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PROFESSOR STRANG: I'm
open for questions

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as always on every
aspect, or comments

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on every aspect of the course.

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Yeah?

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The MATLAB problem, yes.

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I better remember what
that MATLAB problem is.

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So it's solving the equation
-u''+Vu' equal some delta

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function halfway along?

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And I chose boundary
conditions like those

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so that this gets replaced by a
matrix K over delta x squared.

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And this term is V times
some centered difference.

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I don't know whether I should
use C. We called C, earlier

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in the book, C was a circulant.

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This is a centered first
difference over delta x

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equal right-hand side.

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All that multiplies the
discrete solution u.

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I didn't get that very
well for the camera.

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Maybe I'll say it again.

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We have the second difference
that corresponds to -u''.

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And V times the first difference
that corresponds to Vu'.

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And then, the problem is--

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So the physical problem is,
what happens as V gets larger?

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V grows.

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How does the solution
change as V increases?

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So it's a very
practical problem.

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The V measures the importance
of this convection,

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00:02:39,440 --> 00:02:43,490
the velocity of the fluid.

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00:02:43,490 --> 00:02:56,810
Then n, which is, well maybe
delta x is 1/(n+1), maybe.

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So that we would have n
interior points and we know

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the boundary values.

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00:03:06,620 --> 00:03:14,080
So the question is how do we
interpret this increasing n?

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I guess we expect more
and more accuracy, right?

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00:03:17,500 --> 00:03:21,380
As n increases, delta
x getting smaller.

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Our idea is that this should
be a better and better

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approximation to the
differential equation.

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The question about the
eigenvalues of this matrix,

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I don't know exactly the answer.

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So the eigenvalues, did you
discover that the eigenvalues

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suddenly, they change?

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They're real for small
values of V and n.

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But then there's some point
at which they become complex.

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00:04:02,290 --> 00:04:09,390
And a point that involves both
V and delta x actually, V and n.

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It's kind of interesting.

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I guess at this moment,
interesting is the key word.

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So what is the
physics going on here?

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So I have a 1-D flow
and it's blocked

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by these boundary
conditions are sort

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of blocking it at both ends.

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And I'm looking
for a steady state.

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And I'm feeding in
this source here.

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So I'm feeding in a
source halfway along.

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So I think that would be right.

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Now what's going
to happen roughly?

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This term is, I think,
flowing to the right.

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So it's going to carry, suppose
this was smoke or particles

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or whatever.

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00:05:13,280 --> 00:05:16,310
This is like an
environmental differential

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equation or a chemical
engineering equation.

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Just oh, and more and
more applications.

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So I think of it as this term
is carrying the flow with it.

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Like, that way.

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So I'm expecting your solutions
to be larger on this side.

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And how do we get any
action at all on the left?

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That comes from the diffusion.

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The particles, when they enter,
get carried away, and faster

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and faster as V increases.

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But at the same
time they diffuse,

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they bounce back and forth.

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And some bit of
it goes this way.

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But not a lot.

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I'm guessing that the solution
would have a profile that

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would be kind of small.

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And then there's whatever jump
there has to be here, and then

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larger there.

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Oh, but then I have
to get it to zero.

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It's a strange situation.

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00:06:27,330 --> 00:06:31,340
And maybe I'll take this
chance to mention it.

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What happens as V gets
really, really large?

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You could say okay, as
V gets extremely large,

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forget that term.

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This is the important term.

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Vu' equals that.

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Which we could easily solve.

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00:06:52,030 --> 00:06:58,260
But this equation
with just this term

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is only first order, right?

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00:07:00,130 --> 00:07:01,950
It only has a first derivative.

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00:07:01,950 --> 00:07:04,000
And how many boundary
conditions would we

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00:07:04,000 --> 00:07:08,080
expect to have if I gave
you, if this wasn't here

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00:07:08,080 --> 00:07:11,480
and I gave you just a
first order equation?

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00:07:11,480 --> 00:07:12,580
One.

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00:07:12,580 --> 00:07:14,610
And we've got two.

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Which we're imposing.

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00:07:16,160 --> 00:07:19,710
So this sort of
limit as V increases,

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it produces something
called a boundary layer.

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The solution is forced
to satisfy these boundary

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conditions.

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One of them, it's
quite happy about.

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00:07:34,210 --> 00:07:36,720
The other one, it didn't
really want to satisfy.

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00:07:36,720 --> 00:07:41,280
It only satisfied them
because, I had two

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00:07:41,280 --> 00:07:42,950
originally with this part.

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00:07:42,950 --> 00:07:47,330
But as this takes
over, the struggle

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00:07:47,330 --> 00:07:50,150
to satisfy that second boundary
condition that it really

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doesn't want to
satisfy is resolved

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by something, a
layer that just makes

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a sort of exponential correction
at the boundary to get

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00:08:00,870 --> 00:08:03,030
to where it's supposed to go.

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00:08:03,030 --> 00:08:03,860
So, anyway.

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00:08:03,860 --> 00:08:08,850
This is a model problem
that has boundary layers.

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00:08:08,850 --> 00:08:13,940
Those are terrifically
important in aerodynamics,

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00:08:13,940 --> 00:08:17,790
you have layers around
the actual aircraft.

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00:08:17,790 --> 00:08:22,980
So lots of physics and
computational science

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00:08:22,980 --> 00:08:28,430
is hiding in this
type of example.

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00:08:28,430 --> 00:08:29,880
So those are a few words.

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00:08:29,880 --> 00:08:34,790
But not an answer
to your question.

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00:08:34,790 --> 00:08:38,950
I guess I'm happy if
you, for example, let

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00:08:38,950 --> 00:08:43,450
me know where the eigenvalues
change from real to complex.

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00:08:43,450 --> 00:08:46,040
I mean, it happens,
but you can probably,

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00:08:46,040 --> 00:08:47,840
if you look at the
matrix, you can

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00:08:47,840 --> 00:08:58,720
see what happens at the point
where that change takes place.

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00:08:58,720 --> 00:08:59,720
That's some comments.

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00:08:59,720 --> 00:09:05,140
And with the MATLAB, pure
MATLAB part, I guess I'm hoping,

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00:09:05,140 --> 00:09:09,430
and the graders are hoping,
that by providing a code

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00:09:09,430 --> 00:09:15,160
we'll get a pretty
systematic set of answers

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00:09:15,160 --> 00:09:23,390
for the requested V equal,
what was it, 3 and 12 maybe,

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00:09:23,390 --> 00:09:24,310
or something.

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00:09:24,310 --> 00:09:26,330
And the requested n, delta x's.

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00:09:26,330 --> 00:09:29,080


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00:09:29,080 --> 00:09:31,510
So I'm hoping that
everybody's graph

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00:09:31,510 --> 00:09:36,620
is going to look pretty
similar for those requests.

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00:09:36,620 --> 00:09:42,190
And then if you could do a
little exploration yourself

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00:09:42,190 --> 00:09:47,430
and make that a fifth
page, or really less

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00:09:47,430 --> 00:09:53,080
than a page, about take
V larger, take n larger,

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00:09:53,080 --> 00:09:54,790
what happens?

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00:09:54,790 --> 00:09:57,740
I'm very happy.

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00:09:57,740 --> 00:09:59,720
Or any direction.

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00:09:59,720 --> 00:10:03,480
I mean, just think of
it as a mini-project

145
00:10:03,480 --> 00:10:08,060
to do the requested
ones and then

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00:10:08,060 --> 00:10:15,280
to do a little experimentation.

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00:10:15,280 --> 00:10:22,010
I'll just be interested
about anything you observe.

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00:10:22,010 --> 00:10:28,480
So there's no right
answer to that part.

149
00:10:28,480 --> 00:10:33,030
Is that any help
with the MATLAB?

150
00:10:33,030 --> 00:10:37,820
Let me just say, since
you showed up today,

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00:10:37,820 --> 00:10:40,940
if you have trouble
with the MATLAB

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00:10:40,940 --> 00:10:46,330
and you need Peter's
help at noon Friday

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00:10:46,330 --> 00:10:50,740
it would be okay to turn in
the MATLAB later on Friday.

