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Okay. Hello, everybody. Today's a bit different in that I'm going to spend the time showing you fun,

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which has been a centre of my research and that of the people around me for the last ten years or so.

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It's relevant to scientific computing in a very central way.

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It's all about a continuous analogue of MATLAB, and obviously you're all fluent in MATLAB.

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MATLAB became successful because of vectors and matrices.

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The idea of fun is to overload those ideas to functions and operators.

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So I'm curious how many of you have tried to have fun before and how many of you have heard of it before?

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The last day or two? Yeah. Okay. So this should be a life changing experience.

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There's a handout, and I'm actually going to tape a lot of these commands and talk about them as we go.

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So you definitely need a copy of that. Let's begin, however, by looking at the website a little bit.

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So you get Google to have fun and this is what you find.

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We're rather proud of the website. It's pretty.

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If you want to learn how to use the thing, you go to docs and you'll find there a user's guide tab fun guide.

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So if you click on the Chub Fund Guide. You'll see that chapter one is getting started with fun.

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So that's what you need to get started. And then there's 16 or so chapters on different subjects.

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Another key place to go at the Chip Fund website is the tab called Examples.

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Which is a collection of a couple hundred examples of how to use Turbofan in different application areas.

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So on the left you see a table of contents that pretty much looks like a numerical analysis textbook.

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And in each of those areas, you'll find different codes that called have fun interspersed with text to illustrate various things.

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So for example, if we go to ordinary differential equations linear, I find there 18 examples in that category.

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Each one would be an m file that has been published to illustrate this or that.

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So the first one says Fourier Spectral Collocation.

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If I go to that. I see various things going on with various figures all in fun.

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Maybe the other tab that's fun to click on is about.

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So if you go to a bout, you learn something about the history of the project and the people involved.

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For example, if I go to people, you can see photographs of the various regrettably armed men who've been involved in this project over the years.

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Okay. So I want to show you what the thing does, how it works.

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So we go into MATLAB, of course, and oh, I should have just emphasised the download situation.

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So if you're at the website and click on download, it takes you to.

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Some basic information. In the simplest way to download.

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The thing is, as it says there, to paste this text right here into your MATLAB command window, and then it will it will do the right thing.

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That will give you the standard branch that has been released and is much trusted.

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But if you want the latest, you go to GitHub and get the development branch.

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So let's see if I click on that icon there at the upper right.

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It takes me to GitHub, which is the standard place that open source software is the repository.

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So you can see all sorts of things if you want to know what's happening in often

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you can click on issues and find all sorts of the latest things going on.

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I think when I asked you before, many of you had used GitHub, right? Raise your hand if you've used GitHub.

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So that's amazing. Okay. So we certainly use it all the time.

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Okay. So let's play with fun and give you a sense of what it can do.

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So I'll start following the hand out there with basic Jeb funds.

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And the first thing I'll do is create a JAB fund without really telling you what that means,

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because we'll immediately modify it to show something more interesting.

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So we've just created the function X on the interval from 0 to 10 and now let's say ethical sine of x plus sign of x squared.

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So now whatever we've done, it's clearly made something that looks like that function.

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And the point of turbofan is to be able to compute with functions in a fast and accurate way, functions like this one.

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Now, of course, mathematically we know what that means, but in fun, numerically, that's actually represented by a polynomial.

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It's not a sign, it's a polynomial. If we wanted to know something about that polynomial.

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The length of F tells us that it's a polynomial interpolate through 119 points.

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So it's in fact a polynomial of degree, 118.

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And if you wanted to see the points, you could say this.

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So fun is all about interpolation by polynomials on clever grids that are good for interpolation.

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These are chubby chef grids which are clustered near the end points.

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So we're interpolating by a polynomial of degree 118 in a suitably clustered grid.

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So that interpolation is a reliable process all the time.

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Chep fine is doing that, but in principle you don't need to know that you're just using it as a calculator.

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So for example, if I say G equals E to the X each of the F, it makes a new function if I plot it.

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There it is. So you see, it feels like symbolic computing, but it's actually numerical.

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And as you know, from symbolic computing, symbolic computing is fantastic when it works.

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And often it gets sort of grinds to a halt when things get complicated.

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The point of cheap fun is to not grind to a halt. So just to give you a bit of a hint about what's going on here.

