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 --> 00:00:11
To make a donation or to view
additional materials from

7
00:00:11 --> 00:00:15
hundreds of MIT courses, visit
MIT OpenCourseWare

8
00:00:15 --> 00:00:21
at ocw.mit.edu.
 

9
00:00:21 --> 00:00:23
PROFESSOR STRANG: OK.
 

10
00:00:23 --> 00:00:27
Well, we had an election
since I saw you last.

11
00:00:27 --> 00:00:32
I hope you're happy
about the results.

12
00:00:32 --> 00:00:35
I'm very happy.
 

13
00:00:35 --> 00:00:36
Except one thing.
 

14
00:00:36 --> 00:00:40
There was a senator
who was in 18.085.

15
00:00:40 --> 00:00:42
And he lost his seat.
 

16
00:00:42 --> 00:00:48
So we don't have a senator
from 18.085 any more.

17
00:00:48 --> 00:00:51
It was Senator Sununu,
from New Hampshire.

18
00:00:51 --> 00:00:53
So he took 18.085.
 

19
00:00:53 --> 00:00:59
Actually, it was a
funny experience.

20
00:00:59 --> 00:01:05
So, some years ago I had to go
to Washington to testify before

21
00:01:05 --> 00:01:10
the committee about increasing
the funding for mathematics.

22
00:01:10 --> 00:01:13
So I was nervous, of
course, trying to remember

23
00:01:13 --> 00:01:15
what I've got to say.
 

24
00:01:15 --> 00:01:19
More money is the main point,
but you have to say it

25
00:01:19 --> 00:01:22
more delicately than that.
 

26
00:01:22 --> 00:01:28
And so I got in, a lot of
people were taking turns, you

27
00:01:28 --> 00:01:30
only get five minutes to do it.
 

28
00:01:30 --> 00:01:34
And so I went and sat
down, waited for my turn.

29
00:01:34 --> 00:01:38
And here on the committee was
my student, Senator Sununu.

30
00:01:38 --> 00:01:42
Well at that time, I think,
Representative Sununu.

31
00:01:42 --> 00:01:45
So that was nice.
 

32
00:01:45 --> 00:01:48
He's a nice guy.
 

33
00:01:48 --> 00:01:53
Actually, the lobbyist who kind
of guided me there, and found

34
00:01:53 --> 00:01:57
the room for me and sort of
told me what to do, said I

35
00:01:57 --> 00:02:01
hope you gave him an A.
 

36
00:02:01 --> 00:02:06
And the truth is I didn't.
 

37
00:02:06 --> 00:02:09
So I guess you guys should
all get As, just to

38
00:02:09 --> 00:02:11
be on the safe side.
 

39
00:02:11 --> 00:02:18
Or let me know if you're going
to be in the Senate anyway.

40
00:02:18 --> 00:02:20
So there you go.
 

41
00:02:20 --> 00:02:20
Anyway.
 

42
00:02:20 --> 00:02:24
So alright.
 

43
00:02:24 --> 00:02:28
But he wasn't great in 18.085.
 

44
00:02:28 --> 00:02:33
OK, so you know what, this
evening I just called the

45
00:02:33 --> 00:02:38
schedules office to confirm
that it is tonight, in 54-100.

46
00:02:38 --> 00:02:44
It's not a long, difficult,
exam so see you at 7:30, or see

47
00:02:44 --> 00:02:50
you earlier at 4:00 for a
review of everything that

48
00:02:50 --> 00:02:52
you want to ask about.
 

49
00:02:52 --> 00:02:56
So I won't try to do an exam
review this morning; we've

50
00:02:56 --> 00:03:00
got lots to do and we're
into November already.

51
00:03:00 --> 00:03:04
So I want to finish
the discussion of the

52
00:03:04 --> 00:03:05
fast Poisson solver.
 

53
00:03:05 --> 00:03:10
Because it's just a neat
idea which is so simple.

54
00:03:10 --> 00:03:13
Anytime you have a
nice square grid.

55
00:03:13 --> 00:03:17
I guess the message is, anytime
you have a nice square or

56
00:03:17 --> 00:03:21
rectangular grid, you should
be thinking, use the fast

57
00:03:21 --> 00:03:23
Fourier transform somehow.
 

58
00:03:23 --> 00:03:29
And here it turns out that the
eigenvectors are for this

59
00:03:29 --> 00:03:36
problem, this K2D, so we're
getting this matrix K2D.

60
00:03:36 --> 00:03:40
Which is of size n squared by
n squared, so a large matrix.

61
00:03:40 --> 00:03:48
But its eigenvectors will be,
well, actually they'll be

62
00:03:48 --> 00:03:52
sine functions for the
K2D, the fixed-fixed.

63
00:03:52 --> 00:03:56
And they would be cosine
functions for the

64
00:03:56 --> 00:03:58
B2D, free-free.
 

65
00:03:58 --> 00:04:02
And the point is the
fast Fourier transform

66
00:04:02 --> 00:04:05
does all those fast.
 

67
00:04:05 --> 00:04:09
I mean, the fast Fourier
transform is set up for the

68
00:04:09 --> 00:04:15
complex exponentials as we'll
see in a couple of weeks.

69
00:04:15 --> 00:04:18
That's the most important
algorithm I could

70
00:04:18 --> 00:04:21
tell you about.
 

71
00:04:21 --> 00:04:26
After maybe Gaussian
elimination, I don't know.

72
00:04:26 --> 00:04:31
But my point here is that if
you can do complex exponentials

73
00:04:31 --> 00:04:38
fast, you can do sines fast,
you can do cosines fast, and

74
00:04:38 --> 00:04:44
the result will be that instead
of I did an operation count

75
00:04:44 --> 00:04:48
when I did elimination in the
order one, two, three, four,

76
00:04:48 --> 00:04:53
five, six, ordered
by along each row.

77
00:04:53 --> 00:04:56
That gave us a matrix
that I want to look at

78
00:04:56 --> 00:04:59
again, this K2D matrix.
 

79
00:04:59 --> 00:05:02
If I did elimination
as it stands, I think

80
00:05:02 --> 00:05:03
the count was n^4.
 

81
00:05:04 --> 00:05:11
I think the count was size n
squared, I had n squared

82
00:05:11 --> 00:05:16
columns to work on and I think
each column took n squared

83
00:05:16 --> 00:05:17
steps, so I was up to n^4.
 

84
00:05:19 --> 00:05:25
And that's manageable.
 

85
00:05:25 --> 00:05:28
2-D problems, you really
can use backslash.

86
00:05:28 --> 00:05:35
But for this particular
problem, using the FFT way, it

87
00:05:35 --> 00:05:38
comes down to n squared log n.
 

88
00:05:38 --> 00:05:40
So that's way better.
 

89
00:05:40 --> 00:05:41
Way better.
 

90
00:05:41 --> 00:05:42
Big saving.
 

91
00:05:42 --> 00:05:46
In 1-D, you don't see a saving.
 

92
00:05:46 --> 00:05:51
But in 2-D it's a big saving,
and in 3-D an enormous saving.

93
00:05:51 --> 00:05:57
So a lot of people would like
to try to use the FFT also when

94
00:05:57 --> 00:05:59
the region isn't a square.
 

95
00:05:59 --> 00:06:02
You can imagine all the
thinking that goes into this.

96
00:06:02 --> 00:06:05
But we'll focus on the
one where it's a square.

97
00:06:05 --> 00:06:07
Oh, one thing more to add.
 

98
00:06:07 --> 00:06:11
I spoke about reordering
the unknowns.

99
00:06:11 --> 00:06:13
But you probably were
wondering, OK, what's that

100
00:06:13 --> 00:06:16
about because I only
just referred to it.

101
00:06:16 --> 00:06:23
Let me suggest an ordering and
you can tell me what the matrix

102
00:06:23 --> 00:06:26
K2D will look like
in this ordering.

103
00:06:26 --> 00:06:30
I'll call it the red
black ordering.

104
00:06:30 --> 00:06:33
I guess with the election just
behind us I could call it

105
00:06:33 --> 00:06:34
the red blue ordering.
 

106
00:06:34 --> 00:06:40
OK, red blue ordering, OK, so
the ordering, instead of

107
00:06:40 --> 00:06:46
ordering it by, let me draw
this again, I'll put in a

108
00:06:46 --> 00:06:52
couple more, make n=4 here, so
now I've got 16, n

109
00:06:52 --> 00:06:54
squared is now 16.
 

110
00:06:54 --> 00:06:57
So here's the idea.
 

111
00:06:57 --> 00:07:03
The red blue ordering is
like a checkerboard.

