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PROFESSOR STRANG: Starting
with a differential equation.
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00:00:22 --> 00:00:27
So key point here in this
lecture is how do you start
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with a differential equation
and end up with a discrete
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problem that you can solve?
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00:00:38 --> 00:00:40
But simple differential
equation.
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00:00:40 --> 00:00:44
It's got a second derivative
and I put a minus sign for a
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00:00:44 --> 00:00:46
reason that you will see.
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Second derivatives are
essentially negative definite
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00:00:51 --> 00:00:55
things so that minus sign
is to really make it
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00:00:55 --> 00:00:56
positive definite.
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00:00:56 --> 00:01:02
And notice we have boundary
conditions that at one end
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the solution is zero, at
the other end it's zero.
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So this is fixed-fixed.
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And it's a boundary
value problem.
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That's different from an
initial value problem.
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We have x space, not time.
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So we're not starting from some
thing and oscillating or
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00:01:20 --> 00:01:24
growing or decaying in time.
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We have a fixed thing.
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00:01:25 --> 00:01:27
Think of an elastic bar.
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00:01:27 --> 00:01:31
An elastic bar fixed at
both ends, maybe hanging
30
00:01:31 --> 00:01:33
by its own weight.
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00:01:33 --> 00:01:37
So that load f(x) could
represent the weight.
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00:01:37 --> 00:01:40
Maybe the good place to sit is
over there, there are tables
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00:01:40 --> 00:01:43
just, how about that?
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It's more comfortable.
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00:01:48 --> 00:01:51
So we can solve that equation.
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00:01:51 --> 00:01:54
Especially when I
change f(x) to be one.
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00:01:54 --> 00:01:56
As I plan to do.
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00:01:56 --> 00:01:59
So I'm going to change, I'm
going to make it-- it's a
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00:01:59 --> 00:02:03
uniform bar because there's
no variable coefficient in
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00:02:03 --> 00:02:07
there and let me make it
a uniform load, just one.
41
00:02:07 --> 00:02:12
So it actually shows you that,
I mentioned differential
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00:02:12 --> 00:02:15
equations and we'll certainly
get onto Laplace's equation,
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00:02:15 --> 00:02:18
but essentially our
differential equations will
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not-- this isn't a course in
how to solve ODEs or PDEs.
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00:02:23 --> 00:02:25
Especially not ODEs.
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00:02:25 --> 00:02:28
It's a course in how
to compute solutions.
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00:02:28 --> 00:02:32
So the key idea will be to
replace the differential
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equation by a
difference equation.
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00:02:34 --> 00:02:37
So there's the
difference equation.
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00:02:37 --> 00:02:40
And I have to talk about that.
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That's the sort of key point.
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That up here you see what I
would call a second difference.
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00:02:50 --> 00:02:54
Actually with a minus sign.
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00:02:54 --> 00:02:58
And on the right-hand side you
see the load, still f(x).
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Can I move to this board
to explain differences?
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00:03:03 --> 00:03:07
Because this is like, key step
is given the differential
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equation replace it by
difference equation.
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00:03:11 --> 00:03:15
And the interesting point
is you have many choices.
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00:03:15 --> 00:03:19
There's one differential
equation but even for a first
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derivative, so this if you
remember from calculus, how did
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you start with the derivative?
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You started by something before
going to the limit. h or delta
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x goes to zero in the end
to get the derivative.
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But this was a
finite difference.
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You moved a finite amount.
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And this is the one you always
see in calculus courses.
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U(x + h)-U(x) just how
much did that step go.
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You divide by the delta x, the
h and that's approximately
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the derivative, U'(x).
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Let me just continue
with these others.
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I don't remember if calculus
mentions a backward difference.
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But you won't be surprised that
another possibility, equally
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good more or less, would be to
take the point and the point
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before, take the difference,
divide by delta x.
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So again all these
approximate U'.
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And now here's one that
actually is really important.
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A center difference.
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It's the average of
the forward and back.
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If I take that plus that, the
U(x)'s cancel and I'm left
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with, I'm centering it.
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00:04:50 --> 00:04:53
This idea of centering is
a good thing actually.
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00:04:53 --> 00:04:57
And of course I have to divide
by 2h because this step
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00:04:57 --> 00:05:00
is now 2h, two delta x's.
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00:05:00 --> 00:05:02
So that again is going
to represent U'.
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00:05:02 --> 00:05:05
87
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But so we have a choice if
we have a first derivative.
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00:05:10 --> 00:05:15
And actually that's
a big issue.
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You know, one might be called
upwind, one might be called
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downwind, one may be
called centered.
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00:05:21 --> 00:05:24
It comes up constantly
in aero and mechanical
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engineering, everywhere.
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You have these choices to make.
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00:05:28 --> 00:05:30
Especially for the
first difference.
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00:05:30 --> 00:05:36
We don't have, I didn't allow a
first derivative in that
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00:05:36 --> 00:05:41
equation because I wanted to
keep it symmetric and first
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00:05:41 --> 00:05:45
derivatives, first differences
tend to be anti-symmetric.
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00:05:45 --> 00:05:50
So if we want to get our good
matrix K, and I better remember
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00:05:50 --> 00:05:55
to divide by h squared because
the K just has those that I
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00:05:55 --> 00:05:59
introduced last time and will
repeat, the K just has
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00:05:59 --> 00:06:00
the numbers -1, 2, -1.
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00:06:02 --> 00:06:09
Now, first point before
we leave these guys
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00:06:09 --> 00:06:11
what's up with them?
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00:06:11 --> 00:06:13
How do we decide
which one is better?
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00:06:13 --> 00:06:17
There's something called
the order of accuracy.
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00:06:17 --> 00:06:21
How close is the difference
to the derivative?
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00:06:21 --> 00:06:26
And the answer is the
error is of size h.
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00:06:26 --> 00:06:33
So I would call that
first order accurate.
109
00:06:33 --> 00:06:39
And I can repeat here but the
text does it, how you recognize
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00:06:39 --> 00:06:44
what this is the, sort of local
error, truncation error,
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00:06:44 --> 00:06:50
whatever, you've chopped
off the exact answer and
112
00:06:50 --> 00:06:51
just did differences.
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00:06:51 --> 00:06:55
This one is also order of h.
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00:06:55 --> 00:07:00
And in fact, the h terms, the
leading error, which is going
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00:07:00 --> 00:07:05
to multiply the h, has opposite
sign in these two and
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00:07:05 --> 00:07:08
that's the reason center
differences are great.
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00:07:08 --> 00:07:13
Because when you average
them you center things.
118
00:07:13 --> 00:07:19
This is correct to
order h squared.
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00:07:19 --> 00:07:24
And I may come back and find
out why that h squared term is.
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00:07:24 --> 00:07:25
Maybe I'll do that.
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00:07:25 --> 00:07:26
Yeah.
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00:07:26 --> 00:07:27
Why don't I just?
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00:07:27 --> 00:07:29
Second differences
are so important.
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00:07:29 --> 00:07:31
Why don't we just see.
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00:07:31 --> 00:07:32
And center differences.
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00:07:32 --> 00:07:34
So let me see.
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00:07:34 --> 00:07:36
How do you figure U(x + h)?
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00:07:36 --> 00:07:40
129
00:07:40 --> 00:07:48
This is a chance to remember
something called Taylor series.
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00:07:48 --> 00:07:51
But that was in calculus.
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00:07:51 --> 00:07:55
If you forgot it, you're
a normal person.
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00:07:55 --> 00:07:58
So but what does it say?
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00:07:58 --> 00:08:01
That's the whole point
of calculus, in a way.
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00:08:01 --> 00:08:07
That if I move a little bit, I
start from the point x and then
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00:08:07 --> 00:08:11
there's a little correction and
that's given by the derivative
136
00:08:11 --> 00:08:14
and then there's a further
correction if I want to go
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00:08:14 --> 00:08:18
further and that's given by
half of h squared, you see
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00:08:18 --> 00:08:23
the second order correction,
times the second derivative.