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00:10:50,740 --> 00:10:52,730
I shouldn't say that.

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00:10:52,730 --> 00:10:55,580
But I just did.

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00:10:55,580 --> 00:10:57,520
So there you are.

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00:10:57,520 --> 00:11:00,490
Right.

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00:11:00,490 --> 00:11:06,450
I mean, this course,
as you've begun to see,

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00:11:06,450 --> 00:11:10,280
I know that you have lots
of demands on your time.

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00:11:10,280 --> 00:11:13,610
And sometimes you
have job interviews.

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00:11:13,610 --> 00:11:14,910
All sorts of stuff.

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00:11:14,910 --> 00:11:17,320
Conferences that
you have to go to.

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00:11:17,320 --> 00:11:19,210
We deal with that.

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00:11:19,210 --> 00:11:25,260
So just give me the
homeworks as soon as you can.

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00:11:25,260 --> 00:11:29,990
If you're ready Friday at
class time, that's perfect.

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00:11:29,990 --> 00:11:38,120
If are stuck on
MATLAB and you want

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00:11:38,120 --> 00:11:41,740
to go to Peter's discussion,
which follows the Friday

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00:11:41,740 --> 00:11:45,120
class in here, that's fine too.

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00:11:45,120 --> 00:11:48,190
So that's some thoughts
about the MATLAB

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00:11:48,190 --> 00:11:53,200
and it's partly open-ended.

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00:11:53,200 --> 00:11:57,030
What about lots of other
things in this course?

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00:11:57,030 --> 00:11:59,210
Homework came in.

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00:11:59,210 --> 00:12:00,200
How was the homework?

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00:12:00,200 --> 00:12:02,540
Maybe I just take
this chance to ask.

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00:12:02,540 --> 00:12:05,020
Was it a reasonable length?

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00:12:05,020 --> 00:12:07,270
The homework three, the pset?

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00:12:07,270 --> 00:12:09,940


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00:12:09,940 --> 00:12:15,070
Not too bad?

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00:12:15,070 --> 00:12:21,000
I will be able to post solutions
because several people are

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00:12:21,000 --> 00:12:28,240
contributing typed
solutions that could go

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00:12:28,240 --> 00:12:36,360
on on the
math.mit.edu/cse website.

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00:12:36,360 --> 00:12:39,520
Or they could also
go on our 18.085

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00:12:39,520 --> 00:12:41,350
website for this semester.

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00:12:41,350 --> 00:12:42,630
Good.

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00:12:42,630 --> 00:12:44,490
Okay, help me out
with some questions.

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00:12:44,490 --> 00:12:58,150
Thanks.

187
00:12:58,150 --> 00:13:01,400
So the question is
about element matrices.

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00:13:01,400 --> 00:13:12,450
In lecture 8, I guess it was,
where we were assembling this,

189
00:13:12,450 --> 00:13:16,970
I used the word that's used
by finite element people,

190
00:13:16,970 --> 00:13:21,710
we were assembling K.
And one way to do it

191
00:13:21,710 --> 00:13:26,120
was to multiply
the three matrices.

192
00:13:26,120 --> 00:13:30,060
But that's not how
it's actually done.

193
00:13:30,060 --> 00:13:35,860
It's actually assembled
out of small matrices.

194
00:13:35,860 --> 00:13:39,910
Maybe small K would
be the right--

195
00:13:39,910 --> 00:13:43,700
So this would be a K_element.

196
00:13:43,700 --> 00:13:47,580
Out of smaller element matrices.

197
00:13:47,580 --> 00:13:51,360
So for example, what's
the element matrix for,

198
00:13:51,360 --> 00:13:56,770
one for each spring in this
particular problem of springs

199
00:13:56,770 --> 00:13:57,770
and masses.

200
00:13:57,770 --> 00:14:03,660
So let me draw some springs,
some masses, some springs,

201
00:14:03,660 --> 00:14:06,470
another mass, more
springs, fixed,

202
00:14:06,470 --> 00:14:10,460
not fixed, whatever,
maybe fixed-free

203
00:14:10,460 --> 00:14:12,140
the way I've drawn it there.

204
00:14:12,140 --> 00:14:16,940
So here's a spring with
spring constant c_2.

205
00:14:16,940 --> 00:14:22,240
And we could look
at the contribution

206
00:14:22,240 --> 00:14:29,160
to the whole matrix coming
from this piece of the problem.

207
00:14:29,160 --> 00:14:32,570
This was actually, the
finite element method

208
00:14:32,570 --> 00:14:36,560
has a wonderful
history of people,

209
00:14:36,560 --> 00:14:38,890
and it had different
names way back,

210
00:14:38,890 --> 00:14:44,710
of people seeing the
structure as broken in pieces

211
00:14:44,710 --> 00:14:47,520
and then connected together.

212
00:14:47,520 --> 00:14:50,970
And then what did a
typical piece look like?

213
00:14:50,970 --> 00:14:56,180
So a typical piece
there, well, you

214
00:14:56,180 --> 00:14:59,290
let me just write down what
this matrix is going to be.

215
00:14:59,290 --> 00:15:03,750
The little element matrix
is coming from spring two.

216
00:15:03,750 --> 00:15:06,790
So this would be,
like element two will

217
00:15:06,790 --> 00:15:10,260
be, it'll have a c_2 outside.

218
00:15:10,260 --> 00:15:12,630
I'll put the c_2 outside.

219
00:15:12,630 --> 00:15:18,530
And then inside will
be a little, this guy.

220
00:15:18,530 --> 00:15:23,710
And we can talk more
about why it's that one.

221
00:15:23,710 --> 00:15:30,800
But just to have the element
matrix there on the board.

222
00:15:30,800 --> 00:15:38,540
So my claim is that this is
a small piece of the big K.

223
00:15:38,540 --> 00:15:45,280
So the big K matrix, what's the
size of the big K, of K itself?

224
00:15:45,280 --> 00:15:48,080
Three by three in this
case, yeah, three masses.

225
00:15:48,080 --> 00:15:50,780
So it'll be three by three.

226
00:15:50,780 --> 00:15:56,650
So I'm thinking that
this spring which

227
00:15:56,650 --> 00:16:03,340
connects mass one to mass two,
so it's only going to be like,

228
00:16:03,340 --> 00:16:07,930
a two by two piece, a little
local piece, you could say,

229
00:16:07,930 --> 00:16:13,050
that that little
K fits in this--

230
00:16:13,050 --> 00:16:20,430
is assembled into the-- I
better call it K_2, right?

231
00:16:20,430 --> 00:16:26,230
So it's a little K that sits
up in this two by two block

232
00:16:26,230 --> 00:16:29,540
and doesn't contribute
to the rest.

233
00:16:29,540 --> 00:16:32,170
Then let's just draw
the rest of the picture.

234
00:16:32,170 --> 00:16:35,380
So this would produce
an element matrix

235
00:16:35,380 --> 00:16:39,010
that looks just the same,
that's the beauty of this.

236
00:16:39,010 --> 00:16:43,260
That all the elements, apart
from change in the spring

237
00:16:43,260 --> 00:16:44,760
constant, look the same.

238
00:16:44,760 --> 00:16:49,170
So there'd be a little K_3.

239
00:16:49,170 --> 00:16:51,430
And where will it go?

240
00:16:51,430 --> 00:16:55,170
This is the whole
core to the point.

241
00:16:55,170 --> 00:16:57,690
It'll go to the lower right.

242
00:16:57,690 --> 00:17:00,910
Down here?

243
00:17:00,910 --> 00:17:02,820
Overlapping, overlapping.

244
00:17:02,820 --> 00:17:09,350
Because this mass is attached
to that spring and to that one.

245
00:17:09,350 --> 00:17:12,760
So these little element
matrices, they overlap.

246
00:17:12,760 --> 00:17:18,610
And you just need, if you
can imagine the code that's

247
00:17:18,610 --> 00:17:22,480
going to do this, you
need a list of springs

248
00:17:22,480 --> 00:17:25,140
and a list of masses and
a list of the connections

249
00:17:25,140 --> 00:17:26,150
between them.