112
00:07:03 --> 00:07:05
The checkerboard ordering,
you could think of.

113
00:07:05 --> 00:07:09
This will be one, this'll
be two, this'll be three,

114
00:07:09 --> 00:07:13
this'll be four, five,
six, seven, eight.

115
00:07:13 --> 00:07:17
So I've numbered
from one to eight.

116
00:07:17 --> 00:07:21
And now nine to 16 will
be the other guys.

117
00:07:21 --> 00:07:26
These will be nine, ten,
11, 12, 13, 14, 15, 16.

118
00:07:26 --> 00:07:33
So suppose I do that,
what K2D look like?

119
00:07:33 --> 00:07:36
Maybe you could, it's
a neat example.

120
00:07:36 --> 00:07:40
It gives us one more chance
to look at this matrix.

121
00:07:40 --> 00:07:44
So K2D is always going to
have fours on the diagonal.

122
00:07:44 --> 00:07:46
So I have 16 fours.
 

123
00:07:46 --> 00:07:51
Whatever order I take, the
equation at a typical point

124
00:07:51 --> 00:07:55
like say this one, that's
point number one, two.

125
00:07:55 --> 00:07:59
That's point number three, at a
point number three I have my

126
00:07:59 --> 00:08:02
five point molecule with
a four in the middle.

127
00:08:02 --> 00:08:06
So it'd be a four in
this, on that position.

128
00:08:06 --> 00:08:10
And then I have the four
guys around it, the minus

129
00:08:10 --> 00:08:13
ones, and where will they
appear in the matrix?

130
00:08:13 --> 00:08:14
With this ordering.
 

131
00:08:14 --> 00:08:18
So you get to see the
point of an ordering.

132
00:08:18 --> 00:08:21
So this was point one, point
two, point three, I'm

133
00:08:21 --> 00:08:24
focusing on point three.
 

134
00:08:24 --> 00:08:28
It's got a four on the diagonal
and then it's got a minus one,

135
00:08:28 --> 00:08:30
a minus one, a minus one, a
minus one, where

136
00:08:30 --> 00:08:32
do they appear?
 

137
00:08:32 --> 00:08:34
Well, all those guys
are what color?

138
00:08:34 --> 00:08:36
They're all color blue.
 

139
00:08:36 --> 00:08:37
Right?
 

140
00:08:37 --> 00:08:38
All the neighbors
are colored blue.

141
00:08:38 --> 00:08:42
So I won't get to them until
I've gone through the red,

142
00:08:42 --> 00:08:45
the eight reds, and then
I come back to nine.

143
00:08:45 --> 00:08:49
Whatever, 11, 12, 13.
 

144
00:08:49 --> 00:08:53
The point is they'll
all be over here.

145
00:08:53 --> 00:09:03
So this is like, you see that
the only nodes that are next

146
00:09:03 --> 00:09:05
to a red node are all blue.
 

147
00:09:05 --> 00:09:08
And the only nodes next
to a blue node are red.

148
00:09:08 --> 00:09:14
So this is, sorry filled in a
little bit, that's where I'm

149
00:09:14 --> 00:09:17
sort of like four diagonals
of minus ones or something.

150
00:09:17 --> 00:09:19
But there you go.
 

151
00:09:19 --> 00:09:22
That gives you an idea of what
a different ordering could be.

152
00:09:22 --> 00:09:26
And you see what will happen
now with elimination?

153
00:09:26 --> 00:09:29
Elimination is completed
already on this stuff.

154
00:09:29 --> 00:09:33
So all the, with elimination
you want to push the

155
00:09:33 --> 00:09:36
non-zeroes way to the end.
 

156
00:09:36 --> 00:09:42
That's sort of like the
central idea is, it's kind

157
00:09:42 --> 00:09:43
of a greedy approach.
 

158
00:09:43 --> 00:09:45
Greedy algorithm.
 

159
00:09:45 --> 00:09:49
Use up zeroes while
the sun's shining.

160
00:09:49 --> 00:09:54
And then near the end, where
the problem is elimination's

161
00:09:54 --> 00:09:56
reduced it to a much smaller
problem, then you've

162
00:09:56 --> 00:09:58
got some work to do.
 

163
00:09:58 --> 00:10:01
So that's what
would happen here.

164
00:10:01 --> 00:10:06
By the way there's a movie,
on the 18.086 page.

165
00:10:06 --> 00:10:09
I hope some of you guys will
consider taking 18.086.

166
00:10:09 --> 00:10:13
It's a smaller class, people
do projects like creating

167
00:10:13 --> 00:10:16
this elimination movie.
 

168
00:10:16 --> 00:10:18
So that would be math
at MIT.edu/18.086.

169
00:10:23 --> 00:10:30
And with the movie has a movie
of different orderings.

170
00:10:30 --> 00:10:32
That to try to figure out
what's the absolute best

171
00:10:32 --> 00:10:36
ordering with the absolute
least fill-in is, that's

172
00:10:36 --> 00:10:39
one of these NP-Hard
combinatorial problems.

173
00:10:39 --> 00:10:48
But to get an ordering that's
much better than bye rows is

174
00:10:48 --> 00:10:53
not only doable but
should be done.

175
00:10:53 --> 00:10:56
OK, so that's the ordering
for elimination.

176
00:10:56 --> 00:10:58
As this is for elimination.
 

177
00:10:58 --> 00:11:01
OK.
 

178
00:11:01 --> 00:11:03
So that's a whole
world of its own.

179
00:11:03 --> 00:11:07
Ordering the thing for
Gaussian elimination.

180
00:11:07 --> 00:11:15
My focus in this section 3.5,
so this is Section 3.5 of the

181
00:11:15 --> 00:11:20
book and then this is Section
3.6, which is our major one.

182
00:11:20 --> 00:11:21
That's our big deal.
 

183
00:11:21 --> 00:11:26
But my focus in 3.5 is
first to try to see

184
00:11:26 --> 00:11:28
what the 2-D matrix is.
 

185
00:11:28 --> 00:11:31
I mean, a big part of this
course, I think, a big part of

186
00:11:31 --> 00:11:35
what I can do for you, is to,
like, get comfortable

187
00:11:35 --> 00:11:36
with matrices.
 

188
00:11:36 --> 00:11:39
Sort of see what do you look
at when you see a matrix?

189
00:11:39 --> 00:11:40
What's its shape?
 

190
00:11:40 --> 00:11:43
What are its properties?
 

191
00:11:43 --> 00:11:48
And so this K2D is our first
example of a 2-D matrix, and

192
00:11:48 --> 00:11:50
it's highly structured.
 

193
00:11:50 --> 00:11:54
The point is we'll have
K2D matrices out of

194
00:11:54 --> 00:11:55
finite elements.
 

195
00:11:55 --> 00:11:59
But the finite elements might
be, well, they could be those.

196
00:11:59 --> 00:12:01
This could be a finite
element picture.

197
00:12:01 --> 00:12:08
And then the finite element
matrix on such a regular mesh

198
00:12:08 --> 00:12:11
would be quite structured too.
 

199
00:12:11 --> 00:12:16
But if I cut them all up into
triangles and I have a curved

200
00:12:16 --> 00:12:21
region and everything, an
unstructured mesh, then I'll

201
00:12:21 --> 00:12:25
still have the good properties,
this will still be symmetric

202
00:12:25 --> 00:12:27
positive definite,
all the good stuff.

203
00:12:27 --> 00:12:31
But the FFT won't come in.
 

204
00:12:31 --> 00:12:34
OK, so now I want to take
a look again at this

205
00:12:34 --> 00:12:38
K2D matrix, OK?
 

206
00:12:38 --> 00:12:42
So one way to describe the K2D
matrix is the way I did last

207
00:12:42 --> 00:12:45
time, to kind of write down the
typical row, typical

208
00:12:45 --> 00:12:46
row in the middle.
 

209
00:12:46 --> 00:12:50
It's got a four on the
diagonal and four minus ones.

210
00:12:50 --> 00:12:55
And here we've re-ordered it so
the four minus ones came off.

211
00:12:55 --> 00:12:56
Came much later.
 

212
00:12:56 --> 00:13:00
But either way, that's what
the matrix looks like.

213
00:13:00 --> 00:13:03
Let me go back to
this ordering.

214
00:13:03 --> 00:13:08
And let's get 2-D, another
better, clearer look at K2D.

215
00:13:09 --> 00:13:19
I want to construct K2D from K.
 

216
00:13:19 --> 00:13:20
Which is this K1D.
 

217
00:13:22 --> 00:13:26
I don't use that name,
but here it is.

