me about number seven?

412
00:28:02 --> 00:28:02
AUDIENCE: Yeah, I was.
 

413
00:28:02 --> 00:28:03
PROFESSOR STRANG: You were?
 

414
00:28:03 --> 00:28:03
OK.
 

415
00:28:03 --> 00:28:04
Alright.
 

416
00:28:04 --> 00:28:07
We get the answer.
 

417
00:28:07 --> 00:28:12
So, question seven is
about the accuracy.

418
00:28:12 --> 00:28:18
So here's the correct number,
this is my e^(i*omega*t),

419
00:28:18 --> 00:28:24
that's the correct number that
I should be multiplying by.

420
00:28:24 --> 00:28:30
And the actual number that I'm
multiplying by is that much.

421
00:28:30 --> 00:28:34
Or, in the forward Euler
case, it's that one.

422
00:28:34 --> 00:28:40
And so I'm comparing
the one step accuracy.

423
00:28:40 --> 00:28:44
So let me compare
one step accuracy.

424
00:28:44 --> 00:28:49
So this is the topic now,
of order of accuracy.

425
00:28:49 --> 00:28:52
This is question seven.
 

426
00:28:52 --> 00:29:01
And it amounts to comparing
the, so what is one delta t

427
00:29:01 --> 00:29:03
step in the continuous case?
 

428
00:29:03 --> 00:29:08
So how much does the
eigenvector x, what does it

429
00:29:08 --> 00:29:12
get multiplied by if I
take a delta t step in the

430
00:29:12 --> 00:29:14
differential equation?
 

431
00:29:14 --> 00:29:18
So this is the exact delta t
step, what the find a

432
00:29:18 --> 00:29:22
difference won't get exactly
right so the exact

433
00:29:22 --> 00:29:26
step delta t?
 

434
00:29:26 --> 00:29:36
The differential equation, and
of course I'm always looking at

435
00:29:36 --> 00:29:46
Ax=lambda*x, the differential
equation multiplies x by what?

436
00:29:46 --> 00:29:54
What's the exact growth factor,
you could say, if my equation

437
00:29:54 --> 00:29:59
is du/dt=Au, that's the
differential equation, and I'm

438
00:29:59 --> 00:30:04
supposing that I'm on an
eigenvector x, so that the

439
00:30:04 --> 00:30:10
solution is e^(i*omega*t),
or e^(i*lambda*x).

440
00:30:12 --> 00:30:19
Now, what happened
over a delta t step?

441
00:30:19 --> 00:30:23
This is the answer like running
along for all time, all I'm

442
00:30:23 --> 00:30:30
asking you to do is if the step
is delta t, what's that number?

443
00:30:30 --> 00:30:33
I mean that number is telling
us how much it grew in that

444
00:30:33 --> 00:30:37
delta t step, and of course
it's e^(i*omega*t).

445
00:30:37 --> 00:30:41
 
 

446
00:30:41 --> 00:30:44
That's the exact growth
factor, that's G_exact.

447
00:30:44 --> 00:30:48
 
 

448
00:30:48 --> 00:30:51
In one time step, the
eigenvector gets multiplied by

449
00:30:51 --> 00:30:55
that, because that's the amount
of time that elapsed.

450
00:30:55 --> 00:31:02
And what's the approximate
growth, the growth factor from

451
00:31:02 --> 00:31:06
trapezoidal is just what
we wrote down here.

452
00:31:06 --> 00:31:14
One plus lambda delta t,
maybe I'll stay with lambda

453
00:31:14 --> 00:31:19
rather than i omega.
 

454
00:31:19 --> 00:31:23
Just use the lambda delta t,
and this was one plus delta t

455
00:31:23 --> 00:31:33
over two. lambda divided by one
minus delta t over two lambda.

456
00:31:33 --> 00:31:42
So question seven just says
compare that with that.

457
00:31:42 --> 00:31:48
Thinking of delta t as a small
time step, if delta t is zero,

458
00:31:48 --> 00:31:52
then of course either the zero
is one, if delta t is zero I

459
00:31:52 --> 00:31:55
get one here, they're correct
if delta t is zero,

460
00:31:55 --> 00:31:59
that's no big deal.
 

461
00:31:59 --> 00:32:09
How do I understand what
happens for small delta t?

462
00:32:09 --> 00:32:14
I'm comparing this exponential
for a small delta t with this

463
00:32:14 --> 00:32:15
guy for a small delta t.
 