250
00:17:26,150 --> 00:17:29,990
And it'll sit in here
because it's two by two.

251
00:17:29,990 --> 00:17:32,500
So that's two by two,
that's two by two

252
00:17:32,500 --> 00:17:38,980
and in this overlap entry will
be a c_2 from the upper box.

253
00:17:38,980 --> 00:17:43,850
It'll be a c_2+c_3 as
we discovered by direct

254
00:17:43,850 --> 00:17:45,310
multiplication.

255
00:17:45,310 --> 00:17:47,640
So that's that spring.

256
00:17:47,640 --> 00:17:51,280
Now what about
this first spring?

257
00:17:51,280 --> 00:17:54,290
So there's a little K_1.

258
00:17:54,290 --> 00:17:57,970
Now K_1 should look
the same as this.

259
00:17:57,970 --> 00:17:59,480
Except what?

260
00:17:59,480 --> 00:18:04,300
So it's going to be a difference
with this spring because?

261
00:18:04,300 --> 00:18:09,330
Because of this fixed end.

262
00:18:09,330 --> 00:18:12,320
There's no mass zero.

263
00:18:12,320 --> 00:18:17,830
K_1 would normally sit up
here, but actually it's

264
00:18:17,830 --> 00:18:21,190
only going to be one by one.

265
00:18:21,190 --> 00:18:30,210
So the K_1 little element matrix
would look like c_1[1, -1; -1,

266
00:18:30,210 --> 00:18:31,010
1].

267
00:18:31,010 --> 00:18:34,670
And then the boundary
conditions knock those out.

268
00:18:34,670 --> 00:18:39,390
And an interesting point
if you're writing big code

269
00:18:39,390 --> 00:18:40,870
you have to decide.

270
00:18:40,870 --> 00:18:46,240
Do I, as I create K_1,
this little element matrix,

271
00:18:46,240 --> 00:18:49,550
do I pay attention to
the boundary conditions?

272
00:18:49,550 --> 00:18:53,970
And then K_1 would just
be that single number

273
00:18:53,970 --> 00:18:56,350
c_1 which would sit there.

274
00:18:56,350 --> 00:19:01,940
So up there will be c_1
from the K_1 matrix and c_2

275
00:19:01,940 --> 00:19:05,170
from the k_2 matrix.

276
00:19:05,170 --> 00:19:08,690
That's the entry there.

277
00:19:08,690 --> 00:19:12,190
And then, as I say, in
coding finite elements, which

278
00:19:12,190 --> 00:19:17,080
you guys may do at some
point, there's a question.

279
00:19:17,080 --> 00:19:18,020
What's sufficient?

280
00:19:18,020 --> 00:19:23,350
Shall I pay attention
to the boundary

281
00:19:23,350 --> 00:19:25,810
in creating these
element matrices

282
00:19:25,810 --> 00:19:27,690
or shall I do it later?

283
00:19:27,690 --> 00:19:31,900
And the rule seems
to be, do it later.

284
00:19:31,900 --> 00:19:36,860
So what gets created
in finite elements

285
00:19:36,860 --> 00:19:43,200
is a, in our language would
be a, free-free matrix.

286
00:19:43,200 --> 00:19:55,440
It's a matrix where even this
spring has a two by two piece.

287
00:19:55,440 --> 00:20:01,440
So what's the problem
with the free-free matrix?

288
00:20:01,440 --> 00:20:05,400
The free-free matrices,
those with no supports,

289
00:20:05,400 --> 00:20:08,300
those matrices will be singular.

290
00:20:08,300 --> 00:20:10,000
Right, singular.

291
00:20:10,000 --> 00:20:13,350
Because the vector
of all ones, if you

292
00:20:13,350 --> 00:20:17,130
multiply the K, the
free-free matrix

293
00:20:17,130 --> 00:20:20,140
times the vector of
all ones, you get zero.

294
00:20:20,140 --> 00:20:23,520
The whole thing could shift.

295
00:20:23,520 --> 00:20:28,080
And will have other rigid
motions in two dimensions.

296
00:20:28,080 --> 00:20:31,050
Let's just think ahead here.

297
00:20:31,050 --> 00:20:36,550
What are the rigid motions,
the null space of K,

298
00:20:36,550 --> 00:20:41,030
you could say, for a
two-dimensional truss.

299
00:20:41,030 --> 00:20:44,320
So I've got a bunch
of bars and springs.

300
00:20:44,320 --> 00:20:46,160
Think of a mattress.

301
00:20:46,160 --> 00:20:47,775
I'm in two
dimensions, a mattress

302
00:20:47,775 --> 00:20:54,470
is a bunch of springs
connected in a 2-D grid.

303
00:20:54,470 --> 00:20:58,810
And say it's free at both ends.

304
00:20:58,810 --> 00:21:00,850
So what can I do to a mattress?

305
00:21:00,850 --> 00:21:01,580
Oh, my gosh!

306
00:21:01,580 --> 00:21:05,680
I didn't expect that
to be on videotape.

307
00:21:05,680 --> 00:21:06,870
So it's in the plane.

308
00:21:06,870 --> 00:21:09,380
We're in 2-D here.

309
00:21:09,380 --> 00:21:13,630
What can I do if there are
no boundary conditions?

310
00:21:13,630 --> 00:21:15,740
Well I can shift
it to the right.

311
00:21:15,740 --> 00:21:16,240
Right?

312
00:21:16,240 --> 00:21:18,040
That's our [1, 1, 1, 1, 1].

313
00:21:18,040 --> 00:21:20,360
What else can I do?

314
00:21:20,360 --> 00:21:21,760
I can rotate.

315
00:21:21,760 --> 00:21:23,340
And I can shift
it the other way.

316
00:21:23,340 --> 00:21:29,140
So there would be three rigid
motions for the 2-D problem.

317
00:21:29,140 --> 00:21:33,420
Two translations and a rotation.

318
00:21:33,420 --> 00:21:35,460
And when you get up
to three dimensions,

319
00:21:35,460 --> 00:21:40,250
do you want to guess
what's the number in 3-D?

320
00:21:40,250 --> 00:21:41,110
Six.

321
00:21:41,110 --> 00:21:42,240
Number is six.

322
00:21:42,240 --> 00:21:46,580
Three translations and
rotations around three axes.

323
00:21:46,580 --> 00:21:54,010
So those, either one rigid
motion or three rigid motions

324
00:21:54,010 --> 00:21:58,410
or six rigid
motions have to get,

325
00:21:58,410 --> 00:21:59,970
the boundary
conditions eventually

326
00:21:59,970 --> 00:22:05,020
have to remove those, have
to knock out rows and columns

327
00:22:05,020 --> 00:22:05,780
and remove them.

328
00:22:05,780 --> 00:22:13,150
But my point was just that
quite efficient to do it later.

329
00:22:13,150 --> 00:22:15,820
The picture is so clear here.

330
00:22:15,820 --> 00:22:18,026
So the actual
matrix would be four

331
00:22:18,026 --> 00:22:20,900
by four, the unreduced matrix.

332
00:22:20,900 --> 00:22:27,500
And then when we
fix node zero there,

333
00:22:27,500 --> 00:22:30,560
that would knock out that part
and leave the three by three

334
00:22:30,560 --> 00:22:31,630
that we want.

335
00:22:31,630 --> 00:22:38,250
So this is just discussion of
how to think of this matrix K.

336
00:22:38,250 --> 00:22:42,220
So the direct way
to think of it was

337
00:22:42,220 --> 00:22:46,840
as a product of big matrices.

338
00:22:46,840 --> 00:22:53,000
But in reality it's assembled
from these element pieces.

339
00:22:53,000 --> 00:22:56,870
And of course, our goal in
talking about finite elements

340
00:22:56,870 --> 00:23:00,430
will be to see that.

341
00:23:00,430 --> 00:23:04,620
It'll come up again, of course.

342
00:23:04,620 --> 00:23:11,150
Your good question led me there,
but did I answer the question?

343
00:23:11,150 --> 00:23:14,970
And maybe the way I
mentioned it in class was

344
00:23:14,970 --> 00:23:19,890
to notice that these guys,
these element matrices

345
00:23:19,890 --> 00:23:22,770
can be thought of this way.