218
00:13:26 --> 00:13:29
Let me just show it to you.
 

219
00:13:29 --> 00:13:34
OK, so let me take the matrix
that does the second

220
00:13:34 --> 00:13:37
differences in the x direction.
 

221
00:13:37 --> 00:13:42
This'll be an n squared
by n squared matrix.

222
00:13:42 --> 00:13:45
And it'll take second
differences on every row.

223
00:13:45 --> 00:13:47
Let me just write
in what it'll be.

224
00:13:47 --> 00:13:50
It'll be K, it'll
be a block matrix.

225
00:13:50 --> 00:13:53
K, K, n blocks.
 

226
00:13:53 --> 00:13:55
Each n by n.
 

227
00:13:55 --> 00:13:59
You see that that will be the
second difference matrix?

228
00:13:59 --> 00:14:03
K takes the second
differences along a row.

229
00:14:03 --> 00:14:06
This K will take the second x
differences along the next row.

230
00:14:06 --> 00:14:09
Row by row, simple.
 

231
00:14:09 --> 00:14:11
So this is the u_xx part.
 

232
00:14:11 --> 00:14:18
Now what will the u_yy,
the second y derivative.

233
00:14:18 --> 00:14:23
That gives second differences
up the columns, right?

234
00:14:23 --> 00:14:27
So can I see what that matrix
will look like, second

235
00:14:27 --> 00:14:29
differences up the columns?
 

236
00:14:29 --> 00:14:32
Well, I think it will
look like this.

237
00:14:32 --> 00:14:35
It will have twos
on the diagonal.

238
00:14:35 --> 00:14:39
2I, 2I, 2I, this is second
differences, right?

239
00:14:39 --> 00:14:41
Down to 2I.
 

240
00:14:43 --> 00:14:46
And next to it will be a minus
the identity - let me write

241
00:14:46 --> 00:14:48
it, and you see if you
think it looks good.

242
00:14:48 --> 00:14:50
So minus the identity.
 

243
00:14:50 --> 00:14:54
It's like a blown up K.
 

244
00:14:54 --> 00:14:55
Somehow.
 

245
00:14:55 --> 00:14:56
Right?
 

246
00:14:56 --> 00:15:01
I have, do you see, it's
every row has got a two

247
00:15:01 --> 00:15:03
and two minus ones.
 

248
00:15:03 --> 00:15:06
That's from the minus 1one the
two and the minus one in the

249
00:15:06 --> 00:15:08
vertical second difference.
 

250
00:15:08 --> 00:15:12
But you see how the count,
you see how the numbering

251
00:15:12 --> 00:15:14
changed it from here?
 

252
00:15:14 --> 00:15:17
Here the neighbors were right
next to each other, because

253
00:15:17 --> 00:15:19
we're ordering by rows.
 

254
00:15:19 --> 00:15:23
Here I have to wait a whole
n to get the guy above.

255
00:15:23 --> 00:15:28
And I'm also n away
from the guy below.

256
00:15:28 --> 00:15:33
So this is the two minus
one minus one centered

257
00:15:33 --> 00:15:35
there, above and below.
 

258
00:15:35 --> 00:15:39
Do you see that?
 

259
00:15:39 --> 00:15:41
If you think about a
little it's not sort

260
00:15:41 --> 00:15:43
of difficult to see.
 

261
00:15:43 --> 00:15:48
And I guess the thing I want
also to do, is to tell you that

262
00:15:48 --> 00:15:52
there's a neat little MATLAB
command, or neat math idea

263
00:15:52 --> 00:15:55
really, and they just made a
MATLAB command out of it,

264
00:15:55 --> 00:16:02
that produces this matrix
and this one out of K.

265
00:16:02 --> 00:16:03
Can I tell you what this is?
 

266
00:16:03 --> 00:16:12
It's something you don't see in
typical linear algebra courses.

267
00:16:12 --> 00:16:19
So I'm contracting K2D from
K1D by, this is called

268
00:16:19 --> 00:16:20
a Kronecker product.
 

269
00:16:20 --> 00:16:27
It's named after a guy, some
dead German, or sometimes

270
00:16:27 --> 00:16:33
called tensor product.
 

271
00:16:33 --> 00:16:39
The point is, this is always
the simplest thing to do in

272
00:16:39 --> 00:16:41
two dimensions or three
dimensions, or so on.

273
00:16:41 --> 00:16:46
Is like product of 1-D
things, like copy the

274
00:16:46 --> 00:16:48
1-D idea both ways.
 

275
00:16:48 --> 00:16:50
That's all this thing is doing.
 

276
00:16:50 --> 00:16:53
Copying the one idea in
the x direction and

277
00:16:53 --> 00:16:55
in the y direction.
 

278
00:16:55 --> 00:17:00
So the MATLAB command,
I've got two pieces here.

279
00:17:00 --> 00:17:05
For this first piece it's
kron, named after Kronecker,

280
00:17:05 --> 00:17:08
of - now, let's see.
 

281
00:17:08 --> 00:17:13
I'm going to put two n by n
matrices, and Kronecker product

282
00:17:13 --> 00:17:15
is going to be a matrix, a
giant matrix, of

283
00:17:15 --> 00:17:16
size n squared.
 

284
00:17:16 --> 00:17:17
Like these.
 

285
00:17:17 --> 00:17:20
OK, so I want to write the
right thing in there.

286
00:17:20 --> 00:17:28
I think the right thing
is the identity and K.

287
00:17:28 --> 00:17:31
OK, and so I have to explain
what this tensor product,

288
00:17:31 --> 00:17:32
Kronecker product, is.
 

289
00:17:32 --> 00:17:38
And this guy happens to be
also a Kronecker product.

290
00:17:38 --> 00:17:43
But it's K and I.
 

291
00:17:43 --> 00:17:46
So I'm just like
mentioning here.

292
00:17:46 --> 00:17:52
That because if you have 2-D
problems and 3-D problems and

293
00:17:52 --> 00:17:55
they're on a nice square grid
so you can like just take

294
00:17:55 --> 00:18:00
products of things, this is
what you want to know about.

295
00:18:00 --> 00:18:00
OK.
 

296
00:18:00 --> 00:18:03
So what is this
Kronecker product?

297
00:18:03 --> 00:18:07
Kronecker product says
take the first matrix, I.

298
00:18:07 --> 00:18:10
It's n by n.
 

299
00:18:10 --> 00:18:13
Just n by n, that's
the identity.

300
00:18:13 --> 00:18:19
And now take this K and
multiply every one of those

301
00:18:19 --> 00:18:24
numbers, these are all numbers,
let me put more numbers in,

302
00:18:24 --> 00:18:26
so I've got 16 numbers.
 

303
00:18:26 --> 00:18:31
This is going to work for
this mesh that has four

304
00:18:31 --> 00:18:33
guys in each direction.
 

305
00:18:33 --> 00:18:35
And four directions.
 

306
00:18:35 --> 00:18:36
Four levels.
 

307
00:18:36 --> 00:18:42
So I is four by four, and K is
four by four and now, each

308
00:18:42 --> 00:18:45
number here gets
multiplied by K.

309
00:18:45 --> 00:18:50
So because of all those zeroes
I get just K, K, K, K.

310
00:18:50 --> 00:18:53
Which is exactly what I wanted.
 

311
00:18:53 --> 00:18:57
So that Kronecker product just
in one step you've created an n

312
00:18:57 --> 00:19:00
squared by n squared matrix
that you really would not want

313
00:19:00 --> 00:19:05
to type in entry by entry.
 

314
00:19:05 --> 00:19:07
That would be a horrible idea.
 

315
00:19:07 --> 00:19:11
OK, and if I follow the same
principle here, I'll take the

316
00:19:11 --> 00:19:16
Kronecker product here, as I
start with a matrix K, so I

317
00:19:16 --> 00:19:20
start with two, minus one,
minus one, two, minus one,

318
00:19:20 --> 00:19:25
minus one, two, minus
one, minus one, two.

319
00:19:25 --> 00:19:30
That's my K, and now what's
the Kronecker idea?

320
00:19:30 --> 00:19:36
Each of those numbers gets
multiplied by this matrix.

321
00:19:36 --> 00:19:41
So that two becomes 2I, minus
one becomes minus I, minus one

322
00:19:41 --> 00:19:44
becomes minus I, it all clicks.
 

323
00:19:44 --> 00:19:53
And it's producing exactly the
u_yy part, the second part.

324
00:19:53 --> 00:19:57
So that's the good
construction thing.

325
00:19:57 --> 00:19:58
Just to tell you about.
 