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00:08:23 --> 00:08:25
And then, of course, more.
140
00:08:25 --> 00:08:28
But that's all you ever
have to remember.
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00:08:28 --> 00:08:29
It's pretty rare.
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00:08:29 --> 00:08:35
Second order accuracy
is often the goal in
143
00:08:35 --> 00:08:38
scientific computing.
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00:08:38 --> 00:08:41
First order accuracy is,
like, the lowest level.
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00:08:41 --> 00:08:44
You start there, you write a
code, you test it and so on.
146
00:08:44 --> 00:08:48
But if you want production,
if you want accuracy, get to
147
00:08:48 --> 00:08:50
second order if possible.
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00:08:50 --> 00:08:52
Now, what about this U(x - h)?
149
00:08:53 --> 00:08:55
Well, that's a step
backwards, so that's U(x).
150
00:08:56 --> 00:09:02
Now the step is -h, but then
when I square that step I'm
151
00:09:02 --> 00:09:08
back to +h squared,
U''(x) and so on.
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00:09:08 --> 00:09:09
Ooh!
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00:09:09 --> 00:09:14
Am I going to find
even more accuracy?
154
00:09:14 --> 00:09:17
I could tell you what
the next turn is.
155
00:09:17 --> 00:09:22
Plus h cubed upon six,
that's three times
156
00:09:22 --> 00:09:25
two times one, U''' .
157
00:09:25 --> 00:09:30
And this would be, since this
step is -h now, it would be
158
00:09:30 --> 00:09:34
-h cubed upon six U''' .
159
00:09:34 --> 00:09:37
So what happens when I take
the difference of these two?
160
00:09:37 --> 00:09:40
Remember now that center
differences subtract
161
00:09:40 --> 00:09:42
this from this.
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00:09:42 --> 00:09:51
So now that U(x +
h)-U(x-h) is zero.
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00:09:51 --> 00:09:58
2hU' subtracting that from that
is zero, two of these, so I
164
00:09:58 --> 00:10:02
guess that we really have an
h cubed over three, U'''.
165
00:10:02 --> 00:10:05
166
00:10:05 --> 00:10:09
And now when I divide
by the 2h, can I just
167
00:10:09 --> 00:10:11
divide by 2h here?
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00:10:11 --> 00:10:15
Oh yeah, it's coming out right,
divide by 2h, divide this by
169
00:10:15 --> 00:10:19
2h, that'll make it an
h squared over six.
170
00:10:19 --> 00:10:22
I've done what looks like
a messy computation.
171
00:10:22 --> 00:10:26
I'm a little sad to start
a good lecture, important
172
00:10:26 --> 00:10:29
lecture by such grungy stuff.
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00:10:29 --> 00:10:33
But it makes the key point.
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00:10:33 --> 00:10:38
That the center difference
gives the correct derivative
175
00:10:38 --> 00:10:41
with an error of
order h squared.
176
00:10:41 --> 00:10:47
Where the error if for the
first differences the h
177
00:10:47 --> 00:10:49
would have been there.
178
00:10:49 --> 00:10:51
And we can test it.
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00:10:51 --> 00:10:54
Actually we'll test it.
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00:10:54 --> 00:10:55
Okay for that?
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00:10:55 --> 00:10:57
This is first differences.
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00:10:57 --> 00:11:00
And that's a big question; what
do you replace the first
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00:11:00 --> 00:11:03
derivative by if there is one?
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00:11:03 --> 00:11:05
And you've got these
three choices.
185
00:11:05 --> 00:11:08
And usually this is
the best choice.
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00:11:08 --> 00:11:10
Now to second derivatives.
187
00:11:10 --> 00:11:17
Because our equation
has got U'' in it.
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00:11:17 --> 00:11:20
So what's a second derivative?
189
00:11:20 --> 00:11:23
It's the derivative
of the derivative.
190
00:11:23 --> 00:11:24
So what's the
second difference?
191
00:11:24 --> 00:11:27
It's the difference,
first difference of
192
00:11:27 --> 00:11:29
the first difference.
193
00:11:29 --> 00:11:32
So the second difference, the
natural second difference would
194
00:11:32 --> 00:11:39
be-- so now let me use this
space for second differences.
195
00:11:39 --> 00:11:42
Second differences.
196
00:11:42 --> 00:11:46
I could take the forward
difference of the
197
00:11:46 --> 00:11:48
backward difference.
198
00:11:48 --> 00:11:51
Or I could take the
backward difference of
199
00:11:51 --> 00:11:52
the forward difference.
200
00:11:52 --> 00:11:55
Or you may say why don't I
take the center difference
201
00:11:55 --> 00:11:57
of the center difference.
202
00:11:57 --> 00:12:02
All those, in some sense
it's delta squared,
203
00:12:02 --> 00:12:05
but which to take?
204
00:12:05 --> 00:12:12
Well actually those are the
same and that's the good
205
00:12:12 --> 00:12:16
choice, that's the
1, -2, 1 choice.
206
00:12:16 --> 00:12:18
So let me show you that.
207
00:12:18 --> 00:12:20
Let me say what's the
matter with that.
208
00:12:20 --> 00:12:25
Because now having said how
great center differences are,
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00:12:25 --> 00:12:29
first differences, why don't I
just repeat them for
210
00:12:29 --> 00:12:31
second differences?
211
00:12:31 --> 00:12:35
Well the trouble is, let me say
in a word without even writing,
212
00:12:35 --> 00:12:40
well I could even write a
little, the center difference,
213
00:12:40 --> 00:12:43
suppose I'm at a typical
mesh point here.
214
00:12:43 --> 00:12:46
The center difference is going
to take that value minus that
215
00:12:46 --> 00:12:51
value But then if I take the
center difference of that
216
00:12:51 --> 00:12:52
I'm going to be out here.
217
00:12:52 --> 00:12:56
I'm going to take this value,
this value, and this value.
218
00:12:56 --> 00:12:58
I'll get something correct.
219
00:12:58 --> 00:13:00
Its accuracy will be
second order, good.
220
00:13:00 --> 00:13:05
But it stretches too far.
221
00:13:05 --> 00:13:09
We want compact
difference molecules.
222
00:13:09 --> 00:13:15
We don't want this one, minus
two of this, plus one of that.
223
00:13:15 --> 00:13:22
So this would give us
a 1, 0, -2 , 0, 1.
224
00:13:22 --> 00:13:26
I'm just saying this and then
I'll never come back to it
225
00:13:26 --> 00:13:32
because I don't like this one,
these guys give 1, -2,
226
00:13:32 --> 00:13:36
1 without any gaps.
227
00:13:36 --> 00:13:37
And that's the right choice.
228
00:13:37 --> 00:13:40
And that's the
choice made here.
229
00:13:40 --> 00:13:46
So I'm not thinking you can
see it in your head, the
230
00:13:46 --> 00:13:48
difference of the difference.
231
00:13:48 --> 00:13:51
But well, you almost can.
232
00:13:51 --> 00:13:53
If I take this, yeah.
233
00:13:53 --> 00:13:56
Can you sort of see this
without my writing it?
234
00:13:56 --> 00:14:02
If I take the forward
difference and then I subtract
235
00:14:02 --> 00:14:07
the forward difference to the
left, do you see that
236
00:14:07 --> 00:14:08
I'll have minus two.
237
00:14:08 --> 00:14:10
So there is what
I started with.
238
00:14:10 --> 00:14:18
I subtract U(x)-U(x-h)
and I get two -U(x)'s.
239
00:14:19 --> 00:14:22
This is what I get.
240
00:14:22 --> 00:14:23
Now I'm calling that ui.
241
00:14:23 --> 00:14:29
242
00:14:29 --> 00:14:33
I better make completely
clear about the minus sign.
