Hello. Welcome to this big room. I think we're here just today. And then back to the usual room on Tuesday.
Today, I'm going to talk about more mathematics than usual, but it's very applied mathematics.
Fourier Laurent and Chevy chef are three parallel settings, which are the basis of algorithms used in all areas of computational science.
In particular, odes and pdes ultimately rely on these ideas.
And I want to highlight for you the very close relationship between these three in particular.
These are the basis of spectral methods, but all sorts of expansions.
And whenever you try to connect something discrete to something continuous, the chances are you're going to use one of these three settings.
The picture is the unit circle and that's where Z lives.
So you can imagine a variable Z on the unit circle in the complex plane,
and then the diameter of that circle would be the unit interval from -1 to 1, and that's where X lives.
And then theta is the angle to a point Z.
So if that would be a point Z on the unit circle, then that's the corresponding angle theta and this would be the corresponding value of x.
So that picture describes how three variables relate one to another.
You could say, Why do we need all three? Well, the answer is so much mathematics depends on these that all three are truly important.
Theta is Fourier analysis, Z is complex analysis, and X is real analysis.
These are huge, huge territories. Of course, once you see the picture, you immediately see various algebraic relations between the two.
So for example, X is written as one half times Z plus Z inverse.
And we could also say that x is the cosine of I keep writing theta because in my head I usually use theta.
But I think in this lecture I intend to use T instead. So if I ever write theta, it's a mistake I meant to say T of course it's really theta.
But when you're writing codes and so on, theta is an annoying variable.
So we tend to switch to T. So I'm going to spend the first few minutes actually detailing all three of
these contexts in a completely parallel fashion to emphasise the parallels.
So let's begin with Fourier and of course we've all heard of Fourier analysis about 200 years old now.
So the setting here is that we have a periodic function, so not an arbitrary function, a periodic function,
and conventionally on an interval of length to pi, I'll call the function capital F of T and t will be the interval from 0 to 2 pi.
Of course, you could also take minus pi to pi or various other fundamental domains.
Now, whenever we do numerical things, the chances are we're going to sample the function at a set of points.
And of course, for periodic functions, equally spaced points are the obvious choice.
So let's talk about two and plus one.
Equally spaced, equally spaced points.
And I will label them to t sub k and that will be equal to two pi k over two n plus one where K ranges between zero and two n.
So if you draw the picture for that. Notice that because the function is periodic, we only include one end point.
So here we are on our interval going from 0 to 2 pi.
And if we had four points, for example, well, sorry, they're going to be five points.
So if we have five points, for example, they would be here and here and here and here and here, not including there, because that's the same as this.
So of course, for you now with this corresponds to discrete for analysis, which is what an engineer would typically be doing in signal processing.
Now, the basic functions that we expand things in, we loosely speak of signs and cosines,
but it's usually more convenient mathematically to take their complex version.
The complex exponential. So e to the i k t is the complex exponential with wave number k.
And of course, from there you can get the cosine and the sine.
So the cosine would be the real part of this and the sine would be the imaginary part or a better way to say
it is the cosine is this plus E to the minus k over to the sine is this minus C to the minus Oct to II.
Now something that we do explicitly or implicitly all the time is trigonometric interpolation.
And there again, loosely speaking, we're talking about signs and cosines,
but it's often more convenient to write down the formulas in terms of e to the effect.
So a trigonometric interpolate, let's say.
Would be a function I'll call Capital PM and it's a trigonometric polynomial,
which means it's a finite sum from minus m to m of some coefficient c k times e to the i.
K t. So that's some from minus end to end.
Equivalently, you could talk about signs and cosines of Katie.
Now, when people do numerical integration, which we call quadrature,
the obvious idea is to interpolate by trigonometric and then integrate the interpolate, and that's called the trapezoidal root.
It's not obvious that that's the trapezoidal rule, but it is.
So of course we think of the trapezoidal rule as trapezoidal, and then you integrate the trapezoids.
That turns out to be equivalent to interpolating by trigonometric and integrating the trigonometric interpolate.
I won't go through that argument. It's a straightforward one. So the trapezoidal rule effectively integrates the trigonometric and turbulent.
Let's at least make that statement. This is equivalent to integration of P seven.
More precisely, integration of P seven.
Well, there are two N plus one points, so I should say the two and plus one point trapezoidal rule is equivalent to integration of that interpolate.
Now if you want to find roots, let's say root finding, if you want to find the zeros of a trigonometric interpolate.