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If I look at the length of G, it's 914.

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So that means this curve here is a polynomial of degree 913.

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Now suppose I say the length of g times g. Standard MATLAB notation.

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Of course, the times in MATLAB that would mean the component wise product of a vector in tetfund by analogy, instead of a vector we have a function.

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So it means the position, you know, the in g of x times g of x at each value x, so the length of g times g.

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Mathematically, you would expect it to be 1827 because it's a polynomial of degree.

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913. But it's not that.

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Because what goes on all the time is that the degrees of polynomials are reduced in order to round the whole thing to 16 digit precision.

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So the idea of fun is to compute with polynomials, but constantly round them down to 16 digits.

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And you can see a bit of how that happens in the next section here, how the constructor works.

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So here was f again. Remember, the length of death was 119.

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Well, if I say plot coefficients of F. What that will do is show me the coefficients in the actual representation of F and

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I think I'll just digress briefly to write one equation on the board about that.

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So if you have a function as that sufficiently smooth, for example,

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it's enough for it to be Lipschitz Continuous on an interval and by default it's the unit interval.

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But it could be any other than the right way to expand f for numerical purposes is as a chubby chef series.

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So you write F as the sum from zero to infinity of a coefficient times a the case chevy chef polynomial of x.

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So that is a chubby chef series and it's completely analogous to a three series for.

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As for periodic functions, Chevy chef is for non periodic functions for AA and vital signs and cosines.

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Chevy Chef involves polynomials. The coefficients mathematically are given by a formula which involves a constant two

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times an integral from -1 to 1 of f of x t k of x d x divided by one minus x squared.

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This is numerically a very effective series with which to represent functions.

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And what Cheryshev does all the time is truncate the series to machine precision.

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So this function F,

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this particular function F is represented as a sum from 0 to 118 of a k t sum k of x for some numerically computed cheryshev coefficients a sub k.

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And when I execute plot coefficients. What you're going to see is a plot as a function of k of the size of the coefficient on a large scale.

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So mathematically the coefficients go down to zero as K goes to infinity.

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Numerically, we're going to stop when we get to machine precision. Okay.

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So plot coefficients of F. There it is. You see that this is 118 dots.

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I really should plot them as dots. So those are the coefficients of the Chevy Chase expansion.

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Mathematically, it would keep going forever. Down to zero.

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Numerically, we stopped there when it's gone down about 15 orders of magnitude.

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And Tetfund has very slick algorithms for doing this fast and accurately.

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Just for fun. Let's make another version of that. I'll say two, and I'll make it equal to the very same function.

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So that's what is it? It's a sign of x plus sign of x squared on the interval from 0 to 10.

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But now I'm going to give it a special flag double length, which says.

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Make this function, except take twice as many Chevy chef points as usual.

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And I'm now going to plot the coefficients of F two.

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So what you see here is that if you go further, you don't gain anything because of rounding errors.

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It's all garbage anyway on a computer in 16 digits.

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So Tetfund is constantly finding this point in order to represent a function and in fact we could even superimpose those two plots.

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Suppose I say hold on and I'll say plot coefficients of F as red circles.

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Then you can see that the adaptive algorithm chops at the right place.

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So that's the basis of telephone to use polynomials to do fast computing.

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Any questions? I'm going to show you, of course, how we can use them.

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Let's say I see on my handout here. I proposed to show you a snake.

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I'm not very good at snake, but let me try it anyway. Chap, Snake is a game where I'm not ready.

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Really, but let me try. So the idea is that luckily it's going slowly.

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I see. So you're. You're supposed to catch this apple, right?

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Now it's actually illustrating a mathematical point. So the every time you catch an apple, you get one more dot in the curve.

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I'm sorry. It's going so slowly. And the interpolate that's there is based on cheryshev technology so that it is a perfectly good interpolate.

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So you see there's nothing really wrong with that curve. And the point is to compare that.

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With what would happen if we did. Equally spaced polynomial interpolation.

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Why am I spending time on this? But having started, I guess I have to.

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So okay. This is relatively straightforward. Now I'm going to do it again, but this time I'm going to do it with the equity flag.

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So I'll say chub fun. Chub snake. Sorry. And what is it?

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I guess. Equity. So this is the same game except.