243
00:14:33 --> 00:14:36
The forward difference or the
backward difference, what
244
00:14:36 --> 00:14:41
this leads is 1, -2, 1.
245
00:14:41 --> 00:14:44
That's the second difference.
246
00:14:44 --> 00:14:48
Very important to remember, the
second difference of a function
247
00:14:48 --> 00:14:53
is the function, the value
ahead, minus two of the
248
00:14:53 --> 00:14:56
center, plus one of the left.
249
00:14:56 --> 00:14:59
It's centered obviously,
symmetric, right?
250
00:14:59 --> 00:15:02
Second differences
are symmetric.
251
00:15:02 --> 00:15:07
And because I want a minus
sign I want minus the second
252
00:15:07 --> 00:15:11
difference and that's why
you see here -1, 2, -1 .
253
00:15:14 --> 00:15:18
Because I wanted
positive twos there.
254
00:15:18 --> 00:15:18
Are you ok?
255
00:15:18 --> 00:15:24
This is the natural
replacement for -U''.
256
00:15:26 --> 00:15:32
And I claim that this second
difference is like the second
257
00:15:32 --> 00:15:34
derivative, of course.
258
00:15:34 --> 00:15:38
And why don't we just check
some examples to see how like
259
00:15:38 --> 00:15:40
the second derivative it is.
260
00:15:40 --> 00:15:41
So I'm going to take the
second difference or
261
00:15:41 --> 00:15:47
some easy functions.
262
00:15:47 --> 00:15:51
It's very important that
these come out so well.
263
00:15:51 --> 00:15:54
So I'm going to take
the second difference.
264
00:15:54 --> 00:15:56
I'm going to write it
as sort of a matrix.
265
00:15:56 --> 00:15:58
So this is like the
second different.
266
00:15:58 --> 00:16:02
Yeah, because this is good.
267
00:16:02 --> 00:16:04
I'm inside the region, here.
268
00:16:04 --> 00:16:07
I'm not worried about
the boundaries now.
269
00:16:07 --> 00:16:09
Let me just think of
myself as inside.
270
00:16:09 --> 00:16:14
So I have second differences
and suppose I'm applying it
271
00:16:14 --> 00:16:18
to a vector of all ones.
272
00:16:18 --> 00:16:22
What answer should I get?
273
00:16:22 --> 00:16:28
So if I think of calculus it's
the second derivative of one,
274
00:16:28 --> 00:16:30
of the constant function.
275
00:16:30 --> 00:16:32
So what answer am
I going to get?
276
00:16:32 --> 00:16:33
Zero.
277
00:16:33 --> 00:16:34
And do I get zero?
278
00:16:34 --> 00:16:35
Of course.
279
00:16:35 --> 00:16:35
I get zero.
280
00:16:35 --> 00:16:36
Right?
281
00:16:36 --> 00:16:39
All these second
differences are zero.
282
00:16:39 --> 00:16:43
Because I'm not worrying
about the boundary yet.
283
00:16:43 --> 00:16:46
So that's like, check one.
284
00:16:46 --> 00:16:48
It passed that simple test.
285
00:16:48 --> 00:16:57
Now let me move up from
constant to linear.
286
00:16:57 --> 00:16:58
And so on.
287
00:16:58 --> 00:17:01
So let me apply second
differences to a vector
288
00:17:01 --> 00:17:04
that's growing linearly.
289
00:17:04 --> 00:17:08
What answer do I expect
to get for that?
290
00:17:08 --> 00:17:11
So remember I'm doing second
differences, like second
291
00:17:11 --> 00:17:16
derivatives, or minus second
derivatives, actually.
292
00:17:16 --> 00:17:20
So what do second derivatives
do to a linear function?
293
00:17:20 --> 00:17:23
If I take a straight line
I take the-- sorry,
294
00:17:23 --> 00:17:25
second derivatives.
295
00:17:25 --> 00:17:29
If I take second derivatives
of a linear function I get?
296
00:17:29 --> 00:17:30
Zero, right.
297
00:17:30 --> 00:17:34
So I would hope to get
zero again here and I do.
298
00:17:34 --> 00:17:35
Right?
299
00:17:35 --> 00:17:36
-1+4-3=0.
300
00:17:38 --> 00:17:41
Minus one, sorry, let
me do it here, -2+6-4.
301
00:17:44 --> 00:17:47
And actually, that's
consistent with our little
302
00:17:47 --> 00:17:49
Taylor series stuff.
303
00:17:49 --> 00:17:52
The function x should
come out right.
304
00:17:52 --> 00:17:55
Now what about-- now
comes the moment.
305
00:17:55 --> 00:17:57
What about x squared?
306
00:17:57 --> 00:18:00
So I'm going to put
squares in now.
307
00:18:00 --> 00:18:04
Do I expect to get zeroes?
308
00:18:04 --> 00:18:06
I don't think so.
309
00:18:06 --> 00:18:10
Because let me again test
it by thinking about
310
00:18:10 --> 00:18:12
what second derivative.
311
00:18:12 --> 00:18:17
So now I'm sort of copying
second derivative of
312
00:18:17 --> 00:18:20
x squared, which is?
313
00:18:20 --> 00:18:24
Second derivative
of x squared is?
314
00:18:24 --> 00:18:25
Two, right?
315
00:18:25 --> 00:18:29
First derivative's 2x, second
derivative is just two.
316
00:18:29 --> 00:18:31
So it's a constant.
317
00:18:31 --> 00:18:35
And remember I put in a minus
sign so I'm wondering, do I
318
00:18:35 --> 00:18:39
get the answer minus two?
319
00:18:39 --> 00:18:41
All the way down.
320
00:18:42 --> 00:18:43
-4+8-9.
321
00:18:45 --> 00:18:47
Whoops.
322
00:18:47 --> 00:18:50
What's that?
323
00:18:50 --> 00:18:52
What do I get there?
324
00:18:52 --> 00:18:57
What do I get from that second
difference of these squares?
325
00:18:57 --> 00:19:00
-4+8-9 is?
326
00:19:00 --> 00:19:03
Minus two, good.
327
00:19:03 --> 00:19:04
So can we keep going?
328
00:19:04 --> 00:19:05
-4+18-16.
329
00:19:08 --> 00:19:10
What's that?
330
00:19:10 --> 00:19:15
-4+18-16, so I've got -20+18 .
331
00:19:15 --> 00:19:19
332
00:19:19 --> 00:19:20
I got minus two again.
333
00:19:21 --> 00:19:24
-9, 32, -25, it's right.
334
00:19:24 --> 00:19:32
The second differences of the
vector of squares, you could
335
00:19:32 --> 00:19:37
say, is a constant vector
with the right number.
336
00:19:37 --> 00:19:39
And that's because that
second difference is
337
00:19:39 --> 00:19:41
second order accurate.
338
00:19:41 --> 00:19:45
It not only got constants
right and linears right,
339
00:19:45 --> 00:19:48
it got quadratics right.
340
00:19:48 --> 00:19:52
So that's, you're seeing
second differences.
341
00:19:52 --> 00:19:57
We'll soon see that second
differences are also on
342
00:19:57 --> 00:20:01
the ball when you apply
them to other vectors.
343
00:20:01 --> 00:20:04
Like vectors of sines or
vectors of cosines or
344
00:20:04 --> 00:20:08
exponentials, they do well.
345
00:20:08 --> 00:20:12
So that's just a useful check
which will help us over here.
346
00:20:12 --> 00:20:17
Okay, can I come back to the
part of the lecture now?
347
00:20:17 --> 00:20:23
Having prepared
the way for this.
348
00:20:23 --> 00:20:25
Well, let's start right
off by solving the
349
00:20:25 --> 00:20:29
differential equation.
350
00:20:29 --> 00:20:33
So I'm bringing you back years
and years and years, right?
351
00:20:33 --> 00:20:37
Solve that differential
equation with these two
352
00:20:37 --> 00:20:40
boundary conditions.