Well, I'm not going to go into that just now. But effectively what you do is you can compute the eigenvalues of a matrix.
So that turns out to be an effective way to find roots of P sub in its so-called companion matrix.
So root finding and eigenvalue problems are very closely related in numerical practice.
What happens as you take the limit of K going to infinity?
I really should say n going to infinity. The limit as NW Goes to Infinity is a 48 series.
So you have a trigonometric interpolate means a finite degree trigonometric polynomial.
As that degree goes to infinity, you have 48 series.
So in that limit you get this famous idea, assuming F is, let's say Lipschitz Continuous or something,
you can write it as an infinite series where now K goes from minus infinity to infinity of the same thing.
A k. E to the t. What are the 48 coefficients in that series?
So a sub k is equal to one over two pi.
Times the integral from 0 to 2 pi of f little f of t no foreign capital f 50.
Getting my notations confused. Times e to the minus i k.
T. D d t. So 48 theories involves 48 coefficients and then infinite series.
And you all know this from different angles, I'm sure.
Finally, let me mention the basic principle that people use if they're analysing the convergence of the theories.
Assuming that your function is analytic so it has convergent Taylor theories.
The fundamental hypothesis is if you have an analytic function on the interval.
Then it can be analytically continued into the complex plane into some strip.
And I will say a little more about that in a moment.
And that implies that the 48 coefficients decrease geometrically, so they are O of C to the minus absolute value of K for something bigger than one.
So any time you have an analytic function, it's for you. Coefficients decrease at a geometric rate.
So that means its trigonometric and turbulence converge at a geometric rate.
So there on one screen you have a summary of all of 44 theories.
And let me give you a reference.
Of course, there are a thousand work from this, but from a numerical point of view, my favourite references as usual something by myself.
So let me recommend a paper by right and javaid and our to Monto Nellie.
And myself. And that's it's called extension of fun to periodic functions.
It's in the Siam Journal of Numerical Analysis, and it's coming out this year.
You can find it on my web page.
And in particular, we discuss all this stuff in a way that fits very cleanly with the other cases of Laurent and Cherif.
So now let's switch to the Laurent setting for analysis is famous to engineers and real mathematicians.
The Laurent setting is famous to complex mathematicians. This is what you do when you're working in the complex plane.
And incidentally, my opinion is that mathematically, in some sense, the Laurent case is the fundamental ones.
Of course, they're all more or less equivalent. But this conceptually, I think, is the basis from which it's easiest to derive the others.
So now we're going to suppose that we have a function which I will call a fancy F,
so I'm going to call it like that F of Z and Z is in the unit circle.
Now, if you have a function defined on the unit circle, then sort of by definition it's periodic in some sense.
So I don't have to say it's periodic that goes with the territory.
And then in analogy to the equally spaced points, we now have two and plus one roots of unity.
So these are the two and plus first roots of the number one.
And we could write those as z sub k equals e to the i times t some k for k between zero and two m.
Now corresponding to the complex exponential, you have simply Z to the K, which we could call a mano z to the K.
So you begin to see why somehow this is a more fundamental territory.
Z to the K is marginally simpler than E to the T corresponding to a trigonometric and turbulent.
You have what people would call a laurent polynomial interpolate.
And well, you should be able to write down the formula just by this close analogy.
So a laurent polynomial and turbulent I would write as fancy p of z is the sum from minus m
to end of c sub k z to the k so given and plus one roots of two m plus one roots of unity.
There is a unique and turbulent of that form. It's a polynomial, except it has some negative powers as well as some positive powers.
So that's what the word Lawrence polynomial means. Negative as well as positive powers if you want to do quadrature root numerical integration.
Then again, the obvious formula is the appropriately scaled trapezoidal rule,
where basically you add up the samples at Roots of Unity and multiply by the obvious factor.
And that again is equivalent to integrating the interpolate.
And then what comes next in our list is root finding.
So that's the same again, I'm not spelling out the matrix involved,
but if you want to do route finding and for functions that are that sort of live on the unit circle,
then the way to do it is to set up a eigenvalue problem.
You find the eigenvalues of a companion matrix.
A companion matrix is a matrix that's mostly zero except on two diagonals and one row and it contains the coefficient c some k.
What about the limit? Well, as Kate then goes to infinity, you have a long, long series.
So you all know. Taylor Theories. Lauren Theories is the same except with negative as well as positive powers of Z.