464
00:32:15 --> 00:32:19
How do you make comparisons
for a small delta t?

465
00:32:19 --> 00:32:22
Well, that's what Taylor
series is all about.

466
00:32:22 --> 00:32:24
Let's do the Taylor series.
 

467
00:32:24 --> 00:32:27
What's the series for
the exponential?

468
00:32:27 --> 00:32:32
If delta t is small, I have e
to some little number, tell

469
00:32:32 --> 00:32:40
me, start me out on
the exponential.

470
00:32:40 --> 00:32:49
One, thanks, one plus, lambda
delta t plus, this is the

471
00:32:49 --> 00:32:52
exponential series, there are
only two series in this world

472
00:32:52 --> 00:32:54
that are worth knowing.
 

473
00:32:54 --> 00:32:55
Really, that's literally true.
 

474
00:32:55 --> 00:33:00
In calculus you study all these
infinite series, there are two

475
00:33:00 --> 00:33:02
that are important, that
are worth remembering

476
00:33:02 --> 00:33:04
long after calculus.
 

477
00:33:04 --> 00:33:09
And either the x, either the
whatever because one of them.

478
00:33:09 --> 00:33:13
OK, what's the next term?
 

479
00:33:13 --> 00:33:18
Over two, lambda delta t
squared over two, and then

480
00:33:18 --> 00:33:22
there's a cube guy if you don't
mind telling me what's the

481
00:33:22 --> 00:33:25
denominator in that one?
 

482
00:33:25 --> 00:33:26
It's three factorial six.
 

483
00:33:26 --> 00:33:27
Good.
 

484
00:33:27 --> 00:33:28
And onward.
 

485
00:33:28 --> 00:33:29
OK.
 

486
00:33:29 --> 00:33:32
So that's one of the series
that everybody should know.

487
00:33:32 --> 00:33:37
OK, how we going to
deal with this guy?

488
00:33:37 --> 00:33:40
We want to expand that,
so what's my goal?

489
00:33:40 --> 00:33:45
I want you to expand that in
powers of lambda delta t

490
00:33:45 --> 00:33:46
and compare with this.
 

491
00:33:46 --> 00:33:51
And see where, they aren't
going to be equal, right?

492
00:33:51 --> 00:33:54
At some point they're
going to be different.

493
00:33:54 --> 00:33:57
But at least they should
start out equal.

494
00:33:57 --> 00:34:03
So so here's the heart
of problem seven.

495
00:34:03 --> 00:34:08
How do I expand this
in powers of delta t?

496
00:34:08 --> 00:34:11
Do you mind if I just, this is
just a number let me put it

497
00:34:11 --> 00:34:19
times one over, so this is
times one minus delta t over

498
00:34:19 --> 00:34:23
two, lambda inward, right?
 

499
00:34:23 --> 00:34:25
I just bring that
up as a number.

500
00:34:25 --> 00:34:32
So it's this guy times
one over this guy.

501
00:34:32 --> 00:34:34
What do I do?
 

502
00:34:34 --> 00:34:46
This is, here's the moment
when the math tools get used.

503
00:34:46 --> 00:34:52
And I'm well aware that it's
like years since you did

504
00:34:52 --> 00:34:59
calculus or series or whatever,
and those tools get rusty.

505
00:34:59 --> 00:35:02
And the point is that they're
really genuine tools

506
00:35:02 --> 00:35:05
that we can now use.
 

507
00:35:05 --> 00:35:08
So what do you think?
 

508
00:35:08 --> 00:35:11
This is the problem one, this
is the one coming from the

509
00:35:11 --> 00:35:13
denominator; this is 1/(1-x).
 

510
00:35:13 --> 00:35:15
 
 

511
00:35:15 --> 00:35:19
So I have a 1/(1-x) deal.
 

512
00:35:19 --> 00:35:24
And what's the series for that?
 

513
00:35:24 --> 00:35:27
I said there were two series
worth remembering, and sure

514
00:35:27 --> 00:35:30
enough the exponential was
one of them and now we're

515
00:35:30 --> 00:35:32
ready for the other one.
 

516
00:35:32 --> 00:35:35
What's the series for that guy?
 

517
00:35:35 --> 00:35:39
1+x, good start.
 

518
00:35:39 --> 00:35:45
Plus x squared.
 

519
00:35:45 --> 00:35:49
Right, x squared plus
x cubed and so on.

520
00:35:49 --> 00:35:51
Real simple.
 

