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Okay. So today we're going to talk about a lot of things rather hastily.

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And the first is. The question of what is the definition of numerical analysis?

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Now one definition is that it's Professor Profession's field or more broadly,

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it's the field of the numerical analysis group upstairs, whatever that is.

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But of course, it's a serious question, and one reason this has interested me for years is that there's a rather invalid

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definition that people consciously or unconsciously adhere to sometimes,

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which is that it's the study of rounding error is on a computer.

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Now, that's an incredibly boring subject. Well, actually, it's quite interesting, but it's certainly not enough to justify my existence.

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So I think the right definition is the study.

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Of algorithms. Now if we stopped there.

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That would be more or less computer science, I suppose you might say.

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But in particular, it's the study of algorithms for the problems of continuous mathematics.

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And I want to spell out really what continuous means. But the point is it's real and complex numbers, not just integers.

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So for whatever reason, justified or not,

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it's a fact that discrete algorithms and continuous algorithms tend to be carried out rather differently by rather by different people.

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It's a shame that that division is so extreme, but it's a fact that it is pretty extreme.

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And in computer science departments, broadly speaking,

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you find people working on discrete algorithms and numerical analysts are interested in continuous problems.

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So solving a system of equations, solving a differential equation.

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I believe that's what the field is all about. Now, I have been interested in these, if you like, philosophical issues for a long time.

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In the appendix of my book on Linear Algebra, you'll find an essay on the definition of numerical analysis.

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But more recently, one of the handouts today is my essay from the Princeton Companion to Mathematics, published seven or eight years ago.

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And this is a compact description of the whole field.

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And if you have the time, I think it would be very interesting for you to read this.

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This course is, in some sense, samples from this essay, if you like, expanded, of course, mathematically.

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Let me draw your attention to a couple of things in this essay. One is the table at the end.

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Table one on some algorithmic developments in the history of numerical analysis, where I've listed about 30 greatest hit algorithms.

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And I've even gone so far as to list some names of the people who were associated with originating those algorithms.

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We're all human. We like names. To have a name attached to something makes it more full for us.

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But you can imagine it was a bit controversial to make a list like this.

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And of course, I would never claim that the list is exactly right or the names are exactly right.

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But it is an interesting, pretty good approximation to some of the things that have happened in this field over the years.

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More philosophically, back to the definition of numerical analysis.

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The final section of this essay, Section seven, called The Future, is very much about this definition and its implications in particular.

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One of the themes that has fascinated me is that even problems that can be solved with

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a finite number of operations in principle are often not solved that way in practice.

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If the number of operations is n cubed and n is a million, nobody's going to do it that way.

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You use iterative methods like conjugate gradients, even for finite problems.

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That intrigues me. And you'll see that's a theme in this essay.

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The final paragraph of Section seven. Is predicting the future.

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And I think in the seven or eight years since then, some of this has already come to pass.

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I think numerical algorithms more and more are embedded inside artificial intelligence, if you like.

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And it's fascinating for us to watch how things evolve in the 20th century.

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When you computed something that was sort of deterministic, you could run it again and be sure of getting the same answer.

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That's less and less true in the 21st century as more and more things get handled in adaptive, intelligent ways.

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Okay. So that's the end of section 1.8 of the course.

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Now returning to our particular problem of matrix algorithms.

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I want to give you an overview really in the form of a table.

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Of matrix iterations.

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Now the historical overview, as I've mentioned from various angles, is that the mathematical underpinnings are really from the 1950s.

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But this stuff began to be used in the 1970s, and since then it has just grown more and more important.

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Interpretive methods are used all throughout computational science.

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Pre conditioners in particular are just amazingly important.

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Now for my table, I'd like to emphasise the two biggest problems of iterative linear algebra, and those are eigenvalues and systems of equations.

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So it's X equals B, B is given and we're trying to find X or x equals lambda x, which we haven't talked about much, but that's an eigenvalue problem.