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Now it's going to be basing the curve on equally spaced polynomial interpolation, which is a very bad method of interpolation.

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You see, I'm not very good at this game. Okay.

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Now, what you're supposed to see is that as we get more and more dots in our in turbulent.

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It oscillates more and more. And if my playing is up to the challenge.

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You will see. That it's pretty dramatic.

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So in fact, equally spaced interpolation has oscillations that grow by a factor of two.

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Every time you add one point. So if I every time we capture one more apple, the wobbles should get twice as bad.

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Let's try to get about three more. Okay. And then we'll move on.

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Okay. Good. Okay.

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We're doing pretty well here. So this is equity based interpolation.

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That's the wrong phenomenon. It's called. And the runner phenomenon says you should never interpolate data by polynomials in equally spaced points.

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Because the oscillations will be terrible. So if wealthy.

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So if Jeb Fund had been based on equally spaced interpolation, we wouldn't have been able to go past degree 15 or so.

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But in fact, we can go to degree in the thousands. Okay.

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So let's go back to F. There was a F. And let's do some stuff with it, for example, if I say some of that.

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Well, in MATLAB, some would add up the elements of a vector.

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So here we do the continuous analogue, which is to integrate over the interval where F is defined.

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So that will be the integral from 0 to 10 of this function.

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F If I say F prime times g. Well, in MATLAB that would be a row vector times a column vector.

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In Turbofan it's a row function times, a column function, an inner product.

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So again, that's computed by an integral evaluated to 16 digit precision.

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If I say norm of F, the idea is if you're a MATLAB user, you automatically know what this means.

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You know, this will mean the square root of the integral of the square.

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Or if I say norm of f comma, one will get the one norm.

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Or if I say norm of f comma inf, we'll get the inform.

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Suppose I say come some of F well in MATLAB comes is the indefinite integral operation, right?

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Is the discrete cumulative sum. The continuous analogue is the indefinite integral.

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So there, there's that. And on the other direction, if I say diff of F well diff in matlab is a finite difference in TETFUND it's a derivative.

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So back to EF again. One of the key things of Tetfund and one of the most useful features is root finding, which is really wonderful.

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So if I say Roots of F, then it will compute all the points where that function goes through zero and return them as a vector.

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Ah, so there's that vector r. And let's guess plot them.

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I could say plot Avar against FLIR as dots.

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So they're the roots. Very quick way to do global root finding on an interval.

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Let's do another example. Let's say Bess equals Jeb fun of Bessel j of zero x on the interval from 0 to 1000.

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Okay, so there's a vessel function. If I plot it, you want to see its roots.

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Okay. Our equals roots of this plot.

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Hold on. Plot R against roots of a sorry against Bess of R as red dots.

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So there you see, those are the roots. And I could zoom in. How does this room work?

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How is it? I'm not doing very well.

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My zooming here. Sorry. Anyway, the roots are all there to high accuracy.

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So Japan has this global feature because things are represented globally as polynomials.

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It means that any operation you do more or less gets the right answer over the whole interval.

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So let's go back to F. And let's make another function.

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I'll say hat equals the absolute value of one minus x minus five over five, and I'm going to plot hat as a black line.

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Oops, I forgot. What did I do wrong?

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I meant one minus absolute on the inside. So plot f I meant hat equals the equals one minus the absolute value of x minus five over five.

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So in the first few years, Chevy Fun was all about globally smooth functions.

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But of course, in applications, people always have piecewise smooth functions.

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So we had to make it so that it could handle piecewise smooth things.

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And the black curve is an example of a piecewise linear function.

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We call that still a step fun, but each piece we call a fun.

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So a chair fun consists of one or more funds that one has to fund.

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And then all the applique, all the manipulations of tab funds that you might expect we've tried to implement.

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So for example, suppose I say h equals the max of f and hat.

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So H is now another type of fun, which should be the maximum of the black curve and the blue curve.

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So if I plot it, there it is, and I can do all sorts of things.

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Now, what's the maximum of H or what's the standard deviation of H or what's the norm

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of H or what's H evaluated at 5.3 or what's the infinity norm of H and so on.

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So all of these things are implemented by algorithms that do the right thing from all the pieces to find the maximum.

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We find the maximum on each piece and then find the maximum of those numbers.