346
00:23:22,770 --> 00:23:26,450
It is matrix multiplication,
but done differently.

347
00:23:26,450 --> 00:23:30,820
It's a column of this
matrix times the number

348
00:23:30,820 --> 00:23:33,650
c here times a row of this.

349
00:23:33,650 --> 00:23:38,260
So it's matrix multiplication,
columns times rows.

350
00:23:38,260 --> 00:23:49,060
So you can multiply AB,
columns of A times rows of B

351
00:23:49,060 --> 00:23:55,090
and then add over,
from one to n.

352
00:23:55,090 --> 00:24:01,340
So column of A times
row of B. So a column,

353
00:24:01,340 --> 00:24:03,350
then, is a vector like this.

354
00:24:03,350 --> 00:24:10,590
A row is a vector like that.

355
00:24:10,590 --> 00:24:15,700
And the result is
a full size matrix,

356
00:24:15,700 --> 00:24:18,880
but if the column
is concentrated

357
00:24:18,880 --> 00:24:21,690
at a couple of nodes
and the row is,

358
00:24:21,690 --> 00:24:28,630
then it will have zeroes
everywhere except at that.

359
00:24:28,630 --> 00:24:34,230
This is the element matrix,
with the C included.

360
00:24:34,230 --> 00:24:41,010
That would be the element matrix
already blown up to full size

361
00:24:41,010 --> 00:24:43,590
by adding zeroes elsewhere.

362
00:24:43,590 --> 00:24:49,040
So, I mean the heart
of the element matrix

363
00:24:49,040 --> 00:24:51,490
is where the action
is on that spring.

364
00:24:51,490 --> 00:24:55,020
And then, when we
assemble it, of course,

365
00:24:55,020 --> 00:24:59,090
it doesn't contribute
down there.

366
00:24:59,090 --> 00:25:03,850
So the technology of finite
elements is quite interesting

367
00:25:03,850 --> 00:25:05,160
and it fits.

368
00:25:05,160 --> 00:25:11,360
It's a beautiful way to
see the discrete problem.

369
00:25:11,360 --> 00:25:15,770
Ready for another
question of any sort.

370
00:25:15,770 --> 00:25:22,930
Thank you, good.

371
00:25:22,930 --> 00:25:25,090
Yeah, sorry, it got
onto the problem set.

372
00:25:25,090 --> 00:25:29,430
And then I thought--
let me write those words

373
00:25:29,430 --> 00:25:34,500
down first, singular
value decomposition.

374
00:25:34,500 --> 00:25:40,230
Well I won't write
all those words.

375
00:25:40,230 --> 00:25:42,620
That's a wonderful thing.

376
00:25:42,620 --> 00:25:45,410
It's a highlight of
matrix theory except

377
00:25:45,410 --> 00:25:47,150
for the length of its name.

378
00:25:47,150 --> 00:25:53,360
So SVD is what
everybody calls it.

379
00:25:53,360 --> 00:25:57,250
It's only like, every
year now people appreciate

380
00:25:57,250 --> 00:26:00,140
more and more the
importance of this SVD,

381
00:26:00,140 --> 00:26:02,910
this singular value
decomposition.

382
00:26:02,910 --> 00:26:09,030
So shall I say a few
words about it now?

383
00:26:09,030 --> 00:26:14,500
Just a few.

384
00:26:14,500 --> 00:26:18,690
So it's Section 1.7 of the book.

385
00:26:18,690 --> 00:26:26,710
And my thought was, hey we've
done so much matrix theory

386
00:26:26,710 --> 00:26:29,790
including eigenvalues and
positive definiteness,

387
00:26:29,790 --> 00:26:32,700
let's get on and use it.

388
00:26:32,700 --> 00:26:36,260
And then I can come
back to the SVD

389
00:26:36,260 --> 00:26:40,190
in a sort of lighter moment.

390
00:26:40,190 --> 00:26:42,100
Because I'm not
thinking you will

391
00:26:42,100 --> 00:26:44,600
be responsible for the SVD.

392
00:26:44,600 --> 00:26:46,680
Eigenvalues I hope
you're understanding.

393
00:26:46,680 --> 00:26:49,950
Positive definite I hope
you're understanding.

394
00:26:49,950 --> 00:26:52,990
That's heart of
the course stuff.

395
00:26:52,990 --> 00:26:58,360
But SVD is a key idea
in linear algebra,

396
00:26:58,360 --> 00:27:01,050
but we can't do everything.

397
00:27:01,050 --> 00:27:06,770
But we can say what it is.

398
00:27:06,770 --> 00:27:08,990
What's the first point?

399
00:27:08,990 --> 00:27:11,200
The first point is
that every matrix,

400
00:27:11,200 --> 00:27:14,910
even a rectangular matrix,
has got a singular value

401
00:27:14,910 --> 00:27:15,770
decomposition.

402
00:27:15,770 --> 00:27:23,200
So the matrix A can be m by n.

403
00:27:23,200 --> 00:27:26,260
I wouldn't speak
about the eigenvalues

404
00:27:26,260 --> 00:27:29,200
of a rectangular matrix.

405
00:27:29,200 --> 00:27:31,610
Because if I multiply,
you remember,

406
00:27:31,610 --> 00:27:34,230
the eigenvalue equation
wouldn't make sense.

407
00:27:34,230 --> 00:27:39,920
Ax=lambda*x is no good
if A is rectangular.

408
00:27:39,920 --> 00:27:40,730
Right?

409
00:27:40,730 --> 00:27:44,720
The input would be of
length n and the output

410
00:27:44,720 --> 00:27:50,250
would be of length m
and this wouldn't work.

411
00:27:50,250 --> 00:27:54,820
So eigenvectors are
not what I'm after.

412
00:27:54,820 --> 00:27:58,660
But somehow the
goal of eigenvectors

413
00:27:58,660 --> 00:28:00,180
was to diagonalize.

414
00:28:00,180 --> 00:28:04,670
The goal of eigenvectors was to
find these special directions

415
00:28:04,670 --> 00:28:09,440
in which the matrix A
just acted like a number.

416
00:28:09,440 --> 00:28:16,330
And then, as we saw today in the
part that partly still up here,

417
00:28:16,330 --> 00:28:19,050
we could solve equations
by eigenvectors

418
00:28:19,050 --> 00:28:26,890
by looking for these,
following these special guys.

419
00:28:26,890 --> 00:28:29,720
What do we do for a
rectangular matrix?

420
00:28:29,720 --> 00:28:31,130
What replaces this?

421
00:28:31,130 --> 00:28:36,380
So this is now not good.

422
00:28:36,380 --> 00:28:40,960
The idea is simply we
need two sets of vectors.

423
00:28:40,960 --> 00:28:48,910
We need some v's and some u's.

424
00:28:48,910 --> 00:28:52,860
So that's the central
equation of the SVD.

425
00:28:52,860 --> 00:28:56,900
Now what can we get?

426
00:28:56,900 --> 00:29:01,130
So now we have more freedom
because we're getting a bunch

427
00:29:01,130 --> 00:29:05,640
of v's that have,
these guys are in R^n.

428
00:29:05,640 --> 00:29:07,070
They have length n, right?

429
00:29:07,070 --> 00:29:10,840
To multiply A times
v, and the output

430
00:29:10,840 --> 00:29:17,620
is-- so there're
n of these, n v's,

431
00:29:17,620 --> 00:29:19,900
we're in n-dimensional space.

432
00:29:19,900 --> 00:29:22,449
So those are called
singular vectors.

433
00:29:22,449 --> 00:29:24,990
They're called right singular
vectors because they're sitting

434
00:29:24,990 --> 00:29:28,050
to the right of
A. And these guys,

435
00:29:28,050 --> 00:29:35,360
these outputs will be in-- these
are v's in R^n, n dimensions.

436
00:29:35,360 --> 00:29:41,890
I have m of these, m u's
in m-dimensional space.

437
00:29:41,890 --> 00:29:45,020
And these things are
numbers, of course.