521
00:35:51 --> 00:35:54
It's all the same stuff
with no factorials.

522
00:35:54 --> 00:35:57
Those are the two
series to know.

523
00:35:57 --> 00:36:00
The exponential series and
the geometric series.

524
00:36:00 --> 00:36:03
Right, that's the
geometric series.

525
00:36:03 --> 00:36:07
OK, so that's what I've
got out of this stuff.

526
00:36:07 --> 00:36:08
Can I write it below?
 

527
00:36:08 --> 00:36:12
I have one plus delta
t over two lambda.

528
00:36:12 --> 00:36:16
Let me just call that x for
the moment. delta t over

529
00:36:16 --> 00:36:17
two lambda is my x.
 

530
00:36:17 --> 00:36:22
One plus x, and this is
1/(1-x), which you just told me

531
00:36:22 --> 00:36:29
is one plus x plus x squared
plus x cubed and so on.

532
00:36:29 --> 00:36:33
And now I've got to do
that multiplication.

533
00:36:33 --> 00:36:40
OK, x is, remember this is
x, I'm just saving space.

534
00:36:40 --> 00:36:43
Can you multiply those guys?
 

535
00:36:43 --> 00:36:47
So that's one plus x times
a lot of stuff here.

536
00:36:47 --> 00:36:49
What do I have all together?
 

537
00:36:49 --> 00:36:53
Well, the one, what's
the next term?

538
00:36:53 --> 00:36:54
Two x's?
 

539
00:36:54 --> 00:36:57
Everybody spots the
two x's there?

540
00:36:57 --> 00:37:02
And then the next term, you
have to get these terms

541
00:37:02 --> 00:37:05
right because we plan to
compare with this guy

542
00:37:05 --> 00:37:07
and see how many we get.
 

543
00:37:07 --> 00:37:10
How many x squareds
are in there?

544
00:37:10 --> 00:37:12
Is it two?
 

545
00:37:12 --> 00:37:13
Looks like two.
 

546
00:37:13 --> 00:37:14
Two x squareds.
 

547
00:37:14 --> 00:37:16
And two x cubes, and so on.
 

548
00:37:16 --> 00:37:18
Yeah, that looks right, OK.
 

549
00:37:18 --> 00:37:22
Now I'm ready, what
am I ready for?

550
00:37:22 --> 00:37:26
I'm ready to say what
x is, x is this delta

551
00:37:26 --> 00:37:28
t over two lambda.
 

552
00:37:28 --> 00:37:29
So what have I got here, one?
 

553
00:37:29 --> 00:37:32
What is this guy now?
 

554
00:37:32 --> 00:37:37
Two x's is delta t lambda.
 

555
00:37:37 --> 00:37:39
Is this good?
 

556
00:37:39 --> 00:37:40
Yes, right?
 

557
00:37:40 --> 00:37:41
We're pleased.
 

558
00:37:41 --> 00:37:47
Because the two x is the, two
of these is delta t lambda and

559
00:37:47 --> 00:37:49
that's what we wanted to match.
 

560
00:37:49 --> 00:37:52
Absolutely. delta t
lambda, lambda delta t.

561
00:37:52 --> 00:37:54
Now let's keep going.
 

562
00:37:54 --> 00:37:58
By the way if this first term
hadn't matched we would

563
00:37:58 --> 00:38:00
be extremely surprised.
 

564
00:38:00 --> 00:38:06
Because that first matching is
only saying that my difference

565
00:38:06 --> 00:38:12
equation is quite consistent,
it's a reasonable creation out

566
00:38:12 --> 00:38:14
of the differential equation.
 

567
00:38:14 --> 00:38:16
And we knew that.
 

568
00:38:16 --> 00:38:19
The question is how much
further are we going to get?

569
00:38:19 --> 00:38:21
Euler will not get any further.
 

570
00:38:21 --> 00:38:24
With Euler the next
ones will fail.

571
00:38:24 --> 00:38:27
But I think with trapezoidal
the next ones are

572
00:38:27 --> 00:38:28
going to work.
 

573
00:38:28 --> 00:38:31
Does it work?
 

574
00:38:31 --> 00:38:35
It's like we're holding
our breath, right?

575
00:38:35 --> 00:38:38
Two now, I'm going to put in x
squared and see about this

576
00:38:38 --> 00:38:45
term. x is what? x is this guy,
delta t over two lambda. delta

577
00:38:45 --> 00:38:49
t lambda over two squared.
 

578
00:38:49 --> 00:38:52
And now you get the fun.
 

