W01_L01_P04 - Derivatives with the FFT

And then you just have to keep in mind,
when you use that fft you'd better know

that the fft is doing the shifting. And
you could ask, well, why not, why the, why

make it, why not shift it back? Why, why
make me shift it back?" Cuz a lot of times

when you're doing computation, you're not
gonna necessarily go look at the spectrum

all the time. You, the reason it stays
shifted is because, remember, speed is

everything. Speed is everything in terms
of the computing. You want, if you got, if

you got n log n speed, you wanna add you
know, stuff on top of it. You can do that

on your own time. With this computing,
keep it at n log n, fast. Okay? In my last

class, I talked in my last course, 581. I
always tell people you may get the right

answer. But it's wrong if you did it the
slow way. Okay? Part of computing now is

not just about oh, I can compute and I got
this really, I got this method that can

compute it right. It's not about that
anymore, it's about not only can I compute

it right, I can compute it faster than
you. Okay. It's not that, it's not that I

guess you know, competitive. But, but
actually, you know it kind of is. If

you're trying to write a paper in this
area, that's what you're trying to say

actually all the time. It's like oh, I can
compute this faster than you can. And here

is why. Okay? That's, that's how you write
a paper in scientific computing. You tell

people how much awesomer you are. Okay?
Alright. But, see, the nice thing with

your data analysis skills, you might be
working, I don't know, in atmospheric

sciences or geology geologoy or something.
With your skills you'll be able to say how

much better you are also. See, then I'll,
then we'll all have won. Okay? Alright. So

there you go. that's the simple
implementation. Now I'm gonna use it to do

a trick. Okay? Yes? So, the actual
coefficients are positive and negative

alternating. This is. Yeah. This is the
absolute value. If it's complex you know,

most people, by the way, when you get a
complex spectrum, what you normally do

what people call the power spectrum. I
don't know if you've heard this. Okay?

Well that's an aggressive word just like
the competition. It's all, it's all like

smackdown for comp, computing. So the
power spectrum is always the absolute

value of the spectral content, cuz that's
what you want to know is how much

frequency content, not necessarily what
phase it's in. Okay? Alright. So, there's

that. And now, for my next trick, I will
differentiate a function, computation. Are

you ready? I, I don't know if you guys are
ready. Let's, can I have the lights on

real quick, upfront? There's this very
nice relationship, that's actually worked

out in the notes, which says the
following. It says the following, the hat

represent Fourier transform by the way,
and this represents the nth derivative. If

I take the nth derivative of a function
and I wanna find what that is, the Fourier

transform of the nth derivative is the
same as taking the function itself, and

Fourier transforming it, and multiplying
by ik to the nth power. Okay? So if I want

to compute the third derivative, all I
gotta do is take the function, Fourier

transform it, multiply this by k^3 inverse
Fourier transform it. That's it. By the

way, this is one of the most, if, if your
boundary conditions work for you, like

here the boundaries go to zero, right? And
there's supposed to be a periodic

function, but here the boundaries are
pretty much going to zero. This is

probably one of the most accurate ways you
can compute a derivative, spectrally

accurate. Okay? So let's compute a
derivative, and here's the thing I'm gonna

compute. oh actually, we can go back with
the lights down. Okay. So I'm gonna take

this function u here. and I'm gonna do the
dollowing function. Here is my favorite

function, hyperbolic secant x. I don't
know if you have a favorite function yet.

It's just a little bit of time before you
graduate to pick one out. This one's taken

by the way, don't even try. and by the
way. its derivative, it's called ud is

sech * tanh. Now what I'm going to do is
try to compute t hat derivative

accurately. And by the way, second
derivative, let's call it ud2 for the

second derivative will be sech x - two *
sech x^3. something like that, pretty

close. I think I actually have it in the
notes here. sorry, give me just a second

here. yeah, actually yeah, that's right.
Okay. and what I'm gonna do is, now,

compute these derivatives using my Fourier
transform. So let me write at the top.

There is u, second or first derivative.
And first of all, if I wanna calculate the

derivatives, first I have to have Fourier
transform of the function. There it is and

I'm gonna calculate now the first
derivative. Okay? So, to calculate the

first derivative, remember, what I gotta
do is take i * k and multiply it by

Fourier transform. There it is. And then,
what I gotta do with this is inverse

Fourier transform it. Right?'Cuz Cuz what
this is, is the derivative of the it's the

Fourier transform of the first derivative.
So what I can do is just say, okay, if I

want the first derivative, let's call it
my ud s for spectral approximations, first

derivative is spectral is ifft. By the way
that's how you invert a Fourier transform,

ifft and fft are the pairs. Okay? And by
the way, the second derivative, let's call

it ud2s is equal to ifft of i times k.
That being squared now, times ut. There

you go. I, I want to call this u, right?
So I'm kind of done. So, let's make some

plots. I'm going to plot here the function
and for short here I'm going to say, call

it ks for k shift will be the fftshift(k)
and so that way I can make this be just,

first I'm gonna plot the function. And
then I'll plot, in addition to the

function, I'm gonna plot the derivative of
the function, so x versus ud. And my

approximation, let's plot that in a red
line and then I'm gonna plot my

approximation to the derivative uds, and
I'll plot that with magenta circles. what

I'm plotting is here is the exact answer,
right there, x, ud, that's the exact

answer and I'm plotting my approximation
to the exact answer. Oh, that's just the

original function. We don't have to have
that in there. Would you like to take it

away? We can just do that. What's that? Oh
that was the fft, yeah we don't want the

fft of the original function. We just want
the function. Oh, by the way. There you

go, look at that. The red line is exact,
the magenta line is my approximation. Who

likes that approximation? Give it up you
all. Okay. So listen, I know it's the

first day of class, we're just warming up.
We'll have a dance party about midway. We

can get disco balls and you know, we can
shine that light on there. It's pretty

sweet. Okay. Alright so there is my
approximation. By the way, let's zoom in a

little bit. Let's just see how well we
really did. Let's see how long it takes

for that magenta ball to fall off. Oh, oh,
oh, oh, come on, fall off. I'm zooming in,

I'm zooming in, I'm zooming in. Look, it's
pretty dang accurate. Look at that. Oh,

it's finally coming off there. I know it,
I can sense it. Oh, look at that. But look

over here on the left, it's like down to
ten minus five or negative six. Pretty

amazing accuracy. Now, you can screw up.
Let me just show you one other thing.

Suppose I take a smaller domain. Oh, not
quite hold on a little smaller just real

quick. Just give me two seconds for the
final. Oh, crap. Okay. Come on. Let's do

four. wanna, I wanna, I wanna show you
what happens when you kind of start to

violate, yeah, there you go, you start
seeing it. Now, notice what happens here?

Here, remember it's supposed to be a two
pie periodic function. Right? This is

clearly not periodic. Before they both
went to zero, you could say kinda looks

periodic. Here, it's up here above point
two here. It's below negative point two.

There's a jump and any time you have a
jump in a Fourier transform, when you try

to represent, you get what's called a
Gibbs' phenomenon, which is oscillations

near the jump and this is what you see
over here. So it does amazingly well here

and over here it starts to oscillate, not
so well. Okay. We're gonna use this on

Friday to do radar problems. Okay. Ha ve a
good Wednesday. See you guys soon.