So a Lauren series. Means that you have the sum from minus infinity to infinity of a sub k z to the k.
What are the law on coefficients? Well,
it's the same formula as above for 48 coefficients transplanted from the TI variable to the Z variable and you get
a sub k equals one over two pi times the integral over the circle of fancy f of z time z to the minus k minus one.
And that's an integral over the unit circle. So if you know your complex variables, that's a Kochi integral.
So Kochi integrals are the same as for a integrals.
Once you know this stuff, you begin to lose the distinction between the different territories.
Now, what happens if you have an analytic function?
Well, now the appropriate region of an electricity is an annulus.
So let's see on the interval from 0 to 2 pi.
If you have an analytic function, then it's also analytic in some strip in the complex plan.
Well, on the unit circle, if you have an analytic function on the unit circle, then it's also analytic in some annulus.
Surrounding the unit circle, and that would be equivalent to this strip under the E to the i z map, e to the t map.
So if F is analytic in an unknown annulus.
Then that implies that the coefficients a sub k decrease geometric same as above o of C to the minus k for C greater than one.
These two settings are not merely analogous, they're identical.
They really are the same. Everything is the same. We've just changed variables between Z and now the reference on that.
This is harder to find numerical references on a lot of people do numerical Fourier analysis.
Not so many people do numerical lorente analysis, although of course there are some.
And the paper I would suggest you look at is by my student, Anthony Austin and Peter Craven.
Yeah. And myself.
And that's in Siam Journal of Numerical Analysis in 2014.
And that's called numerical algorithms based on analytic function values at roots of Unity.
So as I say, these two are identical. The championship chef setting is not quite identical because it has a further symmetry assumption.
And the picture on that is in these settings I can have an arbitrary function on the circle.
But when I predict to the to the interval, I'm going to require that my function take the same value there as there.
See, here's another value, the inverse. They both project to the same value of X.
And in chubby chef analysis, I want a well-defined function here.
So that entails a symmetry assumption for Z or T.
So here we go. Fauria and TV chef were very famous mathematicians.
I actually don't know anything about. Laurent is not as big a name, I guess.
Okay, so now we suppose we have a function. Little F, a little X.
And X is in the unit interval.
And of course, we don't assume it's periodic. Now,
the analogy to equally spaced points or roots of unity is what you get when you
project those equally spaced points or routes of unity down to the interval.
And those are cheryshev points, and now they're only in plus one of them, not two and plus one of them because pairs project to the same thing.
So we've end plus one chubby chef points.
And those are defined by x sub k equals the cosine of k pi over n zero less than or equal to k less than or equal to n.
Analogous to the binomial or the complex exponential.
We have the Chevy chef polynomial.
And the chubby chef polynomial is written with a capital t t sub k of x and that is one half times z to the k plus z to the minus k.
Another way to say it is, it's the cosine of k times, the arc cosine, the inverse cosine of x.
So it oscillates like a cosine a tabish polynomial on the unit interval oscillates like a cosine,
but it's and then it gets huge outside the interval for a polynomial in turbulent.
Can be written as little pe of nn seven of x is equal to the sum, and now we have a sum from zero to n of c sub k.
T k of x. If you do Quadrature Well, of course you know the natural method is to integrate the interpolate.
I've got some data at VHF points. I integrate that interpolate.
That's my approximate integral. All we need is a name for that algorithm and it's called Crenshaw.
Curtis. So Crenshaw. Curtis Quadrature is the name we give to integrating the Chevy chef interpolate.
Root finding. Well, it turns out now that instead of a big diagonal matrix eigenvalue problem,
you essentially have a tri diagonal matrix eigenvalue problem and the matrix is called a colleague matrix.
So you do this by computing the eigenvalues of a colleague matrix.
These names were given at the height of the Cold War when people thought it was funny to talk about
companion matrices and comrade matrices and colleague matrices in the limit as then goes to infinity.
You, of course, have a Chubby Chef series.
So a Chubby Chef series represents a function F of X as an infinite series from zero to infinity of a sum k.
K of x. I haven't mentioned.
Why do I have C K's and AK? So the interpreter and I'm using C K's, the series I'm using.
Well, these are different numbers. As N goes to infinity, those six will approach these.
But for any finite end, you have coefficients here which will be different from that.
What are those cheryshev coefficients? Well, it's the same as the 48 or Lawrence thing with the change of variables, and when you do your sums,
it turns out it's a sub k equals two over pi times the integral from -1 to 1
of f of x t k of x divided by the square root of one minus x squared the x.