353
00:20:40 --> 00:20:42
How would you do that
in a systematic way?
354
00:20:42 --> 00:20:47
You could almost guess after a
while, but systematically if I
355
00:20:47 --> 00:20:51
have a linear, I notice-- What
do I notice about this thing?
356
00:20:51 --> 00:20:53
It's linear.
357
00:20:53 --> 00:20:55
So what am I expecting?
358
00:20:55 --> 00:20:58
I'm expecting, like, a
particular solution that gives
359
00:20:58 --> 00:21:06
the correct answer one and some
null space solution or whatever
360
00:21:06 --> 00:21:11
I want to call it, homogenous
solution that gives zero and
361
00:21:11 --> 00:21:13
has some arbitrary
constants in it.
362
00:21:13 --> 00:21:15
Give me a particular solution.
363
00:21:15 --> 00:21:18
So this is going
to be our answer.
364
00:21:18 --> 00:21:23
This'll be the general solution
to this differential equation.
365
00:21:23 --> 00:21:27
What functions have minus
the second derivative equal
366
00:21:27 --> 00:21:28
one, that's all I'm asking.
367
00:21:28 --> 00:21:30
What are they?
368
00:21:30 --> 00:21:33
So what is one of them?
369
00:21:33 --> 00:21:38
One function that has its
second derivative as a constant
370
00:21:38 --> 00:21:41
and that constant is minus one.
371
00:21:41 --> 00:21:44
So if I want the second
derivative to be a constant,
372
00:21:44 --> 00:21:47
what am I looking
at? x squared.
373
00:21:47 --> 00:21:49
I'm looking at x squared.
374
00:21:49 --> 00:21:51
And I just want to
figure out how many x
375
00:21:51 --> 00:21:53
squareds to get a one.
376
00:21:53 --> 00:21:58
So some number of x squareds
and how many do I want?
377
00:21:58 --> 00:21:59
-1/2, good.
378
00:21:59 --> 00:22:01
Good.
379
00:22:01 --> 00:22:01
-1/2.
380
00:22:01 --> 00:22:05
Because x squared would give
me two but I want minus
381
00:22:05 --> 00:22:07
one so I need -1/2.
382
00:22:07 --> 00:22:10
Okay that's the
particular solution.
383
00:22:10 --> 00:22:18
Now throw in all the solutions,
I can add in any solution that
384
00:22:18 --> 00:22:22
has a zero on the right side,
so what functions have second
385
00:22:22 --> 00:22:30
derivatives equals
zero? x is good.
386
00:22:30 --> 00:22:34
I'm looking for two because
it's a second derivative,
387
00:22:34 --> 00:22:35
second order equation.
388
00:22:35 --> 00:22:37
What's the other guy?
389
00:22:37 --> 00:22:38
Constant, good.
390
00:22:38 --> 00:22:43
So let me put the constant
first, C, say, and Dx.
391
00:22:43 --> 00:22:46
Two constants that I can
play with and what use am
392
00:22:46 --> 00:22:49
I going to make of them?
393
00:22:49 --> 00:22:52
I'm going to use those
to satisfy the two
394
00:22:52 --> 00:22:56
boundary conditions.
395
00:22:56 --> 00:22:59
And it won't be difficult.
396
00:22:59 --> 00:23:04
You could say plug in the first
boundary condition, get an
397
00:23:04 --> 00:23:07
equation for the constants,
plug in the second, got another
398
00:23:07 --> 00:23:09
equation, we'll have two
boundary conditions, two
399
00:23:09 --> 00:23:11
equations, two constants.
400
00:23:11 --> 00:23:15
Everything's going to come out.
401
00:23:15 --> 00:23:22
So if I plug in U(0)=0,
what do I learn?
402
00:23:22 --> 00:23:23
C is zero, right?
403
00:23:23 --> 00:23:25
If I plug in, is that right?
404
00:23:25 --> 00:23:28
If I plug in zero, then
that's zero already, this
405
00:23:28 --> 00:23:32
is zero already, so I just
learned that C is zero.
406
00:23:32 --> 00:23:36
So C is zero.
407
00:23:36 --> 00:23:42
So I'm down to one constant,
one unused boundary condition.
408
00:23:42 --> 00:23:43
Plug that in.
409
00:23:43 --> 00:23:43
U(1)=-1/2.
410
00:23:48 --> 00:23:50
What's D?
411
00:23:50 --> 00:23:51
It's 1/2, right.
412
00:23:51 --> 00:23:54
D is 1/2.
413
00:23:54 --> 00:23:56
So can I close this up?
414
00:23:56 --> 00:23:58
There's 1/2.
415
00:23:58 --> 00:24:00
Dx is 1/2.
416
00:24:00 --> 00:24:03
Now it just always
pays to look back.
417
00:24:03 --> 00:24:06
At x=0, that's obviously zero.
418
00:24:06 --> 00:24:11
At x=1 it's zero because those
are the same and I get zero.
419
00:24:11 --> 00:24:14
So -1/2x squared plus 1/2x.
420
00:24:14 --> 00:24:19
421
00:24:19 --> 00:24:23
That's the kind of differential
equation and solution
422
00:24:23 --> 00:24:25
that we're looking for.
423
00:24:25 --> 00:24:28
Not complicated
nonlinear stuff.
424
00:24:28 --> 00:24:35
So now I'm ready to move to
the difference equation.
425
00:24:35 --> 00:24:41
So again, this is a major step.
426
00:24:41 --> 00:24:46
I'll draw a picture of
this from zero to one.
427
00:24:46 --> 00:24:51
And if I graph that I think
I get a parabola, right?
428
00:24:51 --> 00:24:53
A parabola that has
to go through here.
429
00:24:53 --> 00:24:56
So it's some
parabola like that.
430
00:24:56 --> 00:25:01
That would be always good, to
draw a graph of the solution.
431
00:25:01 --> 00:25:03
Now, what do I get here?
432
00:25:03 --> 00:25:06
Moving to the
difference equation.
433
00:25:06 --> 00:25:09
So that's the equation,
and notice it's
434
00:25:09 --> 00:25:12
boundary conditions.
435
00:25:12 --> 00:25:16
Those boundary conditions
just copied this one because
436
00:25:16 --> 00:25:20
I've chopped this up.
437
00:25:20 --> 00:25:25
I've got i equal one, two,
three, four, five and this
438
00:25:25 --> 00:25:32
is one, the last point
then is 6h . h is 1/6.
439
00:25:32 --> 00:25:35
440
00:25:35 --> 00:25:38
What's going to be the size
of my matrix and my vector
441
00:25:38 --> 00:25:41
and my unknown u here?
442
00:25:41 --> 00:25:43
How many unknowns am
I going to have?
443
00:25:43 --> 00:25:46
Let's just get the
overall picture right.
444
00:25:46 --> 00:25:47
What are the unknowns
going to be?
445
00:25:47 --> 00:25:49
They're going to be u_1,
u_2, u_3, u_4, u_5.
446
00:25:49 --> 00:25:51
447
00:25:51 --> 00:25:52
Those are unknown.
448
00:25:52 --> 00:25:56
Those will be some values, I
don't know where, maybe
449
00:25:56 --> 00:25:59
something like this because
they'll be sort of
450
00:25:59 --> 00:26:01
like that one.
451
00:26:01 --> 00:26:05
And this is not an unknown, u_6
, this is not an unknown, u_0
452
00:26:05 --> 00:26:08
, those are the ones we know.
453
00:26:08 --> 00:26:13
So this is what the
solution to a difference
454
00:26:13 --> 00:26:15
equation looks like.
455
00:26:15 --> 00:26:18
It gives you a discreet
set of unknowns.
456
00:26:18 --> 00:26:22
And then, of course MATLAB or
any code could connect them up
457
00:26:22 --> 00:26:28
by straight lines and
give you a function.