579
00:38:52 --> 00:38:56
Because you're going to
compare this term with what?

580
00:38:56 --> 00:39:00
With this term.
 

581
00:39:00 --> 00:39:03
And are they the same?
 

582
00:39:03 --> 00:39:04
Yes.
 

583
00:39:04 --> 00:39:05
Yes.
 

584
00:39:05 --> 00:39:10
So that's the way, you see,
that you got the extra accuracy

585
00:39:10 --> 00:39:14
which Euler did not give you,
but that's why the trapezoidal

586
00:39:14 --> 00:39:17
rule is a is a second
order accurate method.

587
00:39:17 --> 00:39:29
OK, you may say that I went
overboard to say all that.

588
00:39:29 --> 00:39:31
You may say I didn't
ask that question.

589
00:39:31 --> 00:39:36
But it's the right question to
ask about order of accuracy,

590
00:39:36 --> 00:39:40
and it's what problem seven
was intending to bring.

591
00:39:40 --> 00:39:50
Maybe I called it h in problem
seven rather than x here.

592
00:39:50 --> 00:39:52
Well.
 

593
00:39:52 --> 00:39:55
Oh gosh, I realize I
I'm supposed to come

594
00:39:55 --> 00:39:57
back to this one.
 

595
00:39:57 --> 00:40:00
But some people might
have other problems that

596
00:40:00 --> 00:40:01
they're interested in.
 

597
00:40:01 --> 00:40:07
But let me, because time is
pushing along, and the solution

598
00:40:07 --> 00:40:11
to this one will post, let me
at least offer the possibility

599
00:40:11 --> 00:40:15
to ask me about something
completely not six or seven

600
00:40:15 --> 00:40:18
here but something entirely
different, like what's the

601
00:40:18 --> 00:40:20
first question on the
quiz or anything.

602
00:40:20 --> 00:40:32
And that, let me say I'll
hope to know by Tuesday.

603
00:40:32 --> 00:40:37
I love to teach, but making
up exams is serious work.

604
00:40:37 --> 00:40:39
Anyway.
 

605
00:40:39 --> 00:40:46
Let me open a board and open to
another question of any sort.

606
00:40:46 --> 00:40:50
Any place, Chapter 1,
Chapter 2, whatever.

607
00:40:50 --> 00:40:52
Is there anything?
 

608
00:40:52 --> 00:40:56
So I know that you're in the
middle of this homework.

609
00:40:56 --> 00:41:04
So I can say a little more here
about that number six if you

610
00:41:04 --> 00:41:07
want, but I wanted
to allow, yep.

611
00:41:07 --> 00:41:14
AUDIENCE: [INAUDIBLE].
 

612
00:41:14 --> 00:41:18
PROFESSOR STRANG: The A, from
today's lecture this was the

613
00:41:18 --> 00:41:22
incidence matrix, and this was
the a transpose a that's

614
00:41:22 --> 00:41:28
probably still on the
board somewhere.

615
00:41:28 --> 00:41:28
Yep.
 

616
00:41:28 --> 00:41:30
Yep.
 

617
00:41:30 --> 00:41:37
So this is the A, which you
should take in and be able to

618
00:41:37 --> 00:41:41
create if I gave you the graph,
and this is the A transpose A,

619
00:41:41 --> 00:41:44
so it's through today's
lecture, yeah.

620
00:41:44 --> 00:41:47
Next lecture I'll be talking
about the A transpose by

621
00:41:47 --> 00:41:50
itself, which involves
Kirchhoff's current

622
00:41:50 --> 00:41:52
law, it's beautiful.
 

623
00:41:52 --> 00:41:55
A transpose w equals zero.
 

624
00:41:55 --> 00:42:00
But I think this part was
straightforward enough to

625
00:42:00 --> 00:42:06
be able to add this to our
list of problems which

626
00:42:06 --> 00:42:08
fit the framework.
 

627
00:42:08 --> 00:42:11
So that's what that was about.
 

628
00:42:11 --> 00:42:15
It doesn't mean that this will
be on but it could be, right.

629
00:42:15 --> 00:42:17
OK, what else?
 

630
00:42:17 --> 00:42:21
You guys are patient, I
come on, yeah, thanks.

631
00:42:21 --> 00:42:21
AUDIENCE: [INAUDIBLE].
 

632
00:42:21 --> 00:42:21
PROFESSOR STRANG: Yep.
 