438
00:29:45,020 --> 00:29:50,190
And actually, they're all
greater or equal to zero.

439
00:29:50,190 --> 00:29:53,650
So we can get, by
allowing ourselves

440
00:29:53,650 --> 00:29:59,640
the freedom of
two, right singular

441
00:29:59,640 --> 00:30:02,960
vectors and left singular
vectors, we can get a lot more,

442
00:30:02,960 --> 00:30:06,870
and we can get,
here's the punch line.

443
00:30:06,870 --> 00:30:13,040
We can get the v's to be
orthogonal, orthonormal.

444
00:30:13,040 --> 00:30:19,030
Just like the eigenvectors
of a symmetric matrix,

445
00:30:19,030 --> 00:30:21,800
we can get these
v's, these singular

446
00:30:21,800 --> 00:30:24,640
vectors to be
perpendicular to each other

447
00:30:24,640 --> 00:30:28,810
and we can get the u's to be
perpendicular to each other.

448
00:30:28,810 --> 00:30:34,200
What we can't, what we're not
shooting for is the v's to be

449
00:30:34,200 --> 00:30:36,340
the same as the u's.

450
00:30:36,340 --> 00:30:38,340
They're not even
in the same space

451
00:30:38,340 --> 00:30:41,700
now, if the matrix
A is rectangular.

452
00:30:41,700 --> 00:30:46,920
And even if the matrix A is
square but not symmetric,

453
00:30:46,920 --> 00:30:49,580
we wouldn't get this
perpendicularity.

454
00:30:49,580 --> 00:30:57,580
But we get it in the SVD by
having different v's and u's.

455
00:30:57,580 --> 00:31:02,550
Here's my picture
of linear algebra.

456
00:31:02,550 --> 00:31:10,470
This is the big picture
of linear algebra.

457
00:31:10,470 --> 00:31:12,650
This over here is
n-dimensional space.

458
00:31:12,650 --> 00:31:17,600
We have v's.

459
00:31:17,600 --> 00:31:19,920
Think of that as
n-dimensional space.

460
00:31:19,920 --> 00:31:21,910
That's kind of a puny picture.

461
00:31:21,910 --> 00:31:29,510
But when I multiply by
A, I take a vector here,

462
00:31:29,510 --> 00:31:33,490
I multiply by A. So
let me just do that.

463
00:31:33,490 --> 00:31:38,140
I multiply by A, I take a
vector v, well, I already put v,

464
00:31:38,140 --> 00:31:43,200
I take Av and I get
something over here.

465
00:31:43,200 --> 00:31:51,330
And this will be
the space of u's.

466
00:31:51,330 --> 00:31:53,730
Now I have to ask you one
thing about linear algebra

467
00:31:53,730 --> 00:31:56,560
that I keep hammering away.

468
00:31:56,560 --> 00:32:04,310
If I take any vector and
multiply by A, What do I get?

469
00:32:04,310 --> 00:32:11,570
If I take any vector
v, like [1,  2].

470
00:32:11,570 --> 00:32:16,850
Here's A, say
[3, 6; 4, 7; 5, 8].

471
00:32:16,850 --> 00:32:20,520
So that's A times
v. What can you

472
00:32:20,520 --> 00:32:24,550
tell me about A times v that
goes a little deeper than just

473
00:32:24,550 --> 00:32:26,890
telling me the numbers?

474
00:32:26,890 --> 00:32:33,930
It's a linear combination
of the columns.

475
00:32:33,930 --> 00:32:36,880
These are the outputs, the Av's.

476
00:32:36,880 --> 00:32:40,030
So this space is called
the column space.

477
00:32:40,030 --> 00:32:43,020
It's all combinations
of the columns.

478
00:32:43,020 --> 00:32:50,530
These are all combinations
of the columns.

479
00:32:50,530 --> 00:32:52,130
That's the column space.

480
00:32:52,130 --> 00:32:56,010
It's a bunch of vectors.

481
00:32:56,010 --> 00:33:04,720
So the point is that these u's,
that I have like, a fantastic

482
00:33:04,720 --> 00:33:06,990
choice of axes.

483
00:33:06,990 --> 00:33:10,760
A linear algebra person
would use the word, bases.

484
00:33:10,760 --> 00:33:13,640
But geometrically
I'm saying they're

485
00:33:13,640 --> 00:33:17,310
fantastic axes in these spaces.

486
00:33:17,310 --> 00:33:23,290
So that if I choose the
right axes in the two spaces,

487
00:33:23,290 --> 00:33:28,900
then one axis will go to that
one when I multiply by A.

488
00:33:28,900 --> 00:33:30,440
The next one will
go to that one.

489
00:33:30,440 --> 00:33:32,370
The third will go to that one.

490
00:33:32,370 --> 00:33:37,920
It just could not be better.

491
00:33:37,920 --> 00:33:43,530
And that's the statement.

492
00:33:43,530 --> 00:33:46,610
Now maybe I'll just say a
word about where it's used

493
00:33:46,610 --> 00:33:51,600
or why it's useful.

494
00:33:51,600 --> 00:33:58,310
Oh, in so many applications.

495
00:33:58,310 --> 00:34:02,970
Well most of you
are not biologists.

496
00:34:02,970 --> 00:34:04,680
And I'm not certainly.

497
00:34:04,680 --> 00:34:09,750
But we know that there's a lot
of interesting math these days

498
00:34:09,750 --> 00:34:15,870
in a lot of interesting
experiments with genes.

499
00:34:15,870 --> 00:34:20,680
So what happens?

500
00:34:20,680 --> 00:34:24,210
So we've got about
40 people here.

501
00:34:24,210 --> 00:34:34,960
And we measure the,
well, we get data.

502
00:34:34,960 --> 00:34:36,990
That's what I'm
really going to say.

503
00:34:36,990 --> 00:34:40,200
Somehow we got a giant
amount of data. right?.

504
00:34:40,200 --> 00:34:41,520
Probably 40 columns.

505
00:34:41,520 --> 00:34:46,490
Everybody here is entitled
to be a column of the matrix.

506
00:34:46,490 --> 00:34:51,600
And the entries will be
like, have you got TB,

507
00:34:51,600 --> 00:34:55,110
what height, all this stuff.

508
00:34:55,110 --> 00:34:57,970
So you got enormous
amount of data.

509
00:34:57,970 --> 00:35:02,180
And the question is, what's
important in that data.

510
00:35:02,180 --> 00:35:05,850
You've got a million
entries in a giant matrix

511
00:35:05,850 --> 00:35:08,590
and you want to extract
the important thing.

512
00:35:08,590 --> 00:35:11,410
Well, it's the SVD that does it.

513
00:35:11,410 --> 00:35:14,230
You take that giant
matrix of data,

514
00:35:14,230 --> 00:35:19,080
you find the v's and
the sigmas and the u's.

515
00:35:19,080 --> 00:35:28,720
And then the largest sigmas are
the most important information,

516
00:35:28,720 --> 00:35:31,580
if things are scaled,
and statistics

517
00:35:31,580 --> 00:35:34,450
is coming into this also.

518
00:35:34,450 --> 00:35:43,200
I could a give sensible, much
more mathematical discussion

519
00:35:43,200 --> 00:35:47,890
of-- One word, one
name it goes under

520
00:35:47,890 --> 00:35:52,120
is principal component analysis.

521
00:35:52,120 --> 00:35:56,220
That's a standard tool
for statisticians looking

522
00:35:56,220 --> 00:35:58,140
at giant amounts of data.

523
00:35:58,140 --> 00:36:01,080
Is to find the
principal components

524
00:36:01,080 --> 00:36:04,960
and those will come
from these eigenvectors.

525
00:36:04,960 --> 00:36:12,080
Well your question about the
SVD set off that discussion.

526
00:36:12,080 --> 00:36:15,460
I'll only add a couple of words.

527
00:36:15,460 --> 00:36:22,870
These v's and these u's
are actually eigenvectors.

528
00:36:22,870 --> 00:36:28,140
But they're not eigenvectors
of A. v's happen

529
00:36:28,140 --> 00:36:31,730
to be the eigenvectors
of A transpose A.