458
00:26:28 --> 00:26:33
But the heart of it is
these five values.
459
00:26:33 --> 00:26:38
Okay, good.
460
00:26:38 --> 00:26:44
And those five values come
from these equations.
461
00:26:44 --> 00:26:47
I'm introducing this subscript
stuff but I won't need it all
462
00:26:47 --> 00:26:49
the time because you'll
see the picture.
463
00:26:49 --> 00:26:56
This equation applies for
i equal one up to five.
464
00:26:56 --> 00:27:01
Five inside points and then you
notice how when i is one, this
465
00:27:01 --> 00:27:04
needs u_0 , but we know u_0.
466
00:27:04 --> 00:27:08
And when I is five, this needs
u_6 , but we know u_6 .
467
00:27:08 --> 00:27:12
So it's a closed five
by five system and it
468
00:27:12 --> 00:27:17
will be our matrix.
469
00:27:17 --> 00:27:21
That -1, 2, -1 is what
sits on the matrix.
470
00:27:21 --> 00:27:25
When we close it with the two
boundary conditions it chops
471
00:27:25 --> 00:27:30
off the zero column, you could
say and chops off the six
472
00:27:30 --> 00:27:37
column and leaves us with a
five by five problem and yeah.
473
00:27:37 --> 00:27:44
I guess this is a step not
to jump past because it
474
00:27:44 --> 00:27:48
takes a little practice.
475
00:27:48 --> 00:27:51
You see I've written the
same thing two ways.
476
00:27:51 --> 00:27:53
Let me write it a third way.
477
00:27:53 --> 00:27:55
Let me write it out clearly.
478
00:27:55 --> 00:27:59
So now here I'm going to
complete this matrix with a two
479
00:27:59 --> 00:28:03
and a minus one and a two and
a minus one and now
480
00:28:03 --> 00:28:05
it's five by five.
481
00:28:05 --> 00:28:10
And those might be u but I
don't know if they are so
482
00:28:10 --> 00:28:18
let me put in u_1, u_2,
u_3, u_4, and u_5.
483
00:28:18 --> 00:28:23
484
00:28:23 --> 00:28:26
Oh and divide by h squared.
485
00:28:26 --> 00:28:31
I'll often forget that.
486
00:28:31 --> 00:28:34
So I'm asking you to see
something that if you haven't,
487
00:28:34 --> 00:28:38
after you get the hang of
it it's like, automatic.
488
00:28:38 --> 00:28:40
But I have to remember
it's not automatic.
489
00:28:40 --> 00:28:42
Things aren't automatic
until you've done them
490
00:28:42 --> 00:28:43
a couple of times.
491
00:28:43 --> 00:28:53
So do you see that that is a
concrete statement of this?
492
00:28:53 --> 00:28:56
This delta x squared
is the h squared.
493
00:28:56 --> 00:29:00
And do you see those
differences when I do that
494
00:29:00 --> 00:29:04
multiplication that they
produce those differences?
495
00:29:04 --> 00:29:07
And now, what's my
right-hand side?
496
00:29:07 --> 00:29:10
Well I've changed the
right-hand side to
497
00:29:10 --> 00:29:12
one to make it easy.
498
00:29:12 --> 00:29:17
So this right-hand
side is all ones.
499
00:29:17 --> 00:29:25
And this is the problem
that MATLAB would solve
500
00:29:25 --> 00:29:27
or whatever code.
501
00:29:27 --> 00:29:30
Find a difference code.
502
00:29:30 --> 00:29:36
I've got to a linear system,
five by five, it's fortunately,
503
00:29:36 --> 00:29:44
the matrix is not singular,
there is a solution.
504
00:29:44 --> 00:29:45
How does MATLAB find it?
505
00:29:45 --> 00:29:50
It does not find it by finding
the inverse of that matrix.
506
00:29:50 --> 00:29:55
Monday's lecture will quickly
review how to solve five
507
00:29:55 --> 00:29:58
equations and five unknowns.
508
00:29:58 --> 00:30:02
It's by elimination, I'll
tell you the key word.
509
00:30:02 --> 00:30:05
And that's what
every code does.
510
00:30:05 --> 00:30:09
And sometimes you would have to
exchange rows, but not for a
511
00:30:09 --> 00:30:10
positive definite
matrix like that.
512
00:30:10 --> 00:30:13
It'll just go bzzz,
right through.
513
00:30:13 --> 00:30:17
When it's tridiagonal it'll go
like with the speed of light
514
00:30:17 --> 00:30:20
and you'll get the answer.
515
00:30:20 --> 00:30:23
And those five answers will
be these five heights. u_1,
516
00:30:23 --> 00:30:23
u_2, u_3, u_4, and u_5.
517
00:30:23 --> 00:30:29
518
00:30:29 --> 00:30:31
And we could figure it out.
519
00:30:31 --> 00:30:36
Actually I think section 1.2
gives the formula for this
520
00:30:36 --> 00:30:42
particular model problem for
any size, and particular
521
00:30:42 --> 00:30:44
for five by five.
522
00:30:44 --> 00:30:52
And there is something
wonderful for this
523
00:30:52 --> 00:30:55
special case.
524
00:30:55 --> 00:31:00
The five points fall right
on the correct parabola,
525
00:31:00 --> 00:31:01
they're exactly right.
526
00:31:01 --> 00:31:06
So for this particular case
when the solution was a
527
00:31:06 --> 00:31:09
quadratic, the exact solution
was a quadratic, a parabola,
528
00:31:09 --> 00:31:14
it will turn out, and that
quadratic matches these
529
00:31:14 --> 00:31:21
boundary conditions, it will
turn out that those points
530
00:31:21 --> 00:31:25
are right on the money.
531
00:31:25 --> 00:31:27
So that's, you could
call, is like, super
532
00:31:27 --> 00:31:29
convergence or something.
533
00:31:29 --> 00:31:35
I mean that won't happen
every time, otherwise life
534
00:31:35 --> 00:31:39
would be like, too easy.
535
00:31:39 --> 00:31:46
It's a good life, but it's
not that good as a rule.
536
00:31:46 --> 00:31:56
So they fall right
on that curve.
537
00:31:56 --> 00:31:59
And we can say what
those numbers are.
538
00:31:59 --> 00:32:01
Actually, we know
what they are.
539
00:32:01 --> 00:32:03
Actually, I guess I
could find them.
540
00:32:03 --> 00:32:09
What are those numbers then?
541
00:32:09 --> 00:32:13
And of course, one over h
squared is-- What's one over h
542
00:32:13 --> 00:32:16
squared, just to not forget?
543
00:32:16 --> 00:32:20
One over h squared
there, h is what?
544
00:32:20 --> 00:32:22
1/6.
545
00:32:22 --> 00:32:24
Squared is going to be a 36.
546
00:32:24 --> 00:32:30
So if I bring it up here,
bring the h squared up here,
547
00:32:30 --> 00:32:33
it would be times a 36.
548
00:32:33 --> 00:32:37
Well let me leave it here, 36.
549
00:32:37 --> 00:32:41
And I'm just saying that these
numbers would come out right.
550
00:32:41 --> 00:32:42
Maybe I'll just do
the first one.
551
00:32:42 --> 00:32:44
What's the exact u_1, u_2?
552
00:32:45 --> 00:32:48
u_1 and u_2 would be what?
553
00:32:48 --> 00:32:51
The exact u_1, ooh!
554
00:32:51 --> 00:32:53
Oh shoot, I've got
to figure it out.
555
00:32:53 --> 00:32:55
If I plug in x=1/6...
556
00:32:55 --> 00:32:58
557
00:32:58 --> 00:33:04
Do we want to do this?
558
00:33:04 --> 00:33:04
Plug in x=1/6?
559
00:33:05 --> 00:33:08
No, we don't.
560
00:33:08 --> 00:33:08
We don't.
561
00:33:08 --> 00:33:11
We've got something better
to do with our lives.