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Or to do the integral, we would integrate over each piece and then add up those integrals.

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So let's do some more like that. Suppose I say HP four h prime equals the derivative of H.

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I could then say extrema. Equals the roots of the derivative.

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And then I could say plot extrema against h of extrema as black dots.

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So there you see, we found all the roots of the derivative, including those where the derivative jumps from one side to the other.

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So here the derivative jumps from negative to positive, and there's a dot.

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There's no dot here because the derivative is jumping from negative to negative.

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So there's no group there. I'll skip a little bit.

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For most of its history, Tetfund was all about cheryshev expansions and non periodic functions.

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We didn't do the periodic case because it seemed too obvious. Everybody knows about Fourier.

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Most people don't know about Gem often. But eventually we realised, even if it's obvious, it's still useful.

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So eventually a couple of years ago we added a capability for periodic functions.

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So for example, if I say E to the sign of pi x, what if I just did this?

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I get a perfectly good function. Bubs.

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Hold off. That's a Chevy chef representation of a periodic function.

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But we could also get a periodic representation. If I say trig, then it's the same thing graphically, but a completely different representation.

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It's now a 48 series. And indeed, if I say plot coefficients of F, then you can see that the 48 series has a zero component there.

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And then as you go negative and positive wave numbers, it decreases down to machine precision in the usual fashion.

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So let me show you again some piecewise smooth jab funds.

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So if we say x equals scribble, there is no fun like jab fund.

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What that does is it makes a jab fun. So let's plot and I'll say plot x axis equal.

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So this is a piecewise linear function, but the catch is it's a piecewise linear, complex function.

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So that's the real part and that's the imaginary part. It's a function of a real variable, as always with fun, but it's a complex function.

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So maybe the first piece would be that stroke of the T, and then the second piece might be that stroke of the T.

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So this has maybe 100 pieces, but often it's just a function.

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So I could do things like what's S at point three or what's the maximum of the real

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part of S or what's the maximum imaginary part of this or what's the norm of this?

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Or What's E to the s? So here we're computing.

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A complex function or what each of the three times as can anyone guess what this will look like.

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I can't hear you. What? Yeah.

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Uh. Yeah.

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So that's a rotation. So several of you said it, but here for you, there's a seven memory stick, eight gigabytes.

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Yeah. So this is just another function.

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It's piecewise smooth, no longer piecewise linear, and each piece would be 16 digit accurate, representing the curve that it's supposed to be.

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It's actually a pretty good way to explore your complex functions.

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You know, if you want to know what does the sign look like or something, you know, anyone know?

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Well, you do that and you can pretty quickly get a sense of things. So what does the hyperbolic sign look like?

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And so on. What does the square look like? Let's try.

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We'll get square s. So lots, lots to play with.

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Okay. Turning over the page now,

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I'm going to show you the stuff that was the original motivation for even though it turned out to be not the most important thing for users.

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So the original idea is one that we mentioned a few lectures ago of taking long skinny matrices and turning their columns into Continua.

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So you imagine instead of an M by N Matrix, an infinity by n matrix, where each column is a function.

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So for example, suppose X is the trivial fun on the unit interval by default.

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And suppose I make a quasi matrix, as we call it, 1xx squared, x cubed, x to the fourth.

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So hopefully you can just from that, see what it's got to mean.

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It's it has five columns and each one is a function.

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So I could do things like evaluate a I could say what's a at point three comma.

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To. Who can tell me the answer to that? Right.

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You get another memory stick. Who was that? That you. Okay. Let's do one more.

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Okay. Let's say. Point five. Karma for that.

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Happy. Somebody.

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Yeah, I think that's right. All right, so I'm running low here, but who was that?

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Sorry, who was that? You. Okay.

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So A is this object which is continuous by five.

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If you say size of A, you get infinity by five.

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So Cliff Muller, the guy who invented MATLAB, like to have fun and this is the thing he likes most.

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Every time he sees infinity by five, he just laughs.

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He loves it because you know, it's such a sweet, natural generalisation of classic MATLAB stuff.

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If I were to plot a Oh, I like spy. If I spy A you can get a sense of continuous by five or if I plot a Well,

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you know, in MATLAB, when you plot a matrix, the default is it plots each column.

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So there you see the columns of A, there's one and X and so on.