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You're given a matrix A and you want to find vectors x and scalar is lambda such that that equation holds.

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And then there are two big classes of matrices and those are simply symmetric and non symmetric.

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So either A equals a transpose or we don't assume that A equals a transpose.

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Of course, if you pick a matrix at random, whatever that means, it will be non symmetric.

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But if you look at applications maybe half the time they are symmetric, they might also be positive, definite, which is a further condition.

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So let me now draw a table, which is merely an attempt to tell you some of the names of the main algorithms.

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A number of which we've talked about, and we're about to talk about one more.

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So let's see. It's a two by two block table.

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And the first column is A, X equals B. The second column is X equals lambda x.

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The first row of the block matrix is symmetric.

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And the second row is not symmetric. So in each of these areas there are several different ideas around which four different problems may be best.

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Conjugate gradients is where we started.

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It's really the mother of all iterative algorithms, and let's remember that that requires the matrix to be symmetric.

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Positive, definite. Actually, it's kind of interesting.

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If you run conjugate gradients on a symmetric, indefinite problem, often it'll do pretty well, at least for a while.

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And then it may fail. If you run it on a non symmetric matrix, it'll blow up properly immediately.

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Then there are other methods, other iterative methods that don't require positive, definite ness.

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And the two famous names are Min Revs and SIM l q.

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So if you have a symmetric indefinite problem, you should learn more about those and related methods.

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Now in the eigenvalue problem, absolutely analogous to C.G. is the method called the Lantos method.

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Of course, land shows, when I say it that way, is not properly pronounced.

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It's a Hungarian name. But the convention in most most people in English, say Lanchester.

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But of course, there are other related things, and a crucial variation on that theme,

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which in practice makes all the difference, is what's called implicitly restarted land shows.

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And it's interesting that mathematically this came along in the fifties computationally in the seventies, but this was more like the nineties.

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That's when a huge advance was taken. And then the other big game in town is JD, if you like, which stands for Jacoby Davidson.

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Now, we're not going to talk about most of these except for the simplest version of land shows.

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But these are big, well-developed technologies.

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And let me just emphasise that it's these two from which everything starts.

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Now non symmetric problems we haven't talked about.

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But just to give you some names, there are really three different mathematical ideas in the non symmetric linear systems business.

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One of them goes by names like CG and LZ Q are.

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But I can tell you immediately what these methods are.

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These methods. Start from the observation that if you're trying to solve X equals B,

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well then you might consider a transpose a x equals A transpose B if you can solve that, of course, it's the same solution.

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Now, this in some respects has a worse problem. The condition number of A is worse.

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On the other hand, it's a symmetric, positive, definite problem.

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So this first class of methods is really all about converting this to that and then using conjugate gradients.

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But then other methods don't do that conversion and the two big classes.

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Well, one of them is called GM Rez, which stands for generalised minimal residual min read stands for minimal residual.

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So I haven't told you what that is, but it's an iterative algorithm invented in the symmetric case,

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which was then generalised to the non symmetric case. The other class of methods is has a lot of names.

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It's really alphabet soup in this business. Kumar and by c g I conjugate a gradient that's called and CGS,

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which stands for conjugate gradient squared and then by c g stab, which stands for by conjugate gradient stabilised and so on.

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When you see all these names, it looks a little untidy intellectually,

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but the fact is these methods are spectacularly important in practical problems.

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Big stab is a remarkable one. I have I got this right in the 1990s.

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The mathematical paper that got the most citations of all was the paper that introduced

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by CGG staff much more important than Fermat's Last Theorem and stuff like that.

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Then of course there are analogues for non symmetric eigenvalue problems, which is a subject I've been very interested in over the years.

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And the basic 1950s mathematics was introduced by somebody called Arno Lee.

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And then in the nineties there's the implicitly restarted version, implicitly restarted only, which made it more and more practical.