530
00:36:31,730 --> 00:36:35,650
And the u's happen to be
eigenvectors of A, A transpose.

531
00:36:35,650 --> 00:36:40,860
And the linear
algebra comes together

532
00:36:40,860 --> 00:36:45,200
to give you this key equation.

533
00:36:45,200 --> 00:36:47,880
And of course, you
and I know that if I'm

534
00:36:47,880 --> 00:36:52,870
looking at the eigenvectors
v of A transpose A,

535
00:36:52,870 --> 00:36:56,900
A transpose A is a
symmetric matrix.

536
00:36:56,900 --> 00:36:58,480
In fact, positive definite.

537
00:36:58,480 --> 00:37:01,800
So that the eigenvectors
will be orthogonal,

538
00:37:01,800 --> 00:37:03,740
the eigenvalues
will be positive.

539
00:37:03,740 --> 00:37:07,580
And then this one is coming
from the eigenvectors of A A

540
00:37:07,580 --> 00:37:12,360
transpose, which is different.

541
00:37:12,360 --> 00:37:15,460
So if the matrix A
happened to be square,

542
00:37:15,460 --> 00:37:18,260
happened to be symmetric,
happened to be positive

543
00:37:18,260 --> 00:37:21,450
definite, this would
just be Ax=lambda*x.

544
00:37:21,450 --> 00:37:24,300
I didn't have to cross it out.

545
00:37:24,300 --> 00:37:26,710
I'll use K for that.

546
00:37:26,710 --> 00:37:30,840
So if the matrix A was one
of our favorite matrices,

547
00:37:30,840 --> 00:37:34,860
was a K, that would be the
case in which the SVD is

548
00:37:34,860 --> 00:37:37,820
no different from eigenvalues.

549
00:37:37,820 --> 00:37:40,840
The v's are the same
as the u's, the sigmas

550
00:37:40,840 --> 00:37:46,460
are the same as the
lambdas, all fine.

551
00:37:46,460 --> 00:37:49,180
This is sort of the
way to get the beauty

552
00:37:49,180 --> 00:37:54,200
of positive definite symmetric
matrices when the matrix itself

553
00:37:54,200 --> 00:37:56,560
that you start with
isn't even square.

554
00:37:56,560 --> 00:38:00,890
Like this one.

555
00:38:00,890 --> 00:38:02,620
More than I wanted
to say, more than you

556
00:38:02,620 --> 00:38:04,220
wanted to hear about the SVD.

557
00:38:04,220 --> 00:38:06,950


558
00:38:06,950 --> 00:38:08,130
What else?

559
00:38:08,130 --> 00:38:17,030
Yes, thank you.

560
00:38:17,030 --> 00:38:19,140
More about, sure.

561
00:38:19,140 --> 00:38:20,330
Let me repeat the question.

562
00:38:20,330 --> 00:38:23,330
So this refers to this
morning's lecture,

563
00:38:23,330 --> 00:38:32,580
lecture 9 about
time-dependent problems.

564
00:38:32,580 --> 00:38:35,080
And the point was that
when I have a differential

565
00:38:35,080 --> 00:38:38,860
equation in time there're
lots of ways to replace it

566
00:38:38,860 --> 00:38:45,560
by difference equations.

567
00:38:45,560 --> 00:38:49,820
So let's take the
equation du/dt,

568
00:38:49,820 --> 00:38:55,860
let's make it first order,
is some function of u, often,

569
00:38:55,860 --> 00:38:57,150
and t.

570
00:38:57,150 --> 00:39:02,420
That would be a first order.

571
00:39:02,420 --> 00:39:07,340
Yeah, it's going to look at
when I write that much down.

572
00:39:07,340 --> 00:39:09,700
What have I got?

573
00:39:09,700 --> 00:39:14,560
I've got a first order
differential equation.

574
00:39:14,560 --> 00:39:17,220
Is it linear?

575
00:39:17,220 --> 00:39:18,120
No.

576
00:39:18,120 --> 00:39:23,150
I'm allowing some, this
function of u could be sin(u).

577
00:39:23,150 --> 00:39:25,190
It could be u to
the tenth power.

578
00:39:25,190 --> 00:39:28,460
It could be e to the u.

579
00:39:28,460 --> 00:39:32,060
So this is how
equations really come.

580
00:39:32,060 --> 00:39:35,530
The linear ones are the model
problems that we can solve.

581
00:39:35,530 --> 00:39:40,610
This is how linear
equations really come.

582
00:39:40,610 --> 00:39:43,320
Euler thought of,
let's give Euler credit

583
00:39:43,320 --> 00:39:51,630
here, so forward Euler
and backward Euler.

584
00:39:51,630 --> 00:39:58,900
And the point will be
that this guy is explicit.

585
00:39:58,900 --> 00:40:01,420
So can I write that word so
I don't forget to write it?

586
00:40:01,420 --> 00:40:02,870
Explicit.

587
00:40:02,870 --> 00:40:04,980
And that this guy
will be implicit.

588
00:40:04,980 --> 00:40:10,190
And this is a big distinction.

589
00:40:10,190 --> 00:40:12,340
And we'll see it.

590
00:40:12,340 --> 00:40:20,290
So this says u_(n+1)-u_n
over delta t is the value

591
00:40:20,290 --> 00:40:33,090
of the slope at the
start of the step.

592
00:40:33,090 --> 00:40:36,450
So that's the most
important, most basic,

593
00:40:36,450 --> 00:40:38,540
first thing you would think of.

594
00:40:38,540 --> 00:40:40,550
It replaces this by this.

595
00:40:40,550 --> 00:40:45,690
You start with u_0, and
from this you get u_1.

596
00:40:45,690 --> 00:40:49,250
And then you know u_1 and
from this you get u_2.

597
00:40:49,250 --> 00:40:55,790
And the point is at every step
this equation is telling you

598
00:40:55,790 --> 00:40:58,130
what u_1 is from u_0.

599
00:40:58,130 --> 00:41:00,860
You just move u_0
over to that side.

600
00:41:00,860 --> 00:41:04,920
You've only got stuff you
know and then you know u_1.

601
00:41:04,920 --> 00:41:07,740
Contrast that with
backward Euler.

602
00:41:07,740 --> 00:41:15,055
So that's u_(n+1)-u_n over delta
t is f at-- Now what am I going

603
00:41:15,055 --> 00:41:18,190
to put there?

604
00:41:18,190 --> 00:41:24,920
I'm going to put the end,
the point we don't know yet.

605
00:41:24,920 --> 00:41:32,040
So it's the slope at u_(n+1)
and the time that goes with it,

606
00:41:32,040 --> 00:41:34,250
t_(n+1).

607
00:41:34,250 --> 00:41:40,140
t_n is just a shorthand
for n*delta t.

608
00:41:40,140 --> 00:41:43,460
n steps forward in
time got us to t_n.

609
00:41:43,460 --> 00:41:45,650
This is one more step.

610
00:41:45,650 --> 00:41:52,560
Now why is this called implicit?

611
00:41:52,560 --> 00:41:57,470
How do I find
u_(n+1) out of this?

612
00:41:57,470 --> 00:41:59,480
I've got to solve for it.

613
00:41:59,480 --> 00:42:01,740
It's much more expensive.

614
00:42:01,740 --> 00:42:04,390
Because it'll be probably
a non-linear system of

615
00:42:04,390 --> 00:42:07,280
equations to solve.

616
00:42:07,280 --> 00:42:09,520
We'll take a little
time on that.

617
00:42:09,520 --> 00:42:12,110
But of course, there's
one outstanding method

618
00:42:12,110 --> 00:42:16,020
to solve a system of
non-linear equations

619
00:42:16,020 --> 00:42:18,190
and that's called
Newton's method.

620
00:42:18,190 --> 00:42:20,120
So Newton is coming in.

621
00:42:20,120 --> 00:42:26,440
Newton's method is sort of
the first, the good way,

622
00:42:26,440 --> 00:42:31,600
if you can do it, to solve.