326
00:19:58 --> 00:20:04
And the beauty is, this is just
like cooked up, set up for

327
00:20:04 --> 00:20:05
separation of variables.
 

328
00:20:05 --> 00:20:08
If I want to know the
eigenvalues and the

329
00:20:08 --> 00:20:16
eigenvectors of this
thing, I start by knowing

330
00:20:16 --> 00:20:20
the eigenvalues and
eigenvectors of K.

331
00:20:20 --> 00:20:22
Can you remind me
what those are?

332
00:20:22 --> 00:20:26
We should remember them.
 

333
00:20:26 --> 00:20:30
So, anybody remember
eigenvectors of K?

334
00:20:30 --> 00:20:34
Well, this is going back
to Section 1.5, way

335
00:20:34 --> 00:20:35
early in the semester.
 

336
00:20:35 --> 00:20:41
And there's a lot of writing
involved and I probably

337
00:20:41 --> 00:20:43
didn't do it all.
 

338
00:20:43 --> 00:20:47
But let me remind
you of the idea.

339
00:20:47 --> 00:20:50
We guessed those eigenvectors.
 

340
00:20:50 --> 00:20:52
And how did we guess them?
 

341
00:20:52 --> 00:20:57
We guessed them by comparing
this difference equation to

342
00:20:57 --> 00:20:59
the differential equation.
 

343
00:20:59 --> 00:21:02
So we're in 1-D now, I'm
just reminding you of what

344
00:21:02 --> 00:21:03
we did a long time ago.
 

345
00:21:03 --> 00:21:06
OK, so what did we do?
 

346
00:21:06 --> 00:21:10
I want to know the
eigenvectors of these guys.

347
00:21:10 --> 00:21:12
So I better know the
eigenvictors of

348
00:21:12 --> 00:21:14
the 1-D one first.
 

349
00:21:14 --> 00:21:17
OK, so what did we do in 1-D?
 

350
00:21:17 --> 00:21:23
We found the eigenvectors of
K by looking first at the

351
00:21:23 --> 00:21:28
differential equation
-u''=lambda*u u with, and

352
00:21:28 --> 00:21:34
remember we're talking about
the fixed-fixed-fixed here.

353
00:21:34 --> 00:21:37
And anybody remember the
eigenvectors of this guy,

354
00:21:37 --> 00:21:40
eigenfunctions I guess
I should say for those?

355
00:21:40 --> 00:21:44
So sines and cosines
look good here, right?

356
00:21:44 --> 00:21:46
Because you take their second
derivative, it brings

357
00:21:46 --> 00:21:48
down the constant.
 

358
00:21:48 --> 00:21:52
And then do I want
sines or cosines?

359
00:21:52 --> 00:21:55
Looking at the boundary
conditions, that's going

360
00:21:55 --> 00:21:56
to tell me everything.
 

361
00:21:56 --> 00:22:00
I want sines, cosines are wiped
out by this first condition

362
00:22:00 --> 00:22:02
that u(0) should be zero.
 

363
00:22:02 --> 00:22:08
And then the sine has to come
back to zero at one, so the

364
00:22:08 --> 00:22:17
eigenfunctions were u equals
the sine of something, I think.

365
00:22:17 --> 00:22:21
I want a multiple of
pi, right? pi*k*x.

366
00:22:21 --> 00:22:24
 
 

367
00:22:24 --> 00:22:26
I think that would be right.
 

368
00:22:26 --> 00:22:30
Because at x=1, that's
come back to zero.

369
00:22:30 --> 00:22:35
And the eigenvalue of lambda,
if I just plug that in, two

370
00:22:35 --> 00:22:38
derivatives bring
down pi*k twice.

371
00:22:38 --> 00:22:41
And with a minus, and
that cancels that minus.

372
00:22:41 --> 00:22:45
So the eigenvalues were
pi squared, K squared.

373
00:22:45 --> 00:22:50
And that's the beautiful
construction for

374
00:22:50 --> 00:22:53
differential equations.
 

375
00:22:53 --> 00:22:56
And these eigenfunctions
are a complete set,

376
00:22:56 --> 00:22:57
it's all wonderful.
 

377
00:22:57 --> 00:23:03
So that's a model problem of
what classical applied math

378
00:23:03 --> 00:23:05
does for Bessel's equation,
Legendre's equation, a

379
00:23:05 --> 00:23:09
whole long list of things.
 

380
00:23:09 --> 00:23:13
Now, note all those equations
that I just mentioned would

381
00:23:13 --> 00:23:15
have finite difference analogs.
 

382
00:23:15 --> 00:23:20
But to my knowledge, it's only
this one, this simplest, best

383
00:23:20 --> 00:23:24
one of all, we could call
it the Fourier equation if

384
00:23:24 --> 00:23:26
we needed a name for it.
 

385
00:23:26 --> 00:23:27
I just thought of that.
 

386
00:23:27 --> 00:23:28
That sounds good to me.
 

387
00:23:28 --> 00:23:29
Fourier equation.
 

388
00:23:29 --> 00:23:30
OK.
 

389
00:23:30 --> 00:23:35
So what's great about this one
is that the eigenvectors in the

390
00:23:35 --> 00:23:41
matrix case are just found
by sampling these functions.

391
00:23:41 --> 00:23:44
You're right on the dot if you
just sample these functions.

392
00:23:44 --> 00:23:49
So, for K, the eigenvectors,
what do I call eigenvectors?

393
00:23:49 --> 00:23:53
Maybe y's, I think I
sometimes call them y.

394
00:23:53 --> 00:24:00
So a typical eigenvector y
would be a sine vectors.

395
00:24:00 --> 00:24:08
I'm sampling it at, let's see,
can I use h for this step size?

396
00:24:08 --> 00:24:10
So h is 1/(n+1).
 

397
00:24:10 --> 00:24:14
 
 

398
00:24:14 --> 00:24:17
As we saw in the
past. h is 1/(n+1).

399
00:24:18 --> 00:24:25
So a typical eigenvector would
sample this thing at h, 2h,

400
00:24:25 --> 00:24:32
sin(2*pi*k*h), and so
on, dot dot dot dot.

401
00:24:32 --> 00:24:34
And that would be
an eigenvector.

402
00:24:34 --> 00:24:37
That would be the eigenvector
number K, actually.

403
00:24:37 --> 00:24:42
And do you see that that sort
of eigenvector is well set up.

404
00:24:42 --> 00:24:46
It's going to work because
it ends, where does it end?

405
00:24:46 --> 00:24:50
Who's the last person, the last
component of this eigenvector?

406
00:24:50 --> 00:24:52
It's sin(N*pi*k*h).
 

407
00:24:52 --> 00:24:57
 
 

408
00:24:57 --> 00:25:00
Sorry about all the symbols,
but compared to most

409
00:25:00 --> 00:25:03
eigenvectors this is,
like, the greatest.

410
00:25:03 --> 00:25:06
OK, now why do I like that?
 

411
00:25:06 --> 00:25:11
Because the pattern, what would
be the next one if there

412
00:25:11 --> 00:25:13
was an n plus first?
 

413
00:25:13 --> 00:25:18
If there was an n plus first
component, what would it be?

414
00:25:18 --> 00:25:19
What would be the
sin((N+1)pi*k*h)?

415
00:25:19 --> 00:25:23
 
 

416
00:25:23 --> 00:25:25
That's the golden question.
 

417
00:25:25 --> 00:25:29
What would be, I'm trying to
say that these have a nice

418
00:25:29 --> 00:25:34
pattern because the guy that's
the next person here would be

419
00:25:34 --> 00:25:40
sin((N+1_pi*k*h),
which is what?

420
00:25:40 --> 00:25:42
So N+1 is in there.
 

421
00:25:42 --> 00:25:45
And h is in there, so what
do I get from those guys?

422
00:25:45 --> 00:25:53
The (N+1)h is just one,
right? (N+1)h carries me

423
00:25:53 --> 00:25:55
all the way to the end.
 

424
00:25:55 --> 00:25:58
That's a guy that's out, it's
not in our matrix because

425
00:25:58 --> 00:26:01
that's the part, that's a
known one, that's a zero in a

426
00:26:01 --> 00:26:03
fixed boundary condition.
 

427
00:26:03 --> 00:26:09
So like the next guy would
be sin((N+1)h), that's

428
00:26:09 --> 00:26:10
one. sin(pi*k).
 

429
00:26:11 --> 00:26:12
And what is sin(pi*k)?
 

430
00:26:12 --> 00:26:14
 
 

431
00:26:14 --> 00:26:17
The sin(pi*k) is always?
 

432
00:26:17 --> 00:26:18
Zero.
 