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And then there's also a Jakobi Davidson set of algorithms in the non symmetric case.

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Now of course, I'm not really telling you very much about these things, but I want you to be aware of them.

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Let me give you a few pointers to where you could learn more.

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So if you wanted to learn more about the X equals B problem, one of the books on our reading list is by Yousef Saad.

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His first book, if you like, and another one of the books on our reading list is the book called Templates.

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And then similarly for the eigenvalue problem, Saad's other book.

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And the other templates book. So those are some remarkable places to look.

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Also references by Vandervoort and many other places.

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Now in MATLAB, there are lots of codes that do lots of these things, and I'll list a few that I happen to have jotted down.

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You know about PC G, right? So we've used that precondition conjugate gradients is the name of the MATLAB code.

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There's also a code called Minions. There's also a code called Simple Q.

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What else have we got in this business? There's a code called LS QR.

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There's a code called G m rez. There's a code called Kumar.

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There's a code called B, C, D. There's a code sigs and basically stab.

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They all have codes with exactly those names.

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If you want to do. I know the Orlan shows both of them.

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Are done by the code called I was. And just to make sure you're with me on I eggs.

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Traditionally in MATLAB, there's a command I which would find all the eigenvalues of a matrix.

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So and by and matrix and eigenvalues that's an n cubed calculation.

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It's grade event is less than 1000. It's impossible and there's 100,000.

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I is an iterative code which aims to get some of the eigenvalues of a matrix, and we're going to do that in a moment.

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By default, it will get you the six biggest eigenvalues, but of course you can specify you want some other method.

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And I as I circled land chosen are nobody but it's really implicitly restarted so I exist you based on this

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implicitly restarted technology and in particular it all came out of a Fortran package which is called a R pack.

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So when the implicitly restarted stuff was being done at Rice University 20 years ago by Danny Sorenson and Richard Luke and others,

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they made a Fortran package air pack and eventually MATLAB incorporated that.

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It's just amazing. And on your assignment problem for you can use args and just it'll it'll work.

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I think that's all I'm going to say in this overview. Are there any questions?

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Okay. So what I'm going to do now is look at one of these.

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The big one that we haven't touched yet are chose, which is mathematically very close to conjugate gradients,

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but for eigenvalues instead of systems of equations. So let's talk about land chose.

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Land chose was a remarkable man. He was, I think, a Hungarian Jew who was born Levi and changed his name.

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And if I remember right, he went to the U.S. but he had been a communist at some point.

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So this was in the early fifties. It wasn't a good time to be a former communist in the US.

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So I guess he ended up in Ireland after he left the US. So lunch with her Asian.

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1950. I'm a little unclear about how much Land chose and Destination Stiefel talked in those early years.

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They're doing such close things now, in particular remain.

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Let me remind you that in the case of conjugate gradients. We had the so-called rediscovery.

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In 1970 by John Reed or was it 71?

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I forget where he noticed that this 20 year old algorithm was more useful in practice than people realised if you regarded it as iterative.

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Now Ranchos, which is the eigenvalue analogue, had just the same history with a rediscovery by somebody else at the same time.

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And this was Chris Page. Very strange history.

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I think maybe even the same year, plus or minus one, happening within a few miles of each other.

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This was in Abingdon. This was in London. So Heston is in the Stiefel and land in the early fifties probably didn't communicate enough.

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And then these guys 20 years later, sort of rediscover the algorithms and practice again independently.

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Eventually, of course, people saw all sorts of connections.

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So I'm going to give you an idea of what nachos is all about and write down the formulas and then we're going to play with it.

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So we're given a symmetric matrix. Real symmetric matrix.

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Now for the eigenvalue problem.

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It doesn't matter whether it's positive definite when you're inverting a matrix, the location of zero matters enormously.

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But if you just want the eigenvalues, it's all translation and variance. So we don't care whether it's positive.

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Definite. So what Mann shows does.