623
00:42:31,600 --> 00:42:37,020
Well when I say solve, I mean
set up an iteration which,

624
00:42:37,020 --> 00:42:43,660
after maybe three loops or
five loops will be accurate

625
00:42:43,660 --> 00:42:47,870
to enough digits that you can
say, okay, that's u_(n+1).

626
00:42:47,870 --> 00:42:49,970
On to the next step.

627
00:42:49,970 --> 00:42:53,100
Then the next step will
have the same equation

628
00:42:53,100 --> 00:42:57,220
but with n up one.

629
00:42:57,220 --> 00:43:00,190
So it'd be u_(n+2)-u_(n+1).

630
00:43:00,190 --> 00:43:05,220
So you see the extra work
here, but it's more stable.

631
00:43:05,220 --> 00:43:08,000
The way this one
spiraled out, this one

632
00:43:08,000 --> 00:43:12,620
spiraled in in
the model problem.

633
00:43:12,620 --> 00:43:16,650
I hope you look at
that Section 2.2 which

634
00:43:16,650 --> 00:43:21,380
was about the model problem
that we discussed today.

635
00:43:21,380 --> 00:43:22,950
There's more to say.

636
00:43:22,950 --> 00:43:29,710
I mean, this is the central
problem of time-dependent,

637
00:43:29,710 --> 00:43:34,430
evolving initial value problems.

638
00:43:34,430 --> 00:43:37,330
You're sort of marching forward.

639
00:43:37,330 --> 00:43:41,830
But here, each step of the
march takes an inner loop

640
00:43:41,830 --> 00:43:49,600
which has to work hard to solve,
to find, where you march to.

641
00:43:49,600 --> 00:43:52,210
Is that a help,
to indicate what?

642
00:43:52,210 --> 00:43:59,040
Because this is a very
fundamental difference.

643
00:43:59,040 --> 00:44:05,250
And Chapter 6 of the book
develops higher order methods.

644
00:44:05,250 --> 00:44:08,540
These are both first order,
first order accurate.

645
00:44:08,540 --> 00:44:14,430
The error that you're making
is of the order of delta t.

646
00:44:14,430 --> 00:44:16,220
That's not great, right?

647
00:44:16,220 --> 00:44:18,270
Because you would
have to take many,

648
00:44:18,270 --> 00:44:24,810
many small steps to
have a reasonable error.

649
00:44:24,810 --> 00:44:26,960
So these higher
order methods allow

650
00:44:26,960 --> 00:44:32,400
you to get to a great
answer with bigger steps.

651
00:44:32,400 --> 00:44:35,390
A lot of thinking
has gone into that

652
00:44:35,390 --> 00:44:43,230
and somehow it's a
pretty basic problem.

653
00:44:43,230 --> 00:44:49,210
Just to mention
for MATLAB, ode45

654
00:44:49,210 --> 00:44:56,070
is maybe the workhorse code to
solve this type of a problem.

655
00:44:56,070 --> 00:45:01,040
And the method is not Euler.

656
00:45:01,040 --> 00:45:04,390
That would not be good enough.

657
00:45:04,390 --> 00:45:07,450
It's called Runge-Kutta.

658
00:45:07,450 --> 00:45:10,350
Two guys, Runge and
Kutta, figured out

659
00:45:10,350 --> 00:45:14,020
a formula that got up to
fourth order accurate.

660
00:45:14,020 --> 00:45:17,980
So that's the thing.

661
00:45:17,980 --> 00:45:22,530
And then if we looked
further about this subject

662
00:45:22,530 --> 00:45:24,620
we would distinguish
some equations

663
00:45:24,620 --> 00:45:27,100
that are called stiff.

664
00:45:27,100 --> 00:45:29,590
So I'll just write
that word down.

665
00:45:29,590 --> 00:45:38,470
Some equations, the iteration
is particularly difficult.

666
00:45:38,470 --> 00:45:40,520
You have two time scales.

667
00:45:40,520 --> 00:45:43,150
Maybe you've got things
happening on a slow scale

668
00:45:43,150 --> 00:45:44,350
and a high scale.

669
00:45:44,350 --> 00:45:49,700
Multi-scale computations,
that's where the subject is now.

670
00:45:49,700 --> 00:45:55,660
And so there would be separate
codes with an S in their names

671
00:45:55,660 --> 00:46:02,920
suitable for these tougher
problems, stiff equation.

672
00:46:02,920 --> 00:46:07,880
I guess one thing you sometimes
get in the review session

673
00:46:07,880 --> 00:46:17,560
is a look outside the scope
of what I can cover and ask

674
00:46:17,560 --> 00:46:20,640
homework problems about.

675
00:46:20,640 --> 00:46:29,130
I'm sure hoping that you're
assembling the elements

676
00:46:29,130 --> 00:46:32,630
of computational science here.

677
00:46:32,630 --> 00:46:34,670
First, what are the questions?

678
00:46:34,670 --> 00:46:35,960
What are some answers?

679
00:46:35,960 --> 00:46:38,970
What are the issues?

680
00:46:38,970 --> 00:46:44,410
What do you have to balance
to make a good decision?

681
00:46:44,410 --> 00:46:47,090
Time for another
question if we like.

682
00:46:47,090 --> 00:46:50,800
Yeah, thank you.

683
00:46:50,800 --> 00:47:05,400
Stability, yeah, right.

684
00:47:05,400 --> 00:47:10,430
This here?

685
00:47:10,430 --> 00:47:12,460
First, if we want
a matrix then this

686
00:47:12,460 --> 00:47:15,500
has to be a vector of unknowns.

687
00:47:15,500 --> 00:47:19,260
I'm thinking now of a system
of, this is n equations.

688
00:47:19,260 --> 00:47:23,650
I've got u is a vector
with n components.

689
00:47:23,650 --> 00:47:25,860
I've got n equations here.

690
00:47:25,860 --> 00:47:29,840
The notation, I can put
a little arrow over it

691
00:47:29,840 --> 00:47:32,740
just to remind myself.

692
00:47:32,740 --> 00:47:38,510
And if I want to get to a matrix
I better do the linear case.

693
00:47:38,510 --> 00:47:48,050
So I'll do the linear case.

694
00:47:48,050 --> 00:47:51,450
The big picture is
that the new values

695
00:47:51,450 --> 00:47:54,030
come from the old values.

696
00:47:54,030 --> 00:47:56,160
There has to be a
matrix multiplier.

697
00:47:56,160 --> 00:48:01,450
Anytime we have
a linear process.

698
00:48:01,450 --> 00:48:06,270
So the new values come from
old values by some linear map.

699
00:48:06,270 --> 00:48:12,260
A matrix is doing it.

700
00:48:12,260 --> 00:48:14,050
So there's a sort
of growth matrix.

701
00:48:14,050 --> 00:48:17,750
Can I just put down
some letters here?

702
00:48:17,750 --> 00:48:22,110
I won't be able to give
you a complete answer.

703
00:48:22,110 --> 00:48:25,930
But this'll do it.

704
00:48:25,930 --> 00:48:34,550
So u_(n+1) is some
matrix times u_n.

705
00:48:34,550 --> 00:48:38,540
That's what I wrote down this
morning for a special case.

706
00:48:38,540 --> 00:48:44,540
It's gotta look like that
for a linear problem.

707
00:48:44,540 --> 00:48:48,480
And let's suppose that we have
this nice situation as today

708
00:48:48,480 --> 00:48:51,930
where it's the same
G at every step.

709
00:48:51,930 --> 00:48:58,170
So the problem is not
changing in time, it's linear,

710
00:48:58,170 --> 00:48:59,340
it's all good.

711
00:48:59,340 --> 00:49:03,680
What is the solution after
n times steps compared

712
00:49:03,680 --> 00:49:06,690
to the start?

713
00:49:06,690 --> 00:49:12,760
So give me a simple formula
for the solution to this step,

714
00:49:12,760 --> 00:49:15,350
step, step equation.

715
00:49:15,350 --> 00:49:19,280
If I started at
initial value u_0

716
00:49:19,280 --> 00:49:24,150
and I looked to see, well what
matrix gets me over n steps,

717
00:49:24,150 --> 00:49:25,840
what do I write there?