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If I do linear algebra, I get the correct continuous analogues.

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So for example, the Q, R Factorisation of A is a product of a matrix that's infinity by five by a matrix that's five by five.

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Or if I plot Q. Well, think about the QR factorisation here we have five functions in the QR.

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Factorisation we orthogonal those functions. So if I plot Q, I'll get the same functions except orthogonal.

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Who knows what those are called?

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They're legendary polynomials because the genre of polynomials are what you get when you are in lies x, x squared and so on.

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And the unit interval. And you can do things like the singular value decomposition of a.

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So what does that mean? Well, you know the picture for the SVD.

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Let's draw that on the board. So you know that if you take a matrix.

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Say an m by n matrix. Then it maps from our end.

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To our M. And it always maps the unit ball in our m to a hyper ellipsoid.

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So origin goes to origin and then the singular value decomposition is a way of getting at the principal axes of that hyper ellipsoid.

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Well, if it was infinity, it's the same thing. So now we have a quasi matrix.

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Whatever our infinity means. So again, it's mapping this end dimensional unit ball onto a hyper ellipsoid.

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This lives in an infinite dimensional space, but it only has dimension end generically.

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So it's a finite, dimensional, hyper ellipsoid in an infinite dimensional space.

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The speed has the obvious interpretation.

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So this here would be the first singular value times, the first singular vector, and this would be, if you like, the next singular value.

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Times, the next singular vector and so on, and their pre images would be vectors over here.

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So everything makes perfectly good sense mathematically in infinite dimensions.

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And chip fun has algorithms that compute that. So the SVR, we just did.

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Those are the singular values, the length of the axes.

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So this is what we were doing when Tetfund was created originally, which is more than ten years ago.

264
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But as far as we can tell, people don't use this stuff very much at school, but who needs it?

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What turned out to be much more important than practice was differential equations.

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So let me show you a bit of that. Let's make what we call a check up.

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So I'll say L equals check up on the interval from -20 to 20.

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So check up, of course, is going to stand for an operator, a differential operator, and let's create one.

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I'll make it define it by an anonymous function, typical MATLAB style.

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So this operator takes the second derivative and then subtracts x times you from that.

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So it's a variable coefficient linear operator in this case.

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Of course we're going to need some boundary conditions. So I'll say l dot LBC equals one held dot RBC equals zero.

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And then in tetfund you can solve this by backslash, just the way you would solve a matrix problem in matlab.

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So if I say you equals l backslash zero.

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So what that's doing is solving l u equals zero for you.

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So l u equals zero would be a differential equation. And we're using whatever algorithm it takes to solve that.

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In fact, the algorithm I can describe easily, it consists of just criticising the problem on a Chevy chef grid of size 17 and then

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33 and then 65 and so on until the constructor gets down to machine precision.

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So it's a standard Chevy chef spectral method in this automatic loop of finer and finer grids until machine precision.

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So if we do that, we have just solved for you. First time you do anything takes forever.

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Let me prove that that's true. Let's do it again. Faster.

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So there we've solved this differential equation. The solution there is an area function.

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You know, this is the area differential equation. And as you can imagine, it's very easy to play with things.

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Suppose I had changed the operator, for example, by putting a point one in front of the derivative.

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So there's a principle of differential equations that whenever you want to make the solution more interesting,

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you reduce the constant on the highest derivative.

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So if I now solve it and then plot you.

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Oops. You'll see it's a higher oscillation, but that's still a polynomial.

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It's a telephone. There's its length. What's the max of you?

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What's you at? -2.3. What's the norm of you?

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All sorts of calculations are in the same fun framework.

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And then you can do other fun, linear stuff. For example, what about the eigenvalues of this operator?

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Now, if you think about MATLAB and Eigenvalues, there's a command eigg, which gives you all the eigenvalues of a matrix.

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Now I give an operator wouldn't make sense because an operator has infinitely many eigenvalues.

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We're never going to compute infinitely many. I was, on the other hand, is the MATLAB command to get a few eigenvalues of a big matrix.

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So we've overloaded that to give eyes of an operator.

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So by default it will give you six of them. And I see because of that small coefficient, it's taking me a while.

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Suppose I only wanted three of them. I could say something like that.

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Let's do another one for fun playing with eigenvalues of Schrödinger operators.