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In principle, the Land Show's algorithm constructs a tri diagonal ization.

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Of a. And let me spell out what I mean by that.

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It finds a tri diagonal matrix that A is similar to.

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So if T equals an orthogonal matrix times a times the inverse of an orthogonal matrix.

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So that's Q inverse. Then these two matrices, A and T are similar, so they have the same eigenvalues.

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So this is what we call a similarity transformation.

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So A and T have the same eigenvalues.

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And T is tri diagonal. So the idea behind land shows is to take your matrix A and turn it into a much simpler matrix, which we can write like this.

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Alpha one one, beta one, alpha two, theta two, Baidu two and so on.

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Everything else is zero. And of course, as we very well know, when a matrix is mostly zero, that opens up all sorts of other possibilities.

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If you want to find the eigenvalues of a tri diagonal matrix, there are lots of ways to do that with relatively little work.

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In fact, you can you can do that pretty much in oh of end work as opposed to end cubed.

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So if you can change a dense matrix into a tri diagonal one, that's a spectacular improvement.

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Now when matlab computes I it try DAG analyses the matrix and then finds the eigenvalues of the tri diagonal.

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When we do AIG's this iterative stuff, it's a different method of tri diagonal lasing.

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And here's the amazing thing. You don't go all the way.

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It's a million by million matrix. Let's say in principle, we would turn it into a million by million tri diagonal matrix.

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But that would be n cubed work. You don't go that far. You just make it maybe 100 by 100 or 200 by 200.

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By 200. You go partway along. Then you look at the eigenvalues of this far from complete tri diagonal matrix,

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and very often they approximate the extreme eigenvalues of a very accurately.

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It's amazing. So that's the key idea. So the big idea, which if you like, became the standard idea after Paige in 1971, don't go all the way.

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Which would be of m cubed operations.

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You go some much lesser distance, which in a typical application might be on the order of hundreds.

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And then you have a small tri diagonal problem.

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You compute its eigenvalues and you see what you've got.

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So let's say to be very loose, take maybe hundreds of steps.

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Of course, with 100 by 100 try diagonal matrix, you'll never get all the eigenvalues of a, but you can get some of them.

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As you might imagine, the theory of this is a little involved, but it's also very beautiful.

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It's again about approximation, as we saw with conjugate gradients.

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With conjugate gradients,

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we had a polynomial whose job in life was to be small on the spectrum while being normalised to take the value one at the origin.

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Well, essentially the same polynomial comes up here,

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and if I were to draw a picture to try to give an idea of how Len chose finds eigenvalues, let me draw that picture.

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Here's the real axis. Now, the location of the origin doesn't really matter.

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Nevertheless, I'll put it in because it helps us see this as a plot of a function.

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Imagine that your matrix has a million eigenvalues, which are, let's say, located somewhere or other.

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And maybe I won't make it definite. There may be located all over the place.

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So that's a million eigenvalues. Okay. Now, what the Lantos iteration does, it turns out, is much like conjugate gradients.

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It constructs a polynomial, which is small on that set.

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The normalisation is a little bit different, but it's the same business of a smaller polynomial.

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So it magically, in some sense, finds a polynomial that's very small on that set.

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And the reason it finds eigenvalues is that, well, let's see if I want to test a parabola.

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I couldn't do very well. It would be sort of, you know, some big parabola.

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It's normalised by a monarch as it happens.

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But as you get higher and higher degree, you can get smaller, better and better agreement to the eigenvalues.

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And the reason land shows finds eigenvalues is it turns out that the optimal polynomials tend to go through approximately the extreme ones,

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and then they do things like this. So I'm just giving you the intuition.

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The monarch polynomial that is smallest at these red dots has the remarkable

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property that typically it goes right through almost exactly the extreme ones.

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So in this picture. Lantos is doing a good job of finding those three eigenvalues.

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It's an amazing connection with polynomials and approximation and linear algebra.