433
00:26:18 --> 00:26:23
So we're getting it right, like
the guy that got knocked off

434
00:26:23 --> 00:26:25
following this pattern
will be zero.

435
00:26:25 --> 00:26:29
And what about the guy that
was knocked off of that end?

436
00:26:29 --> 00:26:32
If this was sin(2pi*k*h), and
sin(pi*k*h), this would be

437
00:26:32 --> 00:26:35
sin(0pi*k*h), which would be?
 

438
00:26:35 --> 00:26:37
Also zero, sin(0).
 

439
00:26:38 --> 00:26:44
So, anyway you could check just
by, it takes a little patience

440
00:26:44 --> 00:26:50
with trig identities, if I
multiply K by that vector, the

441
00:26:50 --> 00:26:54
pattern keeps going, the two
minus one, minus one, you know

442
00:26:54 --> 00:26:57
at a typical point I'm going to
have two of these, minus one of

443
00:26:57 --> 00:26:59
these, minus one of these.
 

444
00:26:59 --> 00:27:01
And it'll look good.
 

445
00:27:01 --> 00:27:05
And when I get to the end I'll
have two minus one of these.

446
00:27:05 --> 00:27:08
The pattern will still hold,
because the minus one of

447
00:27:08 --> 00:27:10
these I can say is there.
 

448
00:27:10 --> 00:27:13
But it is a zero, so it's OK.
 

449
00:27:13 --> 00:27:14
And similarly here.
 

450
00:27:14 --> 00:27:19
Anyway, the pattern's good
and the eigenvalue is?

451
00:27:19 --> 00:27:20
Something.
 

452
00:27:20 --> 00:27:22
OK, it has some formula.
 

453
00:27:22 --> 00:27:25
It's not going to be
exactly this guy.

454
00:27:25 --> 00:27:27
Because we're taking
differences of sines and

455
00:27:27 --> 00:27:29
not derivatives of sines.
 

456
00:27:29 --> 00:27:31
So it has some formula.
 

457
00:27:31 --> 00:27:35
Tell me what you would
know about lambda.

458
00:27:35 --> 00:27:37
I'm not going to ask you
for the formula, I'm

459
00:27:37 --> 00:27:38
going to write it down.
 

460
00:27:38 --> 00:27:42
Tell me, before I write it
down, or I'll write it down

461
00:27:42 --> 00:27:43
and then you can tell me.
 

462
00:27:43 --> 00:27:45
I have two on the diagonal,
so that's just going

463
00:27:45 --> 00:27:47
to give me a two.
 

464
00:27:47 --> 00:27:49
And then the guy on the left
and the guy on the right,

465
00:27:49 --> 00:27:54
two, two with minus ones
I think gives us two.

466
00:27:54 --> 00:27:57
I think it turns out
to be a cos(k).

467
00:27:57 --> 00:28:00
 
 

468
00:28:00 --> 00:28:03
A k has to be in there, a pi
has to be in there, and h.

469
00:28:03 --> 00:28:07
I think that'll be it.
 

470
00:28:07 --> 00:28:09
I think that's the eigenvalue.
 

471
00:28:09 --> 00:28:13
That's the k'th eigenvalue to
go with this eigenvector.

472
00:28:13 --> 00:28:16
What do you notice about
two minus two times the

473
00:28:16 --> 00:28:19
cosine of something?
 

474
00:28:19 --> 00:28:20
What is it?
 

475
00:28:20 --> 00:28:24
Positive, negative,
zero, what's up?

476
00:28:24 --> 00:28:28
Two minus twp times
a cosine is always?

477
00:28:28 --> 00:28:29
Positive.
 

478
00:28:29 --> 00:28:34
What does that tell us about
K that we already knew?

479
00:28:34 --> 00:28:35
It's positive definite, right?
 

480
00:28:35 --> 00:28:38
All eigenvalues, here we
actually know what they all

481
00:28:38 --> 00:28:42
are, they're all positive.
 

482
00:28:42 --> 00:28:43
I'm never going to
get zero here.

483
00:28:43 --> 00:28:53
These k*pi*h's are not hitting
the ones where the cosine is

484
00:28:53 --> 00:28:57
one and, of course,
you can imagine.

485
00:28:57 --> 00:28:59
So I started this sentence
and realized I should

486
00:28:59 --> 00:29:02
add another sentence.
 

487
00:29:02 --> 00:29:04
These are all positive.
 

488
00:29:04 --> 00:29:05
We expected that.
 

489
00:29:05 --> 00:29:08
We knew that k was a
positive definite matrix.

490
00:29:08 --> 00:29:11
All its eigenvalues are
positive, and now we actually

491
00:29:11 --> 00:29:13
know what they are.
 

492
00:29:13 --> 00:29:19
And if I had the matrix B
instead for a free-free one?

493
00:29:19 --> 00:29:22
Then this formula would
change a little bit.

494
00:29:22 --> 00:29:28
And what would be different
about B, the free-free matrix?

495
00:29:28 --> 00:29:31
Its eigenvectors would be
maybe at half angles or

496
00:29:31 --> 00:29:33
some darned thing happened.
 

497
00:29:33 --> 00:29:36
And so we get something
slightly different here.

498
00:29:36 --> 00:29:40
Maybe h is 1/n for
that, I've forgotten.

499
00:29:40 --> 00:29:44
And what's the deal
with the free-free K?

500
00:29:44 --> 00:29:48
One of the eigenvalues is zero.
 

501
00:29:48 --> 00:29:53
The free-free is the positive
semi-definite example.

502
00:29:53 --> 00:30:00
OK, that was like a quick
review of stuff we did earlier.

503
00:30:00 --> 00:30:04
And so I'm coming back to that
early point in the book,

504
00:30:04 --> 00:30:11
because of this great fact that
my eigenvectors are sines.

505
00:30:11 --> 00:30:16
So the point is, my
eigenvectors being sines, that

506
00:30:16 --> 00:30:21
just lights up a light saying
use the fast Fourier transform.

507
00:30:21 --> 00:30:25
You've got a matrix
full of sine vectors.

508
00:30:25 --> 00:30:29
Your eigenvector matrix
is a sine transform.

509
00:30:29 --> 00:30:34
It's golden, so use it.
 

510
00:30:34 --> 00:30:37
So I'll just remember
then how, recall that.

511
00:30:37 --> 00:30:43
So what are the
eigenvectors in 2-D?

512
00:30:43 --> 00:30:47
First of all, so
let's go to 2-D.

513
00:30:47 --> 00:30:51
Let me do the
continuous one first.

514
00:30:51 --> 00:30:52
Yeah, -u_xx-u_yy=lambda*u.
 

515
00:30:52 --> 00:30:57
 
 

516
00:30:57 --> 00:31:00
The eigenvalue problem for
Laplace's equation now in

517
00:31:00 --> 00:31:05
2-D, and again I'm going
to make it on the square.

518
00:31:05 --> 00:31:07
This unit square.
 

519
00:31:07 --> 00:31:10
And I'm going to have zero
boundary conditions.

520
00:31:10 --> 00:31:13
Fixed-fixed.
 

521
00:31:13 --> 00:31:18
Anybody want to propose
an eigenfunction for

522
00:31:18 --> 00:31:20
the 2-D problem?
 

523
00:31:20 --> 00:31:25
So the 2-D one, the whole idea
is hey, this square is like a

524
00:31:25 --> 00:31:27
product of intervals somehow.
 

525
00:31:27 --> 00:31:32
We know the answer
in each direction.

526
00:31:32 --> 00:31:35
What do you figure,
what would be a good

527
00:31:35 --> 00:31:36
eigenfunction, u(x,y)?
 

528
00:31:36 --> 00:31:38
 
 

529
00:31:38 --> 00:31:41
Or eigenfunction, yeah, u(x,y).
 

530
00:31:43 --> 00:31:46
For this problem?
 

531
00:31:46 --> 00:31:49
What do you think?
 

532
00:31:49 --> 00:31:51
This is like this.
 

533
00:31:51 --> 00:31:57
The older courses on applied
math did this until you

534
00:31:57 --> 00:31:58
were blue in the face.
 

535
00:31:58 --> 00:32:02
Because there wasn't
finite elements and good

536
00:32:02 --> 00:32:03
stuff at that time.
 

537
00:32:03 --> 00:32:07
It was exact formulas.
 

538
00:32:07 --> 00:32:10
You wondered about exact
eigenfunctions and for this

539
00:32:10 --> 00:32:13
problem variables separate.
 

540
00:32:13 --> 00:32:19
And you get u(x) is
the product of sine.