Finally, what about analytics?
Well, for periodic functions, if you're analytic on an interval that implies you're analytic in some strip for functions on the unit circle,
if you're analytic on the circle, that implies you're analytic.
In some analysts in the Chevy chef case, where, again, we're analytic on an interval but no longer periodic,
it turns out that the right domain is an ellipse with four at minus one and one.
And that ellipse is what you get when you take a circle and transplant it by this map.
So if I take a circle here in the Z plan and apply this tarkowski map to it,
I get in the lips in the X plane and any function analytic on the unit interval will
be analytic in some ellipse corresponding to some sufficiently narrow annulus.
So that's the analytic city setting. If F is analytic in an ellipse.
With focus at minus one and one. Then that implies that the coefficients decrease geometrically.
Now. It's a little tedious for me to write out all this stuff for 20 minutes.
But let me tell you, the amount of practical mathematics codified in those three screens is just extraordinary.
Here you have the basis of a large part of what people really do on computers, especially when they want high accurate solutions.
The reference on this is my book Approximation Theory and Approximation Practice.
At top, I called it approximation theory and approximation practice and I'll pass that around.
So it's the book with the horse on the cover. This is from Port Meadow, the picture of the horse.
Okay. Let me show a little bit of that stuff on the computer.
So we're now working with the code called M 52, which is going to show a little of Fourier series and championship series in seven for convenience.
So suppose I say Capital F equals ten fun.
And then the function I will choose will be cosine of e to the sign of t,
and that's a periodic function for well I put it on minus pi pi in the handout, but it might as well be two pi I think.
I hope that works. And now in check from the way you do, you signal that you're dealing with something periodic as you type trig.
So there we're constructing a for a series of four and turbulent to that function.
And let's plot it I'll say subplot 1 to 1 plot F and I'll plot it in that mode.
Okay. So there you have a periodic, not especially exciting function.
Now suppose I say subplot one, two, two and I plot the Fourier coefficients of F.
So there you are. And you see.
Zero is the zero wave number, the constant term and you have minus KS and plus K.
So you always have the symmetry when you write 48 coefficients.
So there you see the 48 coefficients of that function on an absolute on a large scale and we're plotting their absolute value in seven.
It always goes down to machine precision and then stops. Of course, mathematically it would keep going forever.
It's an analytic function which implies it's analytic in some strip,
which implies that the 48 coefficients decreased geometrically and sure enough, you see a geometric decrease of the Fourier coefficients.
We have these gallery things built in, so if I say help Cheb gallerie trig, you'll see a collection.
Things are going slowly of all sorts of built in functions you can play with,
and we're now going to play with two of them, the AMP signal and the FM signal.
So suppose I say F equals Chaddock Gallery Trig of F.M. signal.
That is a preloaded Chevy fun. I'll now say subplot.
One, two, two, one. And I'll plotted.
And that is what we see. So it's an FM signal.
You can sort of see that the amplitude is constant, but the frequency is being modulated.
Let's look at it's Fourier transform. So I'll say subplot to two, three plot coefficients of S.
So there you have a lot of coefficients.
It's a smooth signal. So they're decaying nicely down to machine precision.
Now let's compare that with an AM signal. So I'll now say F equals cab, dot, gallerie trig and signal and I'll say subplot.
Two, two, two. So there you have an AM signal where the carrier frequency is constant,
but the amplitude is being modulated and the you should all think in your head.
Now what will its four year transform look like? So subplot 2 to 4.
What you see are peaks at the carrier frequency so that the energy is all near plus or -50 because that's the carrier frequency.
So to an engineer, there's a lot of interesting stuff there.
Of course, I'm of a generation when these analogue signals were very important.
Nowadays everything is so digitised. Who knows how your radio works nowadays?
Finally, let's do a Chevy chaff analogue. So I'll now say F little s equals headphone of the same thing.
So that was cosine of E to the sign of x. And of course by default it's the unit interval minus one one.
If I say subplot one, two, one.
And plot a little less. You see that?
I've made a mistake. I meant to put it on a.
Sorry. I meant to put it on 0 to 2 pi. Yeah.
So there we have the same function, but now it's represented by a Chevy truck series instead of a 48 series.
And if I now look at its coefficients, so that would be 1 to 2.
These will now be Chevy chef rather than 48 coefficients.
So no longer do we have a zero wave number and then plus and minus we have a zero wave
number and then just higher and higher degree Chevy chef polynomials again analytically.