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Land shows probably didn't view it that way, but eventually that became clear.

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So continual gradients and land shows are both about approximation theory.

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They both converge at amazing rates for certain problems.

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And let's show you that. I need to give you the formulas before we play with it.

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It's a loop, just like conjugate gradients.

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So here is the lunch house iteration.

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You start with a scalar better not which is zero and a vector q not which is zero and a vector b which is arbitrary.

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I'm trying to write it in a format that comes close to what we wrote for CG, and then we normalise B and call it Q one.

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So we take B and divide by the two norm of B, which means that root mean square, the square root of the sum of the squares, of the absolute values.

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That gives us a normalised vector, and we call that Q one.

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And then the loop looks like this four equals one, two, three, and so on until we're happy we set the equal to eight times Q And we set Alpha N.

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Equal to the inner product of Q, n and V.

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We update V by saying V equals v minus.

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Beta and minus one. Q and minus one.

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Minus Alpha and Q in. So you can see here the flavour of a three term recurrence relation, which we also had with conjugate gradients.

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So these are very closely related to orthogonal polynomials.

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Three term recurrence relations. We compute the norm of the.

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Call it better end and then we normalise to get the next Q.

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So Q and plus one equals V divided by its norm.

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So you can see it's a iteration very much like conjugate gradients.

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And as with conjugate gradients, the work at each step is basically one matrix vector multiplication.

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So if A is dense, that's n squared work at each step.

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If A is sparse, it's less than n squared work. And now comes the remarkable step.

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Compute the eigenvalues. Of the NBN tri diagonal matrix.

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Of exactly the form that I wrote down.

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So let's call it T sub n and I'll just call it tri diag.

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But I've already written it down so you'll know what it means involving the parameters alpha one through alpha n and theta one through bad n.

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I guess maybe only beta and minus one comes up. No way that I forget.

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This is a small tri diagonal matrix. You can do that by calling matlab iig if you like, which is using other techniques.

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But it's easy because this little n is a small number compared with the dimension of your matrix.

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And that's the end. You stop when you're happy. And now let's see what happens if we do that.

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So I'm going to run the code M6 under bar lunches.

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If you look at the handout, you'll see, just as with conjugate gradients,

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what I've written is a very pure version of Alonzo's code, which does nothing but this.

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Essentially, of course, real software does much more, and in particular the whole business of implicit restarting I haven't told you about at all.

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But it's very important, very powerful. Okay.

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So let's play with the launch code.

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I'll say a equals diag of sparse of 12999.

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So that constructs a 999 dimension matrix whose entries are just one up to 999.

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It's represented as a sparse matrix in MATLAB. And let's run Lantos on that.

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20 steps and look at the result in its last four entries.

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So it has gone through this loop. It's taken 20 steps and outcome these numbers.

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And of course, we don't really know anything about those numbers. Now, you and I know the exact eigenvalues of either 1 to 999.

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So you and I know that those are not very good approximations to the top four eigenvalues.

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So we haven't had anything like Convergence yet. So let's try 40 steps.

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Well, there's a bit of a hint of convergence at the top. Eigenvalue should be 999.

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So you see we're beginning to get there, but it certainly isn't machine precision yet.

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So let's take 60 steps. So now let's see.

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We've got 999 and 997 and so on.

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It's still not there. So let's take 80 steps.

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And now you see it's beginning to converge. So the top one 999, we've got to about six digits.

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We could keep going. But I'm going to make the problem more complicated.

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So there we've got more than six digits. You can see some kind of convergence is happening.

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And we've taken far fewer steps than the dimension of the matrix.

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Even though that dimension isn't that great. Now let's do a hard problem.

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Let's say A equals a random, sparse, symmetric matrix.

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Using MATLAB is built in codes. Spar around a sim.

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So it's a 10,000 by 10,000 matrix with 0.1% of its of its entries non-zero.