633
00:42:21 --> 00:42:25
AUDIENCE: This is only valid
when x is less than one?

634
00:42:25 --> 00:42:29
PROFESSOR STRANG: It's only
valid when x is less than one,

635
00:42:29 --> 00:42:33
so that's now the math point
that this expansion for

636
00:42:33 --> 00:42:37
e^(x) valid for all x's.
 

637
00:42:37 --> 00:42:38
Because you're dividing
by these bigger and

638
00:42:38 --> 00:42:40
bigger numbers.
 

639
00:42:40 --> 00:42:43
But this one is only
valid up to x=1.

640
00:42:44 --> 00:42:48
At x=1 we're getting one plus
one plus one, and we're getting

641
00:42:48 --> 00:42:52
one over one minus one, sort of
infinity matches infinity, but

642
00:42:52 --> 00:42:58
then if x goes up to two, yeah
what happens if x is two?

643
00:42:58 --> 00:43:03
It's sort of not good, but you
know mathematics, it's never

644
00:43:03 --> 00:43:05
completely crazy, right?
 

645
00:43:05 --> 00:43:06
If x is two, what
does this say?

646
00:43:06 --> 00:43:10
What have I got on
the left hand side?

647
00:43:10 --> 00:43:12
Negative one.
 

648
00:43:12 --> 00:43:15
And what have I got on
the right hand side?

649
00:43:15 --> 00:43:21
One plus two plus
four plus eight.

650
00:43:21 --> 00:43:26
I should not allow this to be
videotaped, but that's actually

651
00:43:26 --> 00:43:30
not so completely crazy.
 

652
00:43:30 --> 00:43:37
In some nutty way that could
still make some sense.

653
00:43:37 --> 00:43:39
That's certainly
will not be on the.

654
00:43:39 --> 00:43:44
So you're right that x should
be less than one, and of course

655
00:43:44 --> 00:43:48
it will be here because I'm
looking at little delta t's.

656
00:43:48 --> 00:43:53
Little, so my delta t, x was
this thing and my delta t, the

657
00:43:53 --> 00:43:58
time step was small and somehow
that tells me, actually

658
00:43:58 --> 00:44:01
this is a good indication.
 

659
00:44:01 --> 00:44:05
It gives me the units that
stability and things

660
00:44:05 --> 00:44:09
going right will depend
on lambda delta t.

661
00:44:09 --> 00:44:14
Will depend on lambda delta t,
that's the key parameter there.

662
00:44:14 --> 00:44:19
That's like the dimensionless
parameter that we're, or lambda

663
00:44:19 --> 00:44:21
delta t over two, or whatever.
 

664
00:44:21 --> 00:44:24
But lambda delta t is the key.
 

665
00:44:24 --> 00:44:26
And a highly important key.
 

666
00:44:26 --> 00:44:31
It tells us that as lambda gets
bigger, as the matrix has

667
00:44:31 --> 00:44:35
bigger eigenvalues, delta
t has got to get smaller.

668
00:44:35 --> 00:44:38
And I mentioned
stiff equations.

669
00:44:38 --> 00:44:42
Stiff equations are equations
where the eigenvalues

670
00:44:42 --> 00:44:46
lambda are out of scale.
 

671
00:44:46 --> 00:44:50
You know, you might have two
eigenvalues, one of size one

672
00:44:50 --> 00:44:53
and the other of size ten to
the fourth, because you've

673
00:44:53 --> 00:44:57
got two physical processes
going on at the same time.

674
00:44:57 --> 00:45:00
And those equations are tough,
because that ten to the fourth

675
00:45:00 --> 00:45:06
guy is forcing your delta
t to be really small.

676
00:45:06 --> 00:45:10
Whereas the action might, the
true, real solution might be

677
00:45:10 --> 00:45:12
controlled by the lambda=1 guy.
 

678
00:45:12 --> 00:45:16
So to follow this slow
evolution, you're having to

679
00:45:16 --> 00:45:20
take very small steps because
on top of that slow evolution

680
00:45:20 --> 00:45:24
with the lambda=1, there's some
very fast evolution maybe with

681
00:45:24 --> 00:45:27
lambda equal minus 10,000.
 

682
00:45:27 --> 00:45:32
Yeah, there's a lot
happening here.

683
00:45:32 --> 00:45:36
And always you have to
think OK, is there some

684
00:45:36 --> 00:45:39
way around that box.
 

685
00:45:39 --> 00:45:44
Because forward Euler would
not get you through.