541
00:32:19 --> 00:32:25
Of this guy, sin(pi*k*x),
times the sin(pi*l*y).

542
00:32:25 --> 00:32:30
 
 

543
00:32:30 --> 00:32:34
Well, once you have the idea
that it might look like that,

544
00:32:34 --> 00:32:37
you just plug it in to
see, does it really work?

545
00:32:37 --> 00:32:40
And what's the eigenvalue?
 

546
00:32:40 --> 00:32:42
So I claim that that's a good
eigenfunction function and I

547
00:32:42 --> 00:32:47
need, it's got two indices,
k and l, two frequencies.

548
00:32:47 --> 00:32:50
It's got an x frequency
and a y frequency.

549
00:32:50 --> 00:32:55
So I need double index.
k and l will go, yeah.

550
00:32:55 --> 00:32:57
All the way out to infinity
in the continuous problem.

551
00:32:57 --> 00:32:59
And what's the eigenvector?
 

552
00:32:59 --> 00:33:02
Can you plug this guy in?
 

553
00:33:02 --> 00:33:05
What happens when
you plug that in?

554
00:33:05 --> 00:33:11
Take the second x
derivative, what happens?

555
00:33:11 --> 00:33:14
A constant comes out and
what's the constant?

556
00:33:14 --> 00:33:17
pi squared k squared.
 

557
00:33:17 --> 00:33:19
Then plug it into that term.
 

558
00:33:19 --> 00:33:21
A constant comes out again.
 

559
00:33:21 --> 00:33:25
What's that constant?
pi squared l squared.

560
00:33:25 --> 00:33:28
They come out with a minus and
then you already built in the

561
00:33:28 --> 00:33:31
minus, so that they've
made it a plus.

562
00:33:31 --> 00:33:37
So the lambda_kl is just pi
squared, k squared, plus

563
00:33:37 --> 00:33:41
pi squared l squared.
 

564
00:33:41 --> 00:33:45
You see how it's going?
 

565
00:33:45 --> 00:33:48
If we knew it in 1-D
now we get it in 2-D,

566
00:33:48 --> 00:33:50
practically for free.
 

567
00:33:50 --> 00:33:53
Just the idea of doing this.
 

568
00:33:53 --> 00:33:56
OK, and now I'm going to do
it for finite differences.

569
00:33:56 --> 00:34:04
So now I have K2D and I want to
ask you about its eigenvectors,

570
00:34:04 --> 00:34:05
and what do you think they are?
 

571
00:34:05 --> 00:34:08
Well, of course, they're
just like those.

572
00:34:08 --> 00:34:17
They're just the eigenvectors,
shall I call them z, k, l.

573
00:34:17 --> 00:34:19
So these will be
the eigenvectors.

574
00:34:19 --> 00:34:29
The eigenvectors z_kl, will be,
their components, I don't even

575
00:34:29 --> 00:34:31
know the best way
to write them.

576
00:34:31 --> 00:34:35
The trouble is this matrix
is of size n squared.

577
00:34:35 --> 00:34:38
Its eigenvectors have got
n squared components,

578
00:34:38 --> 00:34:39
and what are they?
 

579
00:34:39 --> 00:34:41
Just you could
tell me in words.

580
00:34:41 --> 00:34:43
What do you figure are going
to be the components of the

581
00:34:43 --> 00:34:51
eigenvectors in K2D when
you know them in 1-D?

582
00:34:51 --> 00:34:54
Products, of course.
 

583
00:34:54 --> 00:35:00
This construction, however I
write it, is just like this.

584
00:35:00 --> 00:35:07
z_kl, a typical component,
typical components so that

585
00:35:07 --> 00:35:14
components of the eigenvectors
are products of these

586
00:35:14 --> 00:35:25
guys, like sin(k*pi*h),
something like that.

587
00:35:25 --> 00:35:28
I've got indices that I
don't want to get into.

588
00:35:28 --> 00:35:30
Just damn, it I'll
just put that.

589
00:35:30 --> 00:35:34
Sine here or something, right.
 

590
00:35:34 --> 00:35:41
We could try to sort out the
indices, the truth is a kron

591
00:35:41 --> 00:35:45
operation does it for us.
 

592
00:35:45 --> 00:35:50
So all I'm saying is we've
got an eigenvector with n

593
00:35:50 --> 00:35:53
components in the x direction.
 

594
00:35:53 --> 00:35:57
We've got another one with
index l and components

595
00:35:57 --> 00:35:58
in the y direction.
 

596
00:35:58 --> 00:36:01
Multiply these end guys
by these end guys, you

597
00:36:01 --> 00:36:02
get n squared guys.
 

598
00:36:02 --> 00:36:04
Whatever order you
write them in.

599
00:36:04 --> 00:36:07
And that's the eigenvector.
 

600
00:36:07 --> 00:36:14
And then the eigenvalue is,
so the eigenvalue will

601
00:36:14 --> 00:36:16
be, well what do we got?
 

602
00:36:16 --> 00:36:22
We have a second x difference,
so it will be a 2-2cos(k*pi*h).

603
00:36:22 --> 00:36:26
 
 

604
00:36:26 --> 00:36:29
From the xx term.
 

605
00:36:29 --> 00:36:33
And it'll plus the other term
will be a 2-2cos(l*pi*h).

606
00:36:33 --> 00:36:38
 
 

607
00:36:38 --> 00:36:40
That's cool, yeah.
 

608
00:36:40 --> 00:36:47
I didn't get into writing the
details of the eigenvector,

609
00:36:47 --> 00:36:48
that's a mess.
 

610
00:36:48 --> 00:36:49
This is not a mess.
 

611
00:36:49 --> 00:36:51
This is nice.
 

612
00:36:51 --> 00:36:52
What do I see again?
 

613
00:36:52 --> 00:36:54
I see four coming
from the diagonal.

614
00:36:54 --> 00:36:59
I see two minus ones coming
from left and right.

615
00:36:59 --> 00:37:03
I see two minus ones
coming from up and below.

616
00:37:03 --> 00:37:07
And do I see a positive number?
 

617
00:37:07 --> 00:37:08
You bet.
 

618
00:37:08 --> 00:37:11
Right, this is positive,
this is positive.

619
00:37:11 --> 00:37:12
Sum is positive.
 

620
00:37:12 --> 00:37:15
The sum of two positive
definite matrices is

621
00:37:15 --> 00:37:17
positive definite.
 

622
00:37:17 --> 00:37:21
I know everything about k.
 

623
00:37:21 --> 00:37:24
And the fast Fourier
transform makes all those

624
00:37:24 --> 00:37:25
calculations possible.
 

625
00:37:25 --> 00:37:31
So maybe I, just to conclude
this subject, would be to

626
00:37:31 --> 00:37:39
remind you, how do you use the
eigenvalues and eigenvectors

627
00:37:39 --> 00:37:43
in solving the equation?
 

628
00:37:43 --> 00:37:46
I guess I'd better
put that on a board.

629
00:37:46 --> 00:37:48
But this is what
we did last time.

630
00:37:48 --> 00:37:53
So the final step would be
how do I solve (K2D)U=F?

631
00:37:53 --> 00:37:58
 
 

632
00:37:58 --> 00:38:01
You remember, what
were the three steps?

633
00:38:01 --> 00:38:05
I do really want you to
learn those three steps.

634
00:38:05 --> 00:38:08
It's the way eigenvectors
and eigenvalues are use.

635
00:38:08 --> 00:38:10
It's the whole point
of finding them.

636
00:38:10 --> 00:38:16
The whole point of finding them
is to split this problem into n

637
00:38:16 --> 00:38:21
squared little 1-D problems,
where each eigenvector is

638
00:38:21 --> 00:38:23
just doing its own thing.
 

639
00:38:23 --> 00:38:24
So what do you do?
 

640
00:38:24 --> 00:38:30
You write F as a combination
with some coefficients, c_kl

641
00:38:30 --> 00:38:32
of the eigenvectors y_kl.
 

642
00:38:33 --> 00:38:35
So these are vectors.
 

643
00:38:35 --> 00:38:38
Put an arrow over them
just to emphasize.

644
00:38:38 --> 00:38:39
Those are vectors.
 

645
00:38:39 --> 00:38:42
Then what's the answer,
let's skip the middle

646
00:38:42 --> 00:38:44
step which was so easy.
 

647
00:38:44 --> 00:38:46
Tell me the answer.
 

648
00:38:46 --> 00:38:49
Supposed the right hand side is
a combination of eigenvectors.

649
00:38:49 --> 00:38:55
Then the left hand side, the
answer, is also a combination

650
00:38:55 --> 00:38:56
of eigenvectors.
 