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And just for fun, let's mess it up a little bit.

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I'm going to make the 500 500 entry equal to ten and I'm going to make the 700 700 entry equal to eight.

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Now we don't know the eigenvalues of this matrix,

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but we know it's a big symmetric matrix and probably those ten and eight are going to appear somewhere approximately.

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But there's no way of knowing exactly what will happen.

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So let's try the same old thing.

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First of all, let's take a look at it. You can see it's a.

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Dense matrix by appearance, but not really so dense.

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If we zoom, we'll discover as usual that it's not so dense.

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Yeah. So it's a sparse matrix. So let's do our nunchucks thing.

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Let's say. E equals land chosen.

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I'll do the same thing. I'll. I'll take 20 steps. Sorry, that was 200 steps.

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Well, let's skip that and we'll come back to that. Let's take 20 steps.

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Let's take 40 steps. So let's ask, do we seem to be converging?

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Well, we do. I'm sorry. You can see the 11.717 is looking pretty good.

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9.8 to 6 is looking pretty good. So to the eye, it does seem that we're converging.

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Let's take more steps. Let's take. 60 steps.

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And now you see the strange effect that land shows. Subject to that.

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Actually, there's only one eigenvalue of 11.717, but it tends to give you multiple copies.

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They're called ghosts for reasons, interestingly, related to the picture I drew.

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So you have to be cautious if you're running a pure chose iteration to exclude multiple eigenvalues that aren't genuine.

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Luckily, the implicit restarting does all that, so real software will not have this problem,

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but a pure lunchbox will have this problem if we keep going, if we make it 80 or 200, as we did a moment ago.

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So there you see we've got two copies of both of the top eigenvalues.

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Let's look at and minus ten. What have we got there?

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We've got two copies of the top eigenvalue, two copies of the next one.

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And then who knows, let's take 200 steps.

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So there we have six copies of the top eigenvalue and four copies of the next eigenvalue.

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So you see, a pure Lantos iteration is great for finding one eigenvalue, but not more than that.

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But once you do the restarting, it's just spectacular. And in particular, if I say I have a let's put a tick tock around it.

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So just to remind you, this is a 10,000 by 10,000 matrix.

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That's not all that it doesn't have a structured sparsity.

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It's got a mess of a sparsity pattern. So in 2.6 seconds on my laptop, it has now found the two eigenvalues we keep seeing,

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11 and nine, also some extreme negative ones, and then the next two.

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So these are the six extreme eigenvalues. And by the way, we could, if we just want the positive ones, of course there are ways to do that.

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I could say eyes of a six.

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And then I believe you say L.A., which stands for largest algebraic.

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So this will now find the six largest algebraic. Or I suppose I want to add more of them.

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I could say 12. So this implicitly restarted land shows.

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Technology is quite amazing. That's all I'm going to say about it.

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Any questions? It's a little surprising how late some of this developed historically.

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In principle, the seventies could have done this, but it took people a long time as problems kept getting bigger and bigger.

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I guess the incentive sort of steadily became stronger over the years.

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So what I want to do in the last bit of this lecture is to show you the other handout,

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the numerical software tools handout, and we're going to play with that online.

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So this is officially section one, 1140.

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The. So of course, you all know how to find things on the web.

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Better than I do, I'm sure. But.

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Maybe you're not so aware of the richness of some of the stuff that's out there related to numerical computation.

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And the goal here is to make you aware of this list of things.

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That is an interesting starting point for seeing many things that are out there.

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So we have about 10 minutes and I'm going to click on 15 or 20 of the things on this sheet of paper to give you a sense of some of what there is.

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So we'll go to. There.

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As you can see, I've organised in various categories.

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Now when I was your age, software libraries were very important and the things were collected together and you use the library.

324
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They're much less important than they used to be. I think things are done more ala carte, I guess you'd say.

325
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For example, net lib was a wonderfully important thing that came along even before the web for collecting numerical software.