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So for example, here I'm on the interval from -3 to 3.

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I could make a Schrödinger operator and find its eigenvalues. But for fun, we've got a special command that does that all for you.

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So if I say, well, let's say quantum states of X squared.

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Now what that does is it sets up a Schrödinger eigenvalue problem with a potential corresponding to x squared.

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And that computes by default the first ten eigen functions, and then it plots them at their appropriate height.

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So the physicists among you will recognise the spacing one three, five, seven, nine.

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The reason it's .1.3.5 is that I think Planck's constant has been set at 2.1 or something.

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So there you have eigen functions of this operator.

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This one you could do on paper, of course, but you don't have to change it very much to get something more challenging mathematically.

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So there we have the first ten eigen functions of the absolute value function.

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And there you can see it's a different shape and the spacing between the eigenvalues is different.

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If you're a physics type, you can have a lot of fun with double well potentials and illustrate all sorts of things here.

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How many people know this kind of picture? That's enough.

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Let's try to put in a double well, potential. Okay. So we'll say let's see if the is x squared.

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And I say quantum states of V. Then I get that.

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Okay, now let's put a little something in the middle. I'll say the equals.

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The max of itself.

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And then I'll say. Uh, 1.5 minus five times X squared, I'm guessing here.

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Will that be a double? Well. Let's see if I pluck V.

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Does that look good? That's what might work. You think it'll work?

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Let's try quantum states of V, see if it's interesting. Huh?

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What happened? Oh, I said absolute X instead of V.

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Oh, it worked. Okay, so here you have a double.

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Well, potential the lowest state.

324
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There are two island modes here with virtually identical eigenvalues.

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The lowest one would be the blue one, which is an even function.

326
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And the next one is the red one, which isn't on function.

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So there you see the kind of line splitting that you get in the spectroscopy are more or less the same thing here, the lowest.

328
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Well, these are the third and fourth eigen functions.

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One of them is approximately even and the other is approximately the green looks even and the blue looks odd.

330
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And then as you get near the top of the double, well, it begins to be less important.

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And once you get way above it, it looks as if there's no barrier at all.

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So, in fact, if we let's compute 30 of them instead of just ten of them.

333
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That's taking too long. That's annoying. So here you can see that the one kind of physics is happening down here and rather different up there.

334
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As a rule, telephone is supposed to be the most convenient method for solving anything available, but it's never the fastest numerically.

335
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I mean, it shouldn't take 5 seconds to solve that problem.

336
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Okay. Let me show you a little more. For a few years, telephones, differential equations were linear.

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But then we realised how we could embed them in a Newton iteration in an automated way.

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Everything done in this sort of continuous mode. So now you can also do nonlinear things, and I could do that all with check ups.

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But to keep the time under control, I'm going to go into the graphical user interface called Jeb Gooey.

340
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So there we are in February. And what you can do here is type a differential equation and see what happens.

341
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So there are a bunch of demos built in and if I went to the area equation, we'd find the kind of thing we looked at before.

342
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Or let's see, let's go to the an advection diffusion equation.

343
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So here we have you double prime plus U, prime plus U, and then some boundary conditions.

344
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If I solve that.

345
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You can see there's a boundary layer at the left end and then the plot down here is showing you the chubby shove coefficients of the solution.

346
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So that's a polynomial of length, about 40. And those are the Chevy chef coefficients.

347
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Let's make the boundary layer narrower. So I'll change point oh to 2.0 to.

348
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So now again, it's a smooth polynomial, but its length has gone up to 122.

349
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Of course, if the boundary layer were of with ten to the minus sixth, this wouldn't be an efficient way to go.

350
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But it's surprisingly good when you're when numbers like 100 are floating around.

351
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If I made it 0.0002, would that still work? See it still worked.

352
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It's a very narrow boundary layer, but no problem. It's only a polynomial of length 322.

353
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On the other hand, if I tried to make it, you know, semitone scale, we'd be in trouble.

354
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Let's show a couple of other examples, some nonlinear ones. So still in the boundary value problem context.

355
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Suppose I say, for example, the non-linear pendulum. So there you have you double prime equals minus sign of you with some boundary

356
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conditions and you can specify an initial guess if you want for the Newton iteration.

357
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And when you press solve, it goes ahead and does a down to Newton iteration.