So geometric convergent, that function is analytic in a ellipse and therefore you have geometric convergence of the Chevy Chef series.
So there's a lot of beautiful mathematics here, which I've been very focussed on in recent years.
Now let me tell you some more of it.
So we're now going to focus on TV chef for the rest of today's lecture and tell you a word about Chevy Chef series and interpret.
Well, the most basic word to say. Is that mathematically cheryshev is essentially the same as for gay.
There's gifts that extra symmetry condition. Otherwise it's the same.
But sociologically it's very different in the 48 territory.
Everyone on earth, given a periodic function and sampling it would of course use equally spaced points.
This is exactly what everyone would obviously do. Nobody would dream of any other method of representing a periodic function.
But in the championship territory with non periodic functions, what people dream of is numerically exponentially bad.
So of course most people don't use championship polynomials.
They would just use mono x to the one, x to the two. Those are exponentially ill conditioned.
So numerical methods that work with polynomials in the Multnomah or basis are catastrophically bad.
If you use TV Chef, everything is just like Fourier analysis and it all works.
So if you're ever working with polynomials on an interval you must use be here unless the degree of your polynomial is like three or four.
Okay, so I've told you what the Chevy chef polynomials are and the general formula,
for example, is t k of x equals the cosine of k times the cosine inverse of x.
So for example, t zero is one and t one is x and T2 is two x squared minus one and t three is four x cubed minus three x and so on.
And as I mentioned, they actually oscillate in the unit interval and grow a great deal outside the unit.
So those are Tabish polynomials. And now as time permits, I want to tell you some theorems about representing functions by Chevy Chef series.
So they were in one. If F is Lipschitz Continuous, and I'm sure many of you know what that means, and maybe some of you don't.
You can look it up. It's a little stronger than continuity. It means that basically the slopes are all bounded by a constant.
Then the TV Chef series, which I've already defined for you, convergence.
And moreover, it converges absolutely and uniformly.
These are technical terms. Absolutely means it works.
Even if you put absolute values on each term, uniformly means that it works even if that the Epsilon two deltas are uniform across the interval.
The consequences of these are that you can work with to have theories however you want, you can reorder them.
That doesn't mess up the convergence. You can differentiate them. You can do all sorts of things.
Turn by turn. Of course, this is an analogue of a theorem in four year analysis,
which says that if Capital F is Lipschitz Continuous, then the Fourier series converges absolutely and uniformly.
Ferrum, too, has to do with the geometric rate of convergence.
Theorem two is that if F is analytic and bounded by some number in absolute value, call it M.
In the Berenstein Rollups. And I'll say in a moment exactly what I mean by that.
Then the Chevy Chase coefficient decreased and here's the rate.
They're bounded by two and row to the minus k.
So I mentioned earlier that analytic functions have geometrically decreasing cheryshev coefficients.
Well, here's a theorem to tell you how fast that goes. And now let me tell you what row is.
There's a unit circle of radius one out here.
I've drawn a circle of radius row bigger than one.
So if you take a circle of radius, row bigger than one and map it by this Tarkovsky map, it turns into an ellipse.
And that's called the Berenstein Rowan. Berenstein worked on this stuff in 1912.
Now from that found on the size of the Chevy chef coefficient, which of course is analogous to a bound on 48 coefficients,
you can reach conclusions about the accuracy of approximations.
So Theorem three. Says if f sub n is the truncated series.
So we take the Chevy Shuffle series, but we truncated a term n.
Then the error between F and it's truncated series.
And when I said the error, I mean the maximum error on the unit interval.
We could call that the infinity normal error. The maximum error on the unit interval is bounded by two and row to the minus n over row minus one.
You get that just by adding up all the terms you've truncated away.
So if you take an infinite TV Chef series with coefficients decreasing at that rate and you chop it at some point, how big is the tail?
Well, you add up all of these bounds and this is what you get.
So Chubby Chef series four analytic functions converge at a geometric rate if you truncate them.
Theorem for. Concerns and turbulence as opposed to truncation.
So suppose ppm is what we've already talked about the Chevy chef in turbulent.
As opposed to the truncated series. Well, it turns out you get the same bound except twice as with that factor too out front.
So f minus pm, the worst error on the unit interval is no greater than four m relative to minus n over rho minus one.
So that tells you how big the error can be when you do interpolation and cheryshev points.
If you have an analytic function, you're guaranteed that the entire plinth converge geometrically.