562
00:33:11 --> 00:33:15
But if we put that number in,
whatever the heck it is, in
563
00:33:15 --> 00:33:19
this one, we would find
out came out right.
564
00:33:19 --> 00:33:22
The fact that it comes
out right is important.
565
00:33:22 --> 00:33:34
But I'd like to move on
to a similar problem.
566
00:33:34 --> 00:33:38
But this one is going
to be free-fixed.
567
00:33:38 --> 00:33:44
So if this problem was like
having an elastic bar hanging
568
00:33:44 --> 00:33:49
under its own weight and these
would be the displacements
569
00:33:49 --> 00:33:54
points on the bar and fixed
at the ends, now I'm
570
00:33:54 --> 00:33:56
freeing up the top end.
571
00:33:56 --> 00:34:02
I'm not making u_0,
zero anymore.
572
00:34:02 --> 00:34:07
I better maybe use a different
blackboard because that's so
573
00:34:07 --> 00:34:12
important that I don't
want to erase it.
574
00:34:12 --> 00:34:18
So let me take the same
problem, uniform bar, uniform
575
00:34:18 --> 00:34:27
load, but I'm going to fix U
over one, that's fixed, but
576
00:34:27 --> 00:34:30
I'm going to free this end.
577
00:34:30 --> 00:34:32
And from a differential
equation point of view, that
578
00:34:32 --> 00:34:39
means I'm going to set the
slope at zero to be zero.
579
00:34:39 --> 00:34:39
U'(0)=0.
580
00:34:39 --> 00:34:44
581
00:34:44 --> 00:34:48
That's going to have a
different solution.
582
00:34:48 --> 00:34:51
Change the boundary conditions
is going to change the answer.
583
00:34:51 --> 00:34:52
Let's find the solution.
584
00:34:52 --> 00:34:56
So here's another
differential equation.
585
00:34:56 --> 00:34:58
Same equation, different
boundary conditions,
586
00:34:58 --> 00:35:00
so how do we go?
587
00:35:00 --> 00:35:04
Well I had the general
solution over there.
588
00:35:04 --> 00:35:05
It still works, right?
589
00:35:05 --> 00:35:10
U(x) is still -1/2
of x squared.
590
00:35:10 --> 00:35:13
The particular solution
that gives me the one.
591
00:35:13 --> 00:35:19
Plus the Cx plus D that gives
me zero, one, zero for second
592
00:35:19 --> 00:35:24
derivatives but gives me the
possibility to satisfy the
593
00:35:24 --> 00:35:27
two boundary conditions.
594
00:35:27 --> 00:35:29
And now again, plug in
the boundary conditions
595
00:35:29 --> 00:35:33
to find C and D.
596
00:35:33 --> 00:35:35
Slope is zero at zero.
597
00:35:35 --> 00:35:37
What does that tell me?
598
00:35:37 --> 00:35:39
I have to plug that in.
599
00:35:39 --> 00:35:43
Here's my solution, I have
to take it's derivative
600
00:35:43 --> 00:35:46
and set x to zero.
601
00:35:46 --> 00:35:50
So it's derivative is
a 2x or a minus x or
602
00:35:50 --> 00:35:53
something which is zero.
603
00:35:53 --> 00:36:00
The derivative of that is C and
the derivative of that is zero.
604
00:36:00 --> 00:36:03
What am I learning from
that left, the free
605
00:36:03 --> 00:36:06
boundary condition?
606
00:36:06 --> 00:36:08
C is zero, right?
607
00:36:08 --> 00:36:12
C is zero because the
slope here is C and it's
608
00:36:12 --> 00:36:12
supposed to be zero.
609
00:36:12 --> 00:36:16
So C is zero.
610
00:36:16 --> 00:36:19
Now the other
boundary condition.
611
00:36:19 --> 00:36:21
Plug in x=1.
612
00:36:21 --> 00:36:24
I want to get the answer zero.
613
00:36:24 --> 00:36:27
The answer I do get is minus
1/2 at x=1 , plus D .
614
00:36:28 --> 00:36:32
So what is D then?
615
00:36:32 --> 00:36:33
What's D?
616
00:36:33 --> 00:36:36
Let me raise that.
617
00:36:36 --> 00:36:40
What do I learn about D?
618
00:36:40 --> 00:36:43
It's 1/2.
619
00:36:43 --> 00:36:45
I need 1/2.
620
00:36:45 --> 00:36:57
So the answer is -1/2
of x squared plus 1/2.
621
00:36:57 --> 00:37:04
Not 1/2x as it was
over there, but 1/2.
622
00:37:04 --> 00:37:06
And now let's graph it.
623
00:37:06 --> 00:37:08
Always pays to graph
these things between x
624
00:37:08 --> 00:37:12
equals zero and one.
625
00:37:12 --> 00:37:15
What does this looks like?
626
00:37:15 --> 00:37:17
It starts at 1/2, right?
627
00:37:17 --> 00:37:18
At x=0.
628
00:37:19 --> 00:37:21
And it's a parabola, right?
629
00:37:21 --> 00:37:22
It's a parabola.
630
00:37:22 --> 00:37:24
And I know it goes
through this point.
631
00:37:24 --> 00:37:28
What else do I know?
632
00:37:28 --> 00:37:32
Slope starts at?
633
00:37:32 --> 00:37:33
The slope starts to zero.
634
00:37:33 --> 00:37:36
The other, the boundary
condition, the free condition
635
00:37:36 --> 00:37:40
at the left-hand end, so slope
starts at zero, so the parabola
636
00:37:40 --> 00:37:42
comes down like that.
637
00:37:42 --> 00:37:46
It's like half a-- where that
was a symmetric bit of a
638
00:37:46 --> 00:37:51
parabola, this is
just half of it.
639
00:37:51 --> 00:37:56
The slope is zero.
640
00:37:56 --> 00:38:00
And so that's a graph of U(x) .
641
00:38:00 --> 00:38:07
Now I'm ready to replace it
by a difference equation.
642
00:38:07 --> 00:38:09
So what'll be the
difference equation?
643
00:38:09 --> 00:38:13
It'll be the same
equation for the -u''.
644
00:38:13 --> 00:38:16
No change.
645
00:38:16 --> 00:38:23
So minus u_(i+1) minus
2u_i , minus u_(i-1)
646
00:38:23 --> 00:38:28
over h squared equals.
647
00:38:28 --> 00:38:34
I'm taking f(x) to be one,
so let's stay with one.
648
00:38:34 --> 00:38:38
Okay, big moment.
649
00:38:38 --> 00:38:40
What boundary conditions?
650
00:38:40 --> 00:38:42
What boundary conditions?
651
00:38:42 --> 00:38:45
Well, this guy is pretty clear.
652
00:38:45 --> 00:38:51
That says u_(n+1) is zero.
653
00:38:51 --> 00:38:55
What do I do for zero slope?
654
00:38:55 --> 00:38:57
What do I do for a zero slope?
655
00:38:57 --> 00:39:00
Okay, let me suggest
one possibility.
656
00:39:00 --> 00:39:04
It's not the greatest, but
one possibility for a zero
657
00:39:04 --> 00:39:06
slope is (u_1-u_0)/h .
658
00:39:06 --> 00:39:08
659
00:39:08 --> 00:39:14
That's the approximate
slope, should be zero.
660
00:39:14 --> 00:39:21
So that's my choice for the
left-hand boundary condition.
661
00:39:21 --> 00:39:24
It says u_1 is u_0 .
662
00:39:24 --> 00:39:28
It says that u_1 is u_0 .
663
00:39:28 --> 00:39:39
So now I've got again five
equations for five unknowns,
664
00:39:39 --> 00:39:39
u_1, u_2, u_3, u_4, and u_5.
665
00:39:39 --> 00:39:44
666
00:39:44 --> 00:39:46
I'll write down what they are.
667
00:39:46 --> 00:39:49
Well, you know what they are.