358
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In the bottom left you can see the corrections that are being added to the step.

359
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This one has converged and there's the converged solution. And you don't have to do this in the Google.

360
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You can export all this to an amp file which turns it into a Jeb fun command.

361
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There's other fun stuff. Let's do a couple more and let's do a let's do the Lorenz equations.

362
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So here we have an initial value problem. There are three bodies.

363
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This is the classic example of a chaotic system.

364
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So if I solve it, it does it thing.

365
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And there you see a chaotic trajectory and you see three curves down here showing the chubby chef coefficients of the three components.

366
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Or we could play with eigenvalue problems. So for example, here's a famous one or Sommerfeld problem.

367
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This comes from fluid mechanics. It's a fourth order complex generalised eigenvalue problem.

368
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And I believe the demo here is computing 50 eigenvalues.

369
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So this is from fluid mechanical stability for flow between a channel.

370
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It's taking a while. But there the eigenvalues in the complex plane the first, the 50 rightmost eigenvalues of this operator.

371
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In the complex plane, there are the eigen functions plotted all on top of one another.

372
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But of course, you could isolate them all if you wanted to.

373
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And there's also some PD stuff built in here, though not quite as reliable.

374
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So, for example, here's the Alan Kahn equation, which is when we like, it's a time dependent PD so Chevy Chef stuff in space and other things in time.

375
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This is an equation whose steady state is going to be the solution minus one.

376
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And when we make it go, those three spikes will decrease to zero at different rates because they have different widths.

377
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So I think on my computer, this goes very slowly, if I remember right.

378
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That's pretty slow. All right. But anyway, you can solve some Pedes.

379
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So that brings us to about three years ago. And then we've been moving into two and soon three dimensions.

380
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So let me just show you a little bit about have fun in two dimensions.

381
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So for example, suppose I say cheap fun to.

382
00:42:18,550 --> 00:42:21,340
Of a function of X and Y, and there's a trivial function.

383
00:42:24,260 --> 00:42:29,810
And similarly, I suppose I say y equals Jeb fun to the function of x and y, another trivial function.

384
00:42:30,050 --> 00:42:35,540
Now I can play with x and Y by default on the unit square from -1 to 1 squared.

385
00:42:35,780 --> 00:42:43,310
So if I say equals cosine of 20 x for example, times e to the minus five times x squared plus y squared.

386
00:42:46,540 --> 00:42:54,490
Then it's doing a similar process of constructing a finite representation to machine precision of this function.

387
00:42:54,820 --> 00:42:58,270
So if I say plot of death, you'll get some kind of surface.

388
00:42:58,270 --> 00:43:05,230
Or if I say contour of F with 50 levels, you'll get a contour plot with a lot of levels.

389
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And of course, you can evaluate you can say F of zero zero.

390
00:43:09,550 --> 00:43:15,420
What will that be? I have one more memory stick now.

391
00:43:15,420 --> 00:43:28,130
I have two more memory sticks. Who said it first.

392
00:43:29,360 --> 00:43:32,820
Did you already have a memory stick? Oh, okay. Mm hmm.

393
00:43:34,490 --> 00:43:39,300
Um, but then, of course, you can do things like compute.

394
00:43:39,320 --> 00:43:46,650
You can take exponentials and so on. If I take the length of this f, it turns out what that gives you is a certain rank.

395
00:43:46,670 --> 00:43:51,620
This turns out to be a function that's separable into an x times a y component.

396
00:43:51,860 --> 00:43:54,950
On the other hand, if I start computing with it, there's G, for example.

397
00:43:55,430 --> 00:44:02,150
Length of G is rank nine. So this function is represented as a sort of a rank nine approximation.

398
00:44:02,360 --> 00:44:09,170
And the mathematics is very much connected to data compression and matrix completion and all that stuff.

399
00:44:09,350 --> 00:44:14,840
I mentioned this a little bit the other day when I mentioned that Gaussian elimination is now an intuitive algorithm.

400
00:44:15,170 --> 00:44:21,260
Well, Tetfund, too, is using an iterative form of Gaussian elimination to construct approximations.

401
00:44:21,500 --> 00:44:27,600
Let's do contour of G. So it looks the same because it's the same shape here.

402
00:44:31,010 --> 00:44:34,520
And of course you can do various things. So if this is G.