686
00:45:44 --> 00:45:46
OK, thanks for that question,
you got another one?

687
00:45:46 --> 00:45:47
OK.
 

688
00:45:47 --> 00:45:49
AUDIENCE: So then if you
weren't using small enough

689
00:45:49 --> 00:45:51
time steps, [INAUDIBLE]?
 

690
00:45:51 --> 00:45:54
PROFESSOR STRANG: If you
weren't using small

691
00:45:54 --> 00:45:56
enough time steps, OK.
 

692
00:45:56 --> 00:45:59
For trapezoidal, let's say?
 

693
00:45:59 --> 00:46:02
AUDIENCE: I mean, that
expansion wouldn't hold if

694
00:46:02 --> 00:46:02
you were using a lambda--
 

695
00:46:02 --> 00:46:04
PROFESSOR STRANG: Well, the
expansion is really intended

696
00:46:04 --> 00:46:05
for a small delta t.
 

697
00:46:05 --> 00:46:06
Yeah.
 

698
00:46:06 --> 00:46:10
It's not intended, I never
added up the whole series.

699
00:46:10 --> 00:46:14
I just compared a couple of
terms to see how am I doing,

700
00:46:14 --> 00:46:19
and I got the extra term to
match from trapezoidal that

701
00:46:19 --> 00:46:22
I didn't get from Euler.
 

702
00:46:22 --> 00:46:26
So what's to say; if you
took delta t too big,

703
00:46:26 --> 00:46:31
what would happen in
the trapezoidal method?

704
00:46:31 --> 00:46:36
Well, you would stay on this
circle because the absolute

705
00:46:36 --> 00:46:39
value of this thing
is truly one.

706
00:46:39 --> 00:46:43
Even if lambda is enormous and
delta t is way too big, we

707
00:46:43 --> 00:46:49
still had complex conjugates
and their ratio was one.

708
00:46:49 --> 00:46:52
So we would not leave the
circle, at least in perfect

709
00:46:52 --> 00:46:54
arithmetic, as everybody says.
 

710
00:46:54 --> 00:46:57
If we didn't make any
round-off error, we would

711
00:46:57 --> 00:46:58
not leave the circle.
 

712
00:46:58 --> 00:47:02
But boy would we skip all over
the place on that circle.

713
00:47:02 --> 00:47:05
So if we took delta t
too big, we would be

714
00:47:05 --> 00:47:06
completely inaccurate.
 

715
00:47:06 --> 00:47:10
We wouldn't be unstable, for
trapezoidal, because it

716
00:47:10 --> 00:47:14
would stay on the circle,
but the phase would be

717
00:47:14 --> 00:47:16
completely wrong, yeah.
 

718
00:47:16 --> 00:47:19
So it would be a complex number
of absolute value one, but it

719
00:47:19 --> 00:47:28
would not be close to the
exact growth factor.

720
00:47:28 --> 00:47:30
Well, so many things to say.
 

721
00:47:30 --> 00:47:37
I realize that the course moves
along pretty quickly but this

722
00:47:37 --> 00:47:41
topic of numerical methods for
differential equations, that's

723
00:47:41 --> 00:47:44
a core part of 18.086.
 

724
00:47:44 --> 00:47:51
So I'm like anticipating here
in just a couple of days what

725
00:47:51 --> 00:47:55
really takes longer is the
stability and the accuracy

726
00:47:55 --> 00:48:05
and the best choices for
time-dependent problems.

727
00:48:05 --> 00:48:07
OK, always good questions.
 

728
00:48:07 --> 00:48:10
Anything else that's on
your mind of any sort?

729
00:48:10 --> 00:48:10
Yes, thanks.
 

730
00:48:10 --> 00:48:11
AUDIENCE: [INAUDIBLE].
 

731
00:48:11 --> 00:48:17
PROFESSOR STRANG: 114?
 

732
00:48:17 --> 00:48:21
AUDIENCE: There
is a figure 2.7.

733
00:48:21 --> 00:48:21
PROFESSOR STRANG: OK.
 

734
00:48:21 --> 00:48:21
OK.
 

735
00:48:21 --> 00:48:24
114 figure 2.7.
 

736
00:48:24 --> 00:48:26
Oh yes, OK.
 

737
00:48:26 --> 00:48:27
Oh yes.
 

738
00:48:27 --> 00:48:31
AUDIENCE: I figure it's about
how these shapes [INAUDIBLE].

739
00:48:31 --> 00:48:32
see.
 