So the final thing I want to mention is the subject of the other handout.
So take a look here. This paper by seltzer.
In 1972, Lagrangian interpolation at the Archbishop Points.
Seltzer pointed out that there's a beautiful, stable way to compute these polynomial, and it's called the Barry centric formula.
And here he set that forth. Somebody should have done this 50 years ago, but nobody did it with seltzer in 1972.
He showed that you really can compute these things even if n is in the millions and this is called the barrier centric interpolation formula.
Let me write it down. So Theorem five says.
That piece of em that's the interpolate and cheryshev points is equal to a quotient of one thing by another thing.
Each of those is a sum from zero to n. There's a prime there, which I'll explain in a second.
You have in the numerator minus one to the J times f at the j championship point
divided by x minus x j and in the denominator you have minus one to the j.
Divided by x. Minus x. J. This is called the virus centric interpolation formula.
So it's a mathematical formula which is kind of a curious one,
because what you seem to see here is a function with a lot of polls in the numerator and a function with a lot of polls in the denominator.
And yet the end result is a polynomial. Well, that is true.
It's a mathematical fact that it's equal to a polynomial.
Moreover, it is provably numerically stable.
So this formula really does work. And this is what Chevy Fun did for many, many years.
This is the formula that led to the creation of Chevy Fun, actually.
I have to explain what the Prime means. So when I say the sum of something from K equals zero to m prime,
I mean that we take one half times D, not plus one half times DRM and then all the others.
Like that. So the prime means you multiply the terms by one half.
Prime. Prime. So this amazing formula enables you to interpolate by polynomials in TV, Chef points.
To any degree, it's as stable as Fourier analysis.
It's just the same, really. So anything you can do with a discrete 48 transform, you can also do with Chevy Chef points for non periodic functions.
If you look at this a little more, the more you look at it, the more puzzling it seems.
Because what happens when x is equal to x j?
Well, in that case, you bypass the formula and you say part of x j is equal to the data value f of x j.
But what if x is very close to x j.
We're going to have something huge here with tremendous cancellation errors and something huge here with tremendous cancellation errors.
But it turns out the cancellation errors cancel and it really is provably numerically stable.
So this is both fast. If there were then operations not in squared and numerically stable.
So this really is the right way to interpolate. And I just want to finish by showing you that.
Oh, so the back page of the handout is a paper by myself and Jean-Paul Baru.
Baru is a professor at the University of Fribourg in Switzerland, and he is a great authority on the various centric methods.
So if you want to read about it, maybe our survey paper is a good place to do that.
So let me finish with this final demonstration showing Ben Stein ellipses.
So this one's called. I'm 53. Ben Stein So what I'll do is I'll set F equal to a famous function.
I run a function it's called. So it's one divided by one plus 25 x squared.
And now I'll say a subplot. One, three, one plot f y lim minus one, two.
So there you have the wrong assumption. It's analytic, but it's troublesome for certain methods of approximation.
Let's look at its TV chef coefficient. So I'll say subplot 132 plot coefficients of f line width.
Point six. Really? Oh. So there you see what looks like a solid triangle.
Actually, all the odd coefficients have zero and the even coefficients are non-zero.
So we're getting this kind of effect. But you see the totally clean geometric rate of convergence of the cheryshev coefficients.
Finally, the last command will show the Bernstein Ellipse.
So suppose I now say subplot. One, three, three.
Plot, region. Plot.
Region of death plots. The Bernstein Ellipse associated with this function axis equal grid arm.
So well, there you see the unit interval and then that ellipse is the region of an electricity of this function
because the wrong function has poles at plus or minus I over five and that ellipse is bounded by that.
Let's now finally confirm the rate of convergence.
It turns out. Oh, well, let's let's plot those polls. I say hold on.
And I'll say plot -0.2.
I'd point to I as red dots. Oops.
What did I do? So you can see the poles of that function are right there on the very fine lips they bound the region of analytically.
So that's the largest ellipse inside which this function is analytic.
Finally, let's compute the Bernstein value of row.
It turns out it's one plus the square root of 26 divided by five.
And let's see how our prediction matches the data.
So I'll go back to the middle plot. And now I'll say plot k against Ro to the minus K as a dashed red line.
So what this is going to do is I've figured out the value of row corresponding to that ellipse of analytically.
And you remember the prediction is the coefficients decrease geometrically and the constant involved is a row.
So there you can see the prediction is beautifully matched by the.