403
00:44:36,200 --> 00:44:44,210
If I say some of, gee, I get a check fund because by default some would plug some long columns.

404
00:44:44,450 --> 00:44:49,610
So if I plot some of G. You get a curve.

405
00:44:49,970 --> 00:44:52,940
On the other hand, if you want to integrate in both directions.

406
00:44:53,090 --> 00:44:58,579
Well, you could say some of some of G and that would integrate in one direction and then the other.

407
00:44:58,580 --> 00:45:08,600
Or we have a built in command some two of G. It's very good at doing things like finding Minima and Maxima.

408
00:45:08,840 --> 00:45:12,860
Let me, however, do this. Explain command. That's kind of new.

409
00:45:15,140 --> 00:45:20,560
So suppose you want to know what's happening when it constructs one of these ten fun tools.

410
00:45:21,640 --> 00:45:30,410
Now. Let's see if this works. This is your Chevy fun movie. So that's the function we're explaining.

411
00:45:35,180 --> 00:45:41,160
First it gets sampled on a grid. And then.

412
00:45:43,240 --> 00:45:50,100
It goes a bit slowly. So then it does this rank one approximation.

413
00:45:50,550 --> 00:45:57,270
So there's a row in a column. So now it's seeing whether F can be represented by rank one, this particular F can.

414
00:45:57,570 --> 00:46:01,050
So it's not a very good choice for the explained function.

415
00:46:01,260 --> 00:46:07,290
We should have done G instead. My apologies. But anyway, it then does this wonderful movie stuff.

416
00:46:09,550 --> 00:46:13,660
And explains how you've thereby reduced your two day function to a couple of

417
00:46:13,660 --> 00:46:17,920
one day functions which are then approximated with turbofan as usual fashion.

418
00:46:18,310 --> 00:46:21,970
Those are some plots of the coefficients of the X function and the Y function.

419
00:46:25,280 --> 00:46:28,490
And then I think it's about to show you that the final representation.

420
00:46:31,030 --> 00:46:35,560
Is a suitable product. But since this is rank one, it'll be a fairly trivial product.

421
00:46:39,980 --> 00:46:49,400
How is it stored? It's stored as two quasi matrices, but these quasi matrices are infinity by one, so they weren't very exciting quasi matrices.

422
00:46:49,700 --> 00:46:53,390
Now, let's see. I'll show you like one more thing. What will I show you?

423
00:46:57,760 --> 00:47:00,280
Well, I'm going to skip to number 14 and just do that.

424
00:47:00,290 --> 00:47:19,720
So suppose I say theta equals 0.9 times pi times x and phi equals pi times x and capital f equals minus one plus point three times cosine of phi.

425
00:47:21,400 --> 00:47:25,600
Stop me if you see a typo, tell me so I can fix it.

426
00:47:26,020 --> 00:47:38,950
Times sign of theta and then I'll say one plus point three times cosine of phi, times cosine theta and point three times sine.

427
00:47:38,950 --> 00:47:42,820
Anyone see any errors here? Hope that's right.

428
00:47:44,490 --> 00:47:49,000
This play. In the first line.

429
00:47:49,240 --> 00:47:52,960
That's a good point. Yes. So this will be garbage. Okay.

430
00:47:52,990 --> 00:47:58,000
Thank you very much. So I should have said five equals pi times y.

431
00:48:00,940 --> 00:48:06,320
Okay. Now, if I do a surface plot of f you see, we can do these.

432
00:48:06,340 --> 00:48:12,400
I need an access. Equal access. Equal can light show.

433
00:48:12,820 --> 00:48:20,660
So we can do this fun stuff with surfaces and through these and we're extending it to methods of computing functions on spheres and disks.

434
00:48:20,920 --> 00:48:23,980
So it's really turned out to be a wonderful organising principle.

435
00:48:24,280 --> 00:48:29,950
Just to remind you, the essential idea is always continuous analogues of familiar, discrete things.

436
00:48:31,840 --> 00:48:36,460
Okay, so that's it. Now, let's see. Mikhail is going to hand out assignment to.

437
00:48:36,670 --> 00:48:39,790
Is that the usual way through him and and through Z.

438
00:48:39,820 --> 00:48:40,660
Okay. Thanks.