651
00:38:56 --> 00:39:00
And just tell me, what
are the coefficients

652
00:39:00 --> 00:39:03
in that combination?
 

653
00:39:03 --> 00:39:10
This is like the whole idea
is on two simple lines here.

654
00:39:10 --> 00:39:14
The coefficient there is?
 

655
00:39:14 --> 00:39:18
What do I have? c_kl,
does that come in?

656
00:39:18 --> 00:39:22
Yes, because c_kl tells
me how much of this

657
00:39:22 --> 00:39:24
eigenvector is here.
 

658
00:39:24 --> 00:39:28
But now, what else comes
in? lambda_kl, and what

659
00:39:28 --> 00:39:30
do I do with that?
 

660
00:39:30 --> 00:39:34
Divide by it, because when
I multiply it'll multiply

661
00:39:34 --> 00:39:36
by it and bring back F.
 

662
00:39:36 --> 00:39:40
So there you go.
 

663
00:39:40 --> 00:39:43
That's the, this u is
a vector, of course.

664
00:39:43 --> 00:39:45
These are all vectors.
 

665
00:39:45 --> 00:39:52
2-D, we'll give them an arrow,
just as a reminder that

666
00:39:52 --> 00:39:55
these are vectors with
n squared components.

667
00:39:55 --> 00:39:59
They're giant vectors, but the
point is this y_kl, its n

668
00:39:59 --> 00:40:03
squared components are just
products, as we saw here,

669
00:40:03 --> 00:40:04
of the 1-D problem.
 

670
00:40:04 --> 00:40:09
So the fast Fourier transform,
the 2-D fast Fourier

671
00:40:09 --> 00:40:11
transform just takes off.
 

672
00:40:11 --> 00:40:17
Takes off and gives you
this fantastic speed.

673
00:40:17 --> 00:40:20
So that's absolutely the right
way to solve the problem.

674
00:40:20 --> 00:40:23
And everybody sees
that picture?

675
00:40:23 --> 00:40:25
Don't forget that picture.
 

676
00:40:25 --> 00:40:30
That's like a nice part of
this subject, is to get

677
00:40:30 --> 00:40:34
it's formulas and not code.
 

678
00:40:34 --> 00:40:37
But it's easily
turned into code.

679
00:40:37 --> 00:40:41
Because the FFT and the sine
transform are all coded.

680
00:40:41 --> 00:40:45
Actually, Professor Johnson in
the Math Department, he created

681
00:40:45 --> 00:40:48
the best FFT code there is.
 

682
00:40:48 --> 00:40:49
Do you know that?
 

683
00:40:49 --> 00:40:51
I'll just mention.
 

684
00:40:51 --> 00:40:56
His code is called FFTW,
Fastest Fourier Transform in

685
00:40:56 --> 00:41:02
the West, and the point is
it's set up to be fast on

686
00:41:02 --> 00:41:06
whatever computer, whatever
architecture you're using.

687
00:41:06 --> 00:41:07
It figures out what that is.
 

688
00:41:07 --> 00:41:11
And optimizes the
code for that.

689
00:41:11 --> 00:41:13
And gives you the answer.
 

690
00:41:13 --> 00:41:14
OK.
 

691
00:41:14 --> 00:41:19
That's Part 2 complete
of Section 3.5.

692
00:41:19 --> 00:41:23
I guess I'm hoping after we get
through the quiz, the next

693
00:41:23 --> 00:41:29
natural step would be
some MATLAB, right?

694
00:41:29 --> 00:41:36
We really should have some
MATLAB case of doing this.

695
00:41:36 --> 00:41:39
I haven't thought of
it, but I'll try.

696
00:41:39 --> 00:41:41
And some MATLAB for
finite elements.

697
00:41:41 --> 00:41:49
And there's a lot of finite
element code on the course

698
00:41:49 --> 00:41:52
page, ready to download.
 

699
00:41:52 --> 00:41:55
But now we have to
understand it.

700
00:41:55 --> 00:41:59
Are you ready to tackle, so in
ten minutes we can get the

701
00:41:59 --> 00:42:04
central idea of finite elements
in 2-D, and then after the

702
00:42:04 --> 00:42:09
quiz, when our minds are clear
again, we'll do it

703
00:42:09 --> 00:42:13
properly Friday.
 

704
00:42:13 --> 00:42:17
This is asking you a
lot, but can I do it?

705
00:42:17 --> 00:42:22
Can I go to finite
elements in 2-D?

706
00:42:22 --> 00:42:25
OK.
 

707
00:42:25 --> 00:42:26
Oh, I have a choice.
 

708
00:42:26 --> 00:42:27
Big, big choice.
 

709
00:42:27 --> 00:42:31
I could use
triangles, or quads.

710
00:42:31 --> 00:42:33
That picture, this
picture would naturally

711
00:42:33 --> 00:42:35
set up for quads.
 

712
00:42:35 --> 00:42:37
Let me start with triangles.
 

713
00:42:37 --> 00:42:46
So I have some region with
lots of triangles here.

714
00:42:46 --> 00:42:48
Got to have some interior
points, or I'm not going

715
00:42:48 --> 00:42:51
to have any unknowns.
 

716
00:42:51 --> 00:42:56
Gosh, that's a big mesh
with very little unknown.

717
00:42:56 --> 00:43:01
Let me put in another few here.
 

718
00:43:01 --> 00:43:02
Have I got enough?
 

719
00:43:02 --> 00:43:06
Oops, that's not a triangle.
 

720
00:43:06 --> 00:43:07
How's that?
 

721
00:43:07 --> 00:43:10
Is that now a triangulation?
 

722
00:43:10 --> 00:43:14
So that's an unstructured
triangular mesh.

723
00:43:14 --> 00:43:16
Its quality is not too bad.
 

724
00:43:16 --> 00:43:25
The angles, some angles are
small but not very small.

725
00:43:25 --> 00:43:29
And they're not near, that
angle's getting a little big

726
00:43:29 --> 00:43:33
too, it's getting toward
a 180 degrees which you

727
00:43:33 --> 00:43:35
have to stay away from.
 

728
00:43:35 --> 00:43:40
It was a 180, the triangle
would squash in.

729
00:43:40 --> 00:43:42
So it's a bit
squashed, that one.

730
00:43:42 --> 00:43:44
This one's a little
long and narrow.

731
00:43:44 --> 00:43:45
But not bad.
 

732
00:43:45 --> 00:43:50
And you need a mesh generator
to generate a mesh like this.

733
00:43:50 --> 00:43:56
And I had a thesis student just
a few years ago who wrote a

734
00:43:56 --> 00:43:59
nice mesh generator in MATLAB.
 

735
00:43:59 --> 00:44:01
So we get a mesh.
 

736
00:44:01 --> 00:44:04
OK.
 

737
00:44:04 --> 00:44:09
Now, what do you remember
the finite element idea?

738
00:44:09 --> 00:44:13
Well, first you have
to use the weak form.

739
00:44:13 --> 00:44:17
Let's save the weak form
for first thing Friday.

740
00:44:17 --> 00:44:20
The weak form of Laplace.
 

741
00:44:20 --> 00:44:22
I need the weak form of
Laplace's equation.

742
00:44:22 --> 00:44:27
I'll just tell you what it is.
 

743
00:44:27 --> 00:44:32
We all have double
integrals, oh, Poisson.

744
00:44:32 --> 00:44:33
I'm making it Poisson.
 

745
00:44:33 --> 00:44:36
I want a right-hand side.
 

746
00:44:36 --> 00:44:39
Double integral,
this will be du/dx.

747
00:44:39 --> 00:44:45
 
 

748
00:44:45 --> 00:44:46
d test function/dx.
 

749
00:44:47 --> 00:44:51
And now what's changed in 2-D,
I'll also have a du/dy*dv/dy.

750
00:44:51 --> 00:44:56
 
 

751
00:44:56 --> 00:44:56
dx/dy.
 

752
00:44:57 --> 00:44:58
That's what I'll get.
 

753
00:44:58 --> 00:45:04
And on the right-hand side
I'll just have F, the load

754
00:45:04 --> 00:45:07
times the test. dx/dy.
 

755
00:45:07 --> 00:45:10
 
 

756
00:45:10 --> 00:45:13
Yeah I guess, I have to write
that down because that's

757
00:45:13 --> 00:45:15
our starting point.
 

758
00:45:15 --> 00:45:18
Do you remember the situation,
and of course that's on the

759
00:45:18 --> 00:45:25
exam this evening, is
remembering about the weak form

760
00:45:25 --> 00:45:32
in 1-D, so in 2-D I expect to
see x's and y's, my functions

761
00:45:32 --> 00:45:34
are functions of x and y.
 