668
00:39:49 --> 00:39:56
So this thing divided by
h squared is all ones,
669
00:39:56 --> 00:39:57
just like before.
670
00:39:57 --> 00:40:05
And of course all these
rows are not changed.
671
00:40:05 --> 00:40:07
But the first row is changed
because we have a new boundary
672
00:40:07 --> 00:40:09
condition at the left end.
673
00:40:09 --> 00:40:11
And it's this.
674
00:40:11 --> 00:40:17
So u_1, well u_0 isn't in the
picture, but previously what
675
00:40:17 --> 00:40:22
happened to u_0 , when i is
one, I'm in the first equation
676
00:40:22 --> 00:40:24
here, that's where I'm
looking. i is one.
677
00:40:24 --> 00:40:25
It had a u_0.
678
00:40:26 --> 00:40:28
Gone.
679
00:40:28 --> 00:40:34
In this case it's not gone. u_0
comes back in, u_0 is u_1.
680
00:40:34 --> 00:40:36
That might-- Ooh!
681
00:40:36 --> 00:40:38
Don't let me do this wrong.
682
00:40:38 --> 00:40:38
Ah!
683
00:40:38 --> 00:40:42
Don't let me do it worse!
684
00:40:42 --> 00:40:43
All right.
685
00:40:43 --> 00:40:43
There we go.
686
00:40:43 --> 00:40:44
Good.
687
00:40:44 --> 00:40:47
Okay.
688
00:40:47 --> 00:40:53
Please, last time I videotaped
lecture 10 had to fix up
689
00:40:53 --> 00:40:56
lecture 9, because
I don't go in.
690
00:40:56 --> 00:41:00
Professor Lewin in the physics
lectures, he cheats, doesn't
691
00:41:00 --> 00:41:03
cheat, but he goes into the
lectures afterwards
692
00:41:03 --> 00:41:05
and fixes them.
693
00:41:05 --> 00:41:10
But you get exactly
what it looks like.
694
00:41:10 --> 00:41:13
So now it's fixed, I hope.
695
00:41:13 --> 00:41:16
But don't let me screw up.
696
00:41:16 --> 00:41:22
So now, what's on this top row?
697
00:41:22 --> 00:41:23
When i is one.
698
00:41:23 --> 00:41:26
I have minus u_2, that's fine.
699
00:41:26 --> 00:41:32
I have 2u_1 as before, but now
I have a minus u_1 because
700
00:41:32 --> 00:41:34
u_0 and u_1 are the same.
701
00:41:34 --> 00:41:37
So I just have a one in there.
702
00:41:37 --> 00:41:40
That's our matrix
that we called T.
703
00:41:40 --> 00:41:44
The top row is changed,
the top row is free.
704
00:41:44 --> 00:41:49
This is the equation T * U
divided by h squared is
705
00:41:49 --> 00:41:52
the right-hand side ones.
706
00:41:52 --> 00:41:57
Ones of five, would call that.
707
00:41:57 --> 00:42:01
Properly I would call it ones
of five one, because the MATLAB
708
00:42:01 --> 00:42:06
command ones wants matrix and
it's a matrix with five
709
00:42:06 --> 00:42:08
rows, one column.
710
00:42:08 --> 00:42:12
But it's T, that's
the important thing.
711
00:42:12 --> 00:42:20
And would you like to guess
what the solution looks like?
712
00:42:20 --> 00:42:25
In particular, is it
again exactly right?
713
00:42:25 --> 00:42:29
Is it right on the money?
714
00:42:29 --> 00:42:35
Or if not, why not?
715
00:42:35 --> 00:42:38
The computer will
tell us, of course.
716
00:42:38 --> 00:42:41
It will tell us whether we
get agreement with this.
717
00:42:41 --> 00:42:48
This is the exact solution here
and this is the exact parabola
718
00:42:48 --> 00:42:51
starting with zero slope.
719
00:42:51 --> 00:42:54
So but I solved this problem.
720
00:42:54 --> 00:42:58
Oh, let me see, I didn't get
u_1, u_2 to u_5 in there.
721
00:42:58 --> 00:43:00
So it didn't look right. u_1,
u_2, u_3, u_4, and u_5.
722
00:43:00 --> 00:43:04
723
00:43:04 --> 00:43:06
And that's the right-hand side.
724
00:43:06 --> 00:43:08
Sorry about that.
725
00:43:08 --> 00:43:13
So that's T divided by h
squared, T with that top
726
00:43:13 --> 00:43:19
row changed times U is
the right-hand side.
727
00:43:19 --> 00:43:25
By the way, I better just say
what was the reason that we
728
00:43:25 --> 00:43:30
came out exactly right
on this problem?
729
00:43:30 --> 00:43:34
Would we come out exactly
right if it was some
730
00:43:34 --> 00:43:36
general load f(x) ?
731
00:43:36 --> 00:43:38
No.
732
00:43:38 --> 00:43:42
Finding differences
can't do miracles.
733
00:43:42 --> 00:43:45
They have no way to know what's
happening to f(x) between
734
00:43:45 --> 00:43:47
the mesh points, right?
735
00:43:47 --> 00:43:52
If I took this to be f(x) and
took this at the five points,
736
00:43:52 --> 00:43:57
at these five points, this
wouldn't know what f(x) is in
737
00:43:57 --> 00:44:00
between, couldn't
be exactly right.
738
00:44:00 --> 00:44:07
It's exactly right in this
lucky special case because, of
739
00:44:07 --> 00:44:11
course, it has the right ones.
740
00:44:11 --> 00:44:15
But also because, the reason
it's exactly right is that
741
00:44:15 --> 00:44:19
second differences of
quadratics are exactly right.
742
00:44:19 --> 00:44:24
That's what we checked on this
board that's underneath there.
743
00:44:24 --> 00:44:29
Second differences of
squares came out perfectly.
744
00:44:29 --> 00:44:34
And that's why the second
differences of this guy give
745
00:44:34 --> 00:44:40
the right answer, so that guy
is the answer to both the
746
00:44:40 --> 00:44:43
differential and the
difference equation.
747
00:44:43 --> 00:44:46
I had to say that word about
why was that exactly right.
748
00:44:46 --> 00:44:48
It was exactly right because
second differences of
749
00:44:48 --> 00:44:50
squares are exactly right.
750
00:44:50 --> 00:44:53
Now, again, we have second
differences of squares.
751
00:44:53 --> 00:44:58
So you could say
exactly right or no?
752
00:44:58 --> 00:44:58
What are you betting?
753
00:44:58 --> 00:45:00
How many think, yeah,
it's going to come
754
00:45:00 --> 00:45:03
out on the parabola?
755
00:45:03 --> 00:45:04
Nobody.
756
00:45:04 --> 00:45:06
Right.
757
00:45:06 --> 00:45:10
Everybody thinks there's
something going to miss here.
758
00:45:10 --> 00:45:11
And why?
759
00:45:11 --> 00:45:14
Why am I going to
miss something?
760
00:45:14 --> 00:45:17
Yes?
761
00:45:17 --> 00:45:19
It's a first order
approximation at
762
00:45:19 --> 00:45:20
the left boundary.
763
00:45:20 --> 00:45:22
Exactly right, exactly right.
764
00:45:22 --> 00:45:27
It's a first order
approximation to take this and
765
00:45:27 --> 00:45:29
I'm not going to get it right.
766
00:45:29 --> 00:45:33
That first order approximation,
that error of size h is
767
00:45:33 --> 00:45:39
going to penetrate over
the whole interval.
768
00:45:39 --> 00:45:40
It'll be biggest here.
769
00:45:40 --> 00:45:43
Actually I think it turns out,
and the book has a graph,
770
00:45:43 --> 00:45:47
I think it comes out
wrong by 1/2h there.
771
00:45:47 --> 00:45:49
1/2h, first order.