740
00:48:32 --> 00:48:38
That has a bunch of figures, so
that in order to say for

741
00:48:38 --> 00:48:41
everybody who's not looking at
the book, those figures are

742
00:48:41 --> 00:48:44
about the problem we've
discussed here with a

743
00:48:44 --> 00:48:48
model problem, where
we're on a circle.

744
00:48:48 --> 00:48:52
So do I have space
to draw a circle?

745
00:48:52 --> 00:48:56
Well, let me just
make space here.

746
00:48:56 --> 00:49:00
OK, so page 114 has that
model problem that

747
00:49:00 --> 00:49:03
we've drawn before.
 

748
00:49:03 --> 00:49:08
There's the exact solution,
here's the phase plane; there's

749
00:49:08 --> 00:49:14
u and there's u', and the u was
cos(t), so the u' was minus

750
00:49:14 --> 00:49:16
sin(t), and we travel
around the circle.

751
00:49:16 --> 00:49:19
On the exact solution.
 

752
00:49:19 --> 00:49:22
Energy constant, u squared
stays one. u squared plus

753
00:49:22 --> 00:49:25
u prime squared stay one.
 

754
00:49:25 --> 00:49:30
Now which figure was it
you wanted me to look at?

755
00:49:30 --> 00:49:30
So.
 

756
00:49:30 --> 00:49:32
AUDIENCE: [INAUDIBLE]
 

757
00:49:32 --> 00:49:33
PROFESSOR STRANG:
Of any of them?

758
00:49:33 --> 00:49:35
AUDIENCE: Yeah.
 

759
00:49:35 --> 00:49:38
PROFESSOR STRANG:
OK, that's fine.

760
00:49:38 --> 00:49:39
Let's see.
 

761
00:49:39 --> 00:49:40
Is trapezoidal on that one?
 

762
00:49:40 --> 00:49:41
Yeah.
 

763
00:49:41 --> 00:49:43
Trapezoidal was the first one.
 

764
00:49:43 --> 00:49:48
OK, so figure 2.6 shows the
trapezoidal method moving

765
00:49:48 --> 00:49:50
around the circle.
 

766
00:49:50 --> 00:49:51
So what happens?
 

767
00:49:51 --> 00:49:55
Yeah, thanks, that's a
very suitable question.

768
00:49:55 --> 00:50:02
OK. and I took, in that figure
I took, how long does it take

769
00:50:02 --> 00:50:07
for the exact solution to get
exactly back where it started?

770
00:50:07 --> 00:50:09
At t equal what do I come back?
 

771
00:50:09 --> 00:50:12
AUDIENCE: 2pi.
 

772
00:50:12 --> 00:50:14
PROFESSOR STRANG: t
equal to 2pi, I'm right

773
00:50:14 --> 00:50:15
back where I was.
 

774
00:50:15 --> 00:50:16
Right?
 

775
00:50:16 --> 00:50:18
Cosine has period 2pi.
 

776
00:50:18 --> 00:50:26
OK, now a single step of
size 2pi would be really

777
00:50:26 --> 00:50:28
ridiculous, right?
 

778
00:50:28 --> 00:50:31
I mean, I want to now delta t.
 

779
00:50:31 --> 00:50:37
So in that figure I took delta
t to be 2pi divided by 32.

780
00:50:37 --> 00:50:43
So I'm taking delta t to be the
2pi that would bring me all the

781
00:50:43 --> 00:50:48
way around but I'm
dividing by 32.

782
00:50:48 --> 00:50:49
So, what does that mean?
 

783
00:50:49 --> 00:50:54
What what does the exact
solution do at those steps?

784
00:50:54 --> 00:50:56
32 steps?
 

785
00:50:56 --> 00:51:02
It goes on the circle, 32 equal
steps, 30, 360, 2pi divided by

786
00:51:02 --> 00:51:06
32 radians every time, comes
back exactly there,

787
00:51:06 --> 00:51:08
the exact solution.
 

788
00:51:08 --> 00:51:12
And right where I started.
 

789
00:51:12 --> 00:51:16
So it's like
following a planet.

790
00:51:16 --> 00:51:20
Now I do it by finding
out differences.

791
00:51:20 --> 00:51:22
So now I'm going to follow the
trapezoidal rule, just what

792
00:51:22 --> 00:51:24
we've been talking about.
 

793
00:51:24 --> 00:51:28
With that time step, and with
the equation, everybody

794
00:51:28 --> 00:51:33
remembers the equation was u,
u' equals, do you remember

795
00:51:33 --> 00:51:36
what the matrix was
in that equation?