718
00:49:25,840 --> 00:49:30,110
G to the n, right, G to the n.

719
00:49:30,110 --> 00:49:36,090
So now comes the question.

720
00:49:36,090 --> 00:49:38,270
The stability question
is whether-- do

721
00:49:38,270 --> 00:49:41,700
the powers of G to the nth grow?

722
00:49:41,700 --> 00:49:43,820
And here's the point.

723
00:49:43,820 --> 00:49:48,150
That the continuous problem
that this came from,

724
00:49:48,150 --> 00:49:52,610
it's got its own growth
or decay or oscillation.

725
00:49:52,610 --> 00:49:56,890
This guy, the discrete
one, has got its.

726
00:49:56,890 --> 00:50:01,000
They're going to be
close for a step or two.

727
00:50:01,000 --> 00:50:05,310
For one or two
steps I'm expecting

728
00:50:05,310 --> 00:50:08,980
that this is a reasonable
consistent approximation

729
00:50:08,980 --> 00:50:10,500
to my problem.

730
00:50:10,500 --> 00:50:14,560
But after I take
a thousand steps,

731
00:50:14,560 --> 00:50:22,030
this one could still be close
or it could have exploded.

732
00:50:22,030 --> 00:50:26,350
And that's the stability thing.

733
00:50:26,350 --> 00:50:32,710
The choice of the difference
method can be be stable or not.

734
00:50:32,710 --> 00:50:36,840
And it's going to be the
eigenvalues of G that

735
00:50:36,840 --> 00:50:38,630
are the best guide.

736
00:50:38,630 --> 00:50:45,930
So in the end people compute
the eigenvalues of the growth

737
00:50:45,930 --> 00:51:10,330
matrix and look to see are
they bigger than one or not.

738
00:51:10,330 --> 00:51:13,660
Let's close with that example
because that's a good example.

739
00:51:13,660 --> 00:51:18,040
So it fits this, but not quite,
because it wasn't completely

740
00:51:18,040 --> 00:51:20,260
forward or completely backward.

741
00:51:20,260 --> 00:51:23,290
And let's just write down what
it was in that model problem

742
00:51:23,290 --> 00:51:26,300
and find the matrix.

743
00:51:26,300 --> 00:51:29,120
So can you remind me
what I wrote today?

744
00:51:29,120 --> 00:51:41,320
So u, was it u first? u_(n+1)
was u_n + delta t*v_n, right?

745
00:51:41,320 --> 00:51:46,950
And then the new velocity
was the old velocity and it

746
00:51:46,950 --> 00:51:50,590
should be minus
delta t times the u.

747
00:51:50,590 --> 00:51:55,390
Because our equation is, these
are representing the equation--

748
00:51:55,390 --> 00:51:59,200
u'=v is the first one.

749
00:51:59,200 --> 00:52:01,740
v'=-u is my second equation.

750
00:52:01,740 --> 00:52:03,550
So and then the
point was I could

751
00:52:03,550 --> 00:52:10,260
take that because I know it
already, I could take it there.

752
00:52:10,260 --> 00:52:14,800
Where is the matrix here?

753
00:52:14,800 --> 00:52:20,640
I want to find the growth
matrix that tells me now--

754
00:52:20,640 --> 00:52:29,330
My growth matrix, of course--
This is [u n, v n] and this is

755
00:52:29,330 --> 00:52:31,780
[u (n+1), v (n+1)].

756
00:52:31,780 --> 00:52:34,550
And this is a two by two matrix.

757
00:52:34,550 --> 00:52:38,170
And I hope we can read it off.

758
00:52:38,170 --> 00:52:40,810
Or find it anyway.

759
00:52:40,810 --> 00:52:45,780
Because that's the key.

760
00:52:45,780 --> 00:52:49,280
How could we get a matrix
out of these two equations?

761
00:52:49,280 --> 00:52:51,200
Let me just ask
you to look at them

762
00:52:51,200 --> 00:52:58,520
and think what are
we going to do.

763
00:52:58,520 --> 00:53:00,830
That's a good question.

764
00:53:00,830 --> 00:53:02,170
What shall I do?

765
00:53:02,170 --> 00:53:07,330
I would like to know the
new u's, u, v from the old.

766
00:53:07,330 --> 00:53:10,720
What'll I do?

767
00:53:10,720 --> 00:53:22,610
Bring stuff onto, a u_(n+1)
on its side and n on its side.

768
00:53:22,610 --> 00:53:25,050
This guy I want
to get over here.

769
00:53:25,050 --> 00:53:29,300
Can I do that with erasing?

770
00:53:29,300 --> 00:53:40,620
So it's going to come
over with a plus sign.

771
00:53:40,620 --> 00:53:44,730
So I guess I see here an
implicit matrix acting

772
00:53:44,730 --> 00:53:48,710
on the left and an explicit
matrix acting on the right.

773
00:53:48,710 --> 00:53:52,350
So I see a [u n, v n].

774
00:53:52,350 --> 00:53:54,580
And I see my explicit matrix.

775
00:53:54,580 --> 00:53:56,730
Shall I call it E?

776
00:53:56,730 --> 00:53:58,580
That's the right sides.

777
00:53:58,580 --> 00:54:02,560
I see a one and a
one and a delta t.

778
00:54:02,560 --> 00:54:09,610
And now my left side of my
equation, the n+1 stuff.

779
00:54:09,610 --> 00:54:14,170
What's my implicit matrix on
the left sides of the equation?

780
00:54:14,170 --> 00:54:23,600
Well, a one and a one and a
plus delta t is here, right?

781
00:54:23,600 --> 00:54:31,080
And let's call that
I for implicit.

782
00:54:31,080 --> 00:54:34,040
Are we close?

783
00:54:34,040 --> 00:54:36,110
I think that looks pretty good.

784
00:54:36,110 --> 00:54:38,550
What's G?

785
00:54:38,550 --> 00:54:40,980
What's the matrix G now?

786
00:54:40,980 --> 00:54:44,480
You just have to begin
to develop a little faith

787
00:54:44,480 --> 00:54:46,600
that yeah, I can
deal with matrices,

788
00:54:46,600 --> 00:54:51,660
I can move them around,
they're under my control.

789
00:54:51,660 --> 00:54:55,140
I wanted this picture.

790
00:54:55,140 --> 00:54:58,940
I want to move
everything to the right.

791
00:54:58,940 --> 00:55:03,480
What do I do to move everything
to the right-hand side?

792
00:55:03,480 --> 00:55:06,040
I want that to be over there.

793
00:55:06,040 --> 00:55:07,570
It's an inverse.

794
00:55:07,570 --> 00:55:08,840
It's the inverse.

795
00:55:08,840 --> 00:55:14,720
So G, the matrix G there
is, I bring I over here.

796
00:55:14,720 --> 00:55:22,040
It's the inverse
of I times the E.

797
00:55:22,040 --> 00:55:25,360
And I can figure out
what that matrix is.

798
00:55:25,360 --> 00:55:28,110
And I can find its eigenvalues.

799
00:55:28,110 --> 00:55:31,370
And it'll be interesting.

800
00:55:31,370 --> 00:55:33,590
So actually that's
the perfect problem.

801
00:55:33,590 --> 00:55:37,010
I mean, if you want to
see what's going on,

802
00:55:37,010 --> 00:55:40,580
well the book will be
a good help I think,

803
00:55:40,580 --> 00:55:45,670
but that's the growth
matrix for leapfrog.

804
00:55:45,670 --> 00:55:53,210
And I'll tell you the
result. The eigenvalues are

805
00:55:53,210 --> 00:55:56,890
right of magnitude
one, as you hope, up

806
00:55:56,890 --> 00:55:58,840
to a certain value of delta t.

807
00:55:58,840 --> 00:56:01,350
And then when delta t
passes that stability

808
00:56:01,350 --> 00:56:04,710
limit the eigenvalues take off.

809
00:56:04,710 --> 00:56:08,800
So that this method
is great provided

810
00:56:08,800 --> 00:56:15,410
you're not too ambitious and
take too large a delta t.

811
00:56:15,410 --> 00:56:18,340
Thanks for that last
question, thanks for coming.