772
00:45:49 --> 00:45:54
And then it, of course, it's
discrete and of course it's
773
00:45:54 --> 00:45:57
straight across because that's
the boundary condition, right?
774
00:45:57 --> 00:46:02
And then it starts down, it
gets sort of closer, closer,
775
00:46:02 --> 00:46:06
closer and gets, of course,
that's right at the end.
776
00:46:06 --> 00:46:08
But there's an error.
777
00:46:08 --> 00:46:13
The difference between
U, the true U and the
778
00:46:13 --> 00:46:22
computed U is of order h.
779
00:46:22 --> 00:46:28
So you could say alright, if h
is small I can live with that.
780
00:46:28 --> 00:46:33
But as I said in the end you
really want to get second
781
00:46:33 --> 00:46:34
order accuracy if you can.
782
00:46:34 --> 00:46:36
And in a simple problem
like this we should
783
00:46:36 --> 00:46:39
be able to do it.
784
00:46:39 --> 00:46:46
What I've done already covers
section 1.2 but then there's a
785
00:46:46 --> 00:46:52
note, a worked example at the
end of 1.2 that tells you how
786
00:46:52 --> 00:46:56
to upgrade to second order.
787
00:46:56 --> 00:47:00
And maybe we've got a moment
to see how would we do it.
788
00:47:00 --> 00:47:04
What would you suggest
that I do differently?
789
00:47:04 --> 00:47:07
I'll get a different matrix.
790
00:47:07 --> 00:47:09
I'll get a different
discrete problem.
791
00:47:09 --> 00:47:10
But that'll be ok.
792
00:47:10 --> 00:47:12
I can solve that just as well.
793
00:47:12 --> 00:47:17
And what shall I replace that
by? because that was the
794
00:47:17 --> 00:47:19
guilty party, as you said.
795
00:47:19 --> 00:47:21
That was guilty.
796
00:47:21 --> 00:47:23
That's only a first
order approximation to
797
00:47:23 --> 00:47:28
zero slope at zero.
798
00:47:28 --> 00:47:30
A couple of ways we could go.
799
00:47:30 --> 00:47:33
This is a correct second
order approximation
800
00:47:33 --> 00:47:38
at what mesh point?
801
00:47:38 --> 00:47:44
That is a correct second order
approximation to U'=0, but not
802
00:47:44 --> 00:47:48
at that point or at
that point where?
803
00:47:48 --> 00:47:50
Halfway between.
804
00:47:50 --> 00:47:53
If I was looking at a point
halfway between that, that
805
00:47:53 --> 00:47:56
would be centered there, that
would be a centered difference
806
00:47:56 --> 00:47:57
and it would be good.
807
00:47:57 --> 00:48:00
But we're not looking there.
808
00:48:00 --> 00:48:02
So I'm looking here.
809
00:48:02 --> 00:48:06
So what do you suggest I do?
810
00:48:06 --> 00:48:10
Well I've got to center it.
811
00:48:10 --> 00:48:15
Essentially I'm going
to use U, minus one.
812
00:48:15 --> 00:48:17
I'm going to use U, minus one.
813
00:48:17 --> 00:48:23
And let me just say
what the effect is.
814
00:48:23 --> 00:48:27
You remember we started with
the usual second difference
815
00:48:27 --> 00:48:31
here, 2, -1, -1.
816
00:48:31 --> 00:48:34
This is what got chopped
off for the fixed method.
817
00:48:34 --> 00:48:37
It got brought back here by
our first order method.
818
00:48:37 --> 00:48:42
Our second order method
will-- You see what's
819
00:48:42 --> 00:48:43
likely to happen?
820
00:48:43 --> 00:48:48
That -1 is going
to show up where?
821
00:48:48 --> 00:48:50
Over here.
822
00:48:50 --> 00:48:52
To center it around zero.
823
00:48:52 --> 00:48:59
So that guy will make
this into a minus two.
824
00:48:59 --> 00:49:03
Now that matrix is still fine.
825
00:49:03 --> 00:49:07
It's not one of our
special matrices.
826
00:49:07 --> 00:49:10
When I say fine, it's
not beautiful is it?
827
00:49:10 --> 00:49:18
It's got one, like, flaw, it
needs what do you call it when
828
00:49:18 --> 00:49:22
you have your face-- cosmetic
surgery or something.
829
00:49:22 --> 00:49:25
It needs a small improvement.
830
00:49:25 --> 00:49:27
So what's the matter with it?
831
00:49:27 --> 00:49:30
It's not symmetric.
832
00:49:30 --> 00:49:34
It's not symmetric and a person
isn't happy with a un-symmetric
833
00:49:34 --> 00:49:38
problem, approximation to a
perfectly symmetric thing.
834
00:49:38 --> 00:49:41
So I could just divide
that row by two.
835
00:49:41 --> 00:49:45
If I divide that row by 2,
which you won't mind if I do
836
00:49:45 --> 00:49:50
that, make that one, minus
one and makes this 1/2.
837
00:49:50 --> 00:49:55
I divided the first
equation by two.
838
00:49:55 --> 00:50:02
Look in the notes in
the text if you can.
839
00:50:02 --> 00:50:05
And the result is
now it's right on.
840
00:50:05 --> 00:50:07
It's exactly on.
841
00:50:07 --> 00:50:12
Because again, the solution,
the true solution is squares.
842
00:50:12 --> 00:50:17
This is now second order and
we'll get it exactly right.
843
00:50:17 --> 00:50:21
And I say all this
for two reasons.
844
00:50:21 --> 00:50:25
One is to emphasize again that
the boundary conditions are
845
00:50:25 --> 00:50:30
critical and that they
penetrate into the region.
846
00:50:30 --> 00:50:34
The second reason for my saying
this is looking forward
847
00:50:34 --> 00:50:37
way into October.
848
00:50:37 --> 00:50:42
So let me just say the finite
element method, which you may
849
00:50:42 --> 00:50:45
know a little about, you may
have heard about, it's
850
00:50:45 --> 00:50:48
another-- this was
finite differences.
851
00:50:48 --> 00:50:51
Courses starting with finite
differences, because that's
852
00:50:51 --> 00:50:52
the most direct way.
853
00:50:52 --> 00:50:53
You just go for it.
854
00:50:53 --> 00:50:57
You've got derivatives, you
replace them by differences.
855
00:50:57 --> 00:51:07
But another approach which
turns out to be great for big
856
00:51:07 --> 00:51:13
codes and also turns out to be
great for making, for keeping
857
00:51:13 --> 00:51:16
the properties of the problem,
the finite element method,
858
00:51:16 --> 00:51:22
you'll see it, it's weeks away,
but when it comes, notice, the
859
00:51:22 --> 00:51:25
finite element method
automatically produces
860
00:51:25 --> 00:51:27
that first equation.
861
00:51:27 --> 00:51:29
Automatically gets it right.
862
00:51:29 --> 00:51:33
So that's pretty special.
863
00:51:33 --> 00:51:37
And so, the finite element
method just has, it produces
864
00:51:37 --> 00:51:44
that second order accuracy that
we didn't get automatically
865
00:51:44 --> 00:51:48
for finite differences.
866
00:51:48 --> 00:51:52
Ok, questions on today
or on the homework.
867
00:51:52 --> 00:51:55
So the homework is
really wide open.
868
00:51:55 --> 00:51:59
It's really just a
chance to start to see.
869
00:51:59 --> 00:52:03
I mean, the real homework is
read those two sections of the
870
00:52:03 --> 00:52:09
book to capture what these
two lectures have done.
871
00:52:09 --> 00:52:11
So Monday I'll see.
872
00:52:11 --> 00:52:13
We'll do elimination.
873
00:52:13 --> 00:52:17
We'll solve these equations
quickly and then move on
874
00:52:17 --> 00:52:21
to the inverse matrix.
875
00:52:21 --> 00:52:25
More understanding
of these problems.
876
00:52:25 --> 00:52:26
Thanks.