762
00:45:34 --> 00:45:37
I have my, this is my solution.
 

763
00:45:37 --> 00:45:42
The V's are all possible test
functions, and this hold for

764
00:45:42 --> 00:45:47
every, this is like
the virtual work.

765
00:45:47 --> 00:45:50
You could call it the equation
of virtual work, if your

766
00:45:50 --> 00:45:53
were a little old-fashioned.
 

767
00:45:53 --> 00:45:56
Those V's are virtual
displacements, they're

768
00:45:56 --> 00:45:57
test functions.
 

769
00:45:57 --> 00:45:59
That's what it looks like.
 

770
00:45:59 --> 00:46:04
I'll come back to that
at the start of Friday.

771
00:46:04 --> 00:46:07
What's the finite element idea?
 

772
00:46:07 --> 00:46:11
Just as in 1-D, the finite
element idea is choose some

773
00:46:11 --> 00:46:13
trial functions, right?
 

774
00:46:13 --> 00:46:15
Choose some trial functions.
 

775
00:46:15 --> 00:46:20
And our approximate guy is
going to be some combination

776
00:46:20 --> 00:46:22
of these trial functions.
 

777
00:46:22 --> 00:46:26
Some coefficient U_1 times the
first trial function plus so

778
00:46:26 --> 00:46:35
on, plus U_n times the
nth trial function.

779
00:46:35 --> 00:46:38
So this will be our, and
these will also be our

780
00:46:38 --> 00:46:39
test functions again.
 

781
00:46:39 --> 00:46:47
These will also be, the
phis are also the V's.

782
00:46:47 --> 00:46:49
I'll get an equation.
 

783
00:46:49 --> 00:46:53
By using n tests, by using
the weak form n times

784
00:46:53 --> 00:46:55
for these n functions.
 

785
00:46:55 --> 00:46:59
And so each equation comes.
 

786
00:46:59 --> 00:47:03
V is one of the phis, U is
this combination of phis.

787
00:47:03 --> 00:47:04
Plug it in.
 

788
00:47:04 --> 00:47:07
You get an equation.
 

789
00:47:07 --> 00:47:11
What do I got to do in three
minutes, is speak about

790
00:47:11 --> 00:47:13
the most important point.
 

791
00:47:13 --> 00:47:14
The phis.
 

792
00:47:14 --> 00:47:17
Choosing the phis
is what matters.

793
00:47:17 --> 00:47:20
That's what made
finite elements win.

794
00:47:20 --> 00:47:27
The fact that I choose nice
simple functions, which are

795
00:47:27 --> 00:47:30
polynomials, little easy
polynomials on each

796
00:47:30 --> 00:47:33
element, on each triangle.
 

797
00:47:33 --> 00:47:35
And the ones I'm going
to speak about today

798
00:47:35 --> 00:47:37
are the linear ones.
 

799
00:47:37 --> 00:47:39
They're like hat
functions, right?

800
00:47:39 --> 00:47:41
But now we're in 2-D.
 

801
00:47:41 --> 00:47:43
They're going to be pyramids.
 

802
00:47:43 --> 00:47:48
So if I take, this is
unknown number one.

803
00:47:48 --> 00:47:50
Unknown number one, it's
got its hat function.

804
00:47:50 --> 00:47:52
Pyramid function, sorry.
 

805
00:47:52 --> 00:47:53
Pyramid function.
 

806
00:47:53 --> 00:47:56
So its pyramid function
is a one there, right?

807
00:47:56 --> 00:48:01
The pyramid, top of the
pyramid is over that point.

808
00:48:01 --> 00:48:06
And it goes down to zero so
it's zero at all these points.

809
00:48:06 --> 00:48:08
Well, at all the other points.
 

810
00:48:08 --> 00:48:16
That's the one beauty of finite
elements, is that then this U,

811
00:48:16 --> 00:48:19
this coefficient U_1 that we
compute, will actually be our

812
00:48:19 --> 00:48:21
approximation at this point.
 

813
00:48:21 --> 00:48:25
Because all the others
are zero at that point.

814
00:48:25 --> 00:48:28
So can you just, this is all
you have to do in the last 60

815
00:48:28 --> 00:48:32
seconds is visualize
this function.

816
00:48:32 --> 00:48:35
You see it, maybe tent would
be better than pyramid?

817
00:48:35 --> 00:48:36
Either one.
 

818
00:48:36 --> 00:48:39
What's the base of the pyramid?
 

819
00:48:39 --> 00:48:45
The base of the pyramid, I'm
thinking of the surface as a

820
00:48:45 --> 00:48:48
way to visualize this function.
 

821
00:48:48 --> 00:48:50
It's a function that's
one here, it goes down

822
00:48:50 --> 00:48:53
to zero linearly.
 

823
00:48:53 --> 00:48:57
So these are linear
triangular elements.

824
00:48:57 --> 00:48:59
What's the base of the pyramid?
 

825
00:48:59 --> 00:49:03
Just this, right?
 

826
00:49:03 --> 00:49:05
That's the base.
 

827
00:49:05 --> 00:49:10
Because over in these
other triangles, nothing

828
00:49:10 --> 00:49:11
ever got off zero.
 

829
00:49:11 --> 00:49:13
Nothing got off the ground.
 

830
00:49:13 --> 00:49:15
It's zero, zero,
zero at all three.

831
00:49:15 --> 00:49:18
If I know it at all three
corners, I know it everywhere.

832
00:49:18 --> 00:49:22
For linear functions, it's
just a piece of flat roof.

833
00:49:22 --> 00:49:23
Piece of flat roof.
 

834
00:49:23 --> 00:49:25
So I have how many pieces have
I got around this point?

835
00:49:25 --> 00:49:27
Oh, where is the point?
 

836
00:49:27 --> 00:49:32
I guess I've got one, two,
three, four, five, six, is it?

837
00:49:32 --> 00:49:37
It's a six, it's a pyramid with
six sloping faces sloping down

838
00:49:37 --> 00:49:41
to zero along the sides, right?
 

839
00:49:41 --> 00:49:47
So it's like a six-sided
teepee, six-sided tent.

840
00:49:47 --> 00:49:53
And here are the rods that
hold the tent up would

841
00:49:53 --> 00:49:55
be these six things.
 

842
00:49:55 --> 00:50:00
And then the six sides
would be six flat pieces.

843
00:50:00 --> 00:50:05
If you would see what I've
tried to draw there,

844
00:50:05 --> 00:50:06
that's phi_1.
 

845
00:50:07 --> 00:50:12
And you can imagine that when
I've got that, I can take

846
00:50:12 --> 00:50:14
the x derivative and
the y derivative.

847
00:50:14 --> 00:50:17
What will I know about the x
derivative and the y derivative

848
00:50:17 --> 00:50:20
in a typical triangle?
 

849
00:50:20 --> 00:50:26
If my function is in a typical
triangle, say in this triangle,

850
00:50:26 --> 00:50:27
my function looks like a+bx+cy.
 

851
00:50:27 --> 00:50:31
 
 

852
00:50:31 --> 00:50:34
a+bx, it's flat.
 

853
00:50:34 --> 00:50:35
It's linear.
 

854
00:50:35 --> 00:50:39
And these three numbers, a and
b, are decided by the fact that

855
00:50:39 --> 00:50:41
the function should be one
there, zero there,

856
00:50:41 --> 00:50:43
and zero there.
 

857
00:50:43 --> 00:50:47
Three facts about the function,
three coefficients to

858
00:50:47 --> 00:50:49
match those facts.
 

859
00:50:49 --> 00:50:54
And what's the deal
on the x derivative?

860
00:50:54 --> 00:50:54
It's just b.
 

861
00:50:54 --> 00:50:56
It's a constant.
 

862
00:50:56 --> 00:50:58
The y derivative is
just c, a constant.

863
00:50:58 --> 00:51:04
So the integrals are easy and
the the whole finite element

864
00:51:04 --> 00:51:08
system just goes smoothly.
 

865
00:51:08 --> 00:51:12
So then what the finite element
system has to do, and we'll

866
00:51:12 --> 00:51:16
talk about it Friday, is gotta
keep track of which triangles

867
00:51:16 --> 00:51:17
go to which nodes.
 

868
00:51:17 --> 00:51:20
How do you assemble, where
do these pieces go in?

869
00:51:20 --> 00:51:26
But every finite, every element
matrix is going to be simple.

870
00:51:26 --> 00:51:27
OK.