796
00:51:36 --> 00:51:39
This is the derivative of
it and this is u, u'.

797
00:51:39 --> 00:51:44
Sorry to squeeze this in,
but what I'm, u' is u'.

798
00:51:45 --> 00:51:50
u'' is minus u.
 

799
00:51:50 --> 00:51:54
Now we know why that
matrix was good, right?

800
00:51:54 --> 00:51:55
Why is that?
 

801
00:51:55 --> 00:51:59
That's my matrix A,
why is it good?

802
00:51:59 --> 00:52:02
Because it's exactly, it fits.
 

803
00:52:02 --> 00:52:05
A transpose is minus A.
 

804
00:52:05 --> 00:52:06
It's anti-symmetric.
 

805
00:52:06 --> 00:52:08
Keeps me right on the circle.
 

806
00:52:08 --> 00:52:12
OK now, trapezoidal method
keeps me right on the

807
00:52:12 --> 00:52:14
circle, 32 steps.
 

808
00:52:14 --> 00:52:20
And so the picture just shows
where it goes after 32 steps.

809
00:52:20 --> 00:52:25
And 32, 32 does it
come back there?

810
00:52:25 --> 00:52:28
Well, not exactly, right?
 

811
00:52:28 --> 00:52:32
We don't expect to find
a different solution to

812
00:52:32 --> 00:52:37
be exactly in sync with
cos(t), the real one.

813
00:52:37 --> 00:52:38
But it's really close.
 

814
00:52:38 --> 00:52:44
I think in that figure I can
see that that's sort of a

815
00:52:44 --> 00:52:46
double point there, at 2pi.
 

816
00:52:46 --> 00:52:50
I put a little arrow
indicating small phase error.

817
00:52:50 --> 00:52:55
It misses by a little bit.
 

818
00:52:55 --> 00:53:01
And actually, roughly
what does it miss by?

819
00:53:01 --> 00:53:05
This was the point of the
order of accuracy stuff.

820
00:53:05 --> 00:53:09
Roughly what size is
that little error?

821
00:53:09 --> 00:53:12
That's what we did over here.
 

822
00:53:12 --> 00:53:18
The term that we got wrong
was a delta t cubed.

823
00:53:18 --> 00:53:21
At each step.
 

824
00:53:21 --> 00:53:23
Can I just tell you the answer?
 

825
00:53:23 --> 00:53:27
The error here is of
size delta t squared.

826
00:53:27 --> 00:53:30
Because over here we match
those series and we found the

827
00:53:30 --> 00:53:33
error was delta t cubed.
 

828
00:53:33 --> 00:53:35
That's in a single step.
 

829
00:53:35 --> 00:53:39
But now we've got one
over delta t steps, you

830
00:53:39 --> 00:53:40
see what I'm saying?
 

831
00:53:40 --> 00:53:46
That if the error was delta t
cubed per step, and I have one

832
00:53:46 --> 00:53:52
over delta t steps, to get
somewhere, or 2pi over delta

833
00:53:52 --> 00:53:56
t or whatever, then that
gives me delta t squared.

834
00:53:56 --> 00:54:00
So that little error there
is my error of size

835
00:54:00 --> 00:54:02
delta t squared.
 

836
00:54:02 --> 00:54:05
And that square tells me
I've got a good method.

837
00:54:05 --> 00:54:07
At least, decent.
 

838
00:54:07 --> 00:54:09
Second order accurate.
 

839
00:54:09 --> 00:54:16
And the trapezoidal rule is
sort of the natural one.

840
00:54:16 --> 00:54:21
Well, OK, so that's a full
hour mostly devoted to

841
00:54:21 --> 00:54:22
two or three things.
 

842
00:54:22 --> 00:54:26
Actually the eigenvectors
came into it.

843
00:54:26 --> 00:54:31
And the energy conservation
came into it, the stability

844
00:54:31 --> 00:54:34
matching series came into it.
 

845
00:54:34 --> 00:54:36
And the picture.
 

846
00:54:36 --> 00:54:44
OK, I'll see you Friday for
more about these guys, and then

847
00:54:44 --> 00:54:49
Monday evening please ask me
everything you want to

848
00:54:49 --> 00:54:50
on Monday evening.
 

849
00:54:50 --> 00:54:51
OK.
 

850
00:54:51 --> 00:54:53
Thank you.