326
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It's still important, but not not as important as it was. If you go there, you'll see an old fashioned feel to it.

327
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Perhaps it's but it's it's important. It's hosted at Oak Ridge National Laboratory and the University of Tennessee.

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I won't mention all of these things. Traditionally, in the early days, a lot of this software would have been in Fortran and then C became important.

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Those have certainly been the dominant languages of numerical computing over the years.

330
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Of course, a lot of things are done in MATLAB and Python and other languages too.

331
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Fortran 90 was a big change to a sort of more matrix oriented way of thinking.

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And then, of course, you all know probably better than I do about GitHub,

333
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which is the absolutely huge system for storing open source stuff on the web.

334
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And for example, our software project TAB Fund is all hosted on GitHub.

335
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I'm curious how many people here have used GitHub? That's impressive.

336
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And I'll bet even last year it would have been half as many. It's just exploding.

337
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I haven't checked on the web lately, but the number of projects hosted there is spectacular and you know, it's only a few years old.

338
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It's just amazing. Okay.

339
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But of course, GitHub is not numerical in general, although there is some numerical stuff there.

340
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If you go to information portals, the next category here.

341
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For people like me in the profession of numerical analysis, there's a weekly digest which means a lot to us.

342
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These other things are also important in their way. Let's click on the one called S.W. Math, which is relatively new.

343
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This comes out of the people in Germany who run Central Blot, and it's a software search engine, which seems to be pretty interesting.

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I've only played with it a bit, but this is relatively recent and I think if you're interested in finding things, it you may find them there.

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Let's just try something at random that I haven't tried. If I try, well, let's say fast for a transform.

346
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Let's just see what happens. I don't know what will happen. So it gives you a lot of information about various things out there.

347
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I don't know which hedge fund is there that was not intentional, but that's mine.

348
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Interestingly the most famous of their 12 pages. I see.

349
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But on each of these projects you could then find much more information.

350
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And it's pretty impressive. By the way, I have some sort of virus in my computer, so occasionally odd things happen, and that's more than most.

351
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I hope we're all right here. Yeah.

352
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See, I get ads for things that I'm not supposed to get, right.

353
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Let's hope we can recover here. So far, all of the ads have been perfectly clean.

354
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But, you know, I expect any morning that the virus will have mutated if you go to numerical analysis journals.

355
00:41:34,680 --> 00:41:38,730
I've already had a hand out on them. But of course, these days everything is available online.

356
00:41:39,000 --> 00:41:44,129
So that's, again, a list of some of the ones that mean a lot to us. General purpose libraries.

357
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I mentioned these were much more important years ago than they are now, but still there's some impressive stuff there.

358
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Wilkinson Prizes. This is a very interesting one. 20 years or so ago, a prize in numerical software was founded.

359
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And it's I think it's been awarded seven times and they're all remarkable.

360
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So these seven projects are really very special things and differential algebraic equations.

361
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Automatic differentiation, the fastest 48 transform in the West Triangle mess generation deal to finite elements

362
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IP opt for interior point methods and most recently one by a guy here in our building,

363
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Patrick Farrell is Dolphin Adjoint, which is an automatic system for generating add joints of finite element models.

364
00:42:26,040 --> 00:42:29,970
They're all interesting and worth looking at if you're in those areas.

365
00:42:31,450 --> 00:42:36,100
Now, of course, sometimes we do extended precision or symbolic things.

366
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Maple and Mathematica are the most famous in that category.

367
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Some people love Pari. MATLAB has its symbolic math toolbox, which is okay, but I think you can do much more in Maple and Mathematica.

368
00:42:50,240 --> 00:42:59,850
Well, lots of things there. Linear algebra has always been the most developed part of computational science scientific computing.

369
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It's extraordinary what is known there.

370
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In the seventies you had these packages Ice Pack and Linpack, which really created the modern software business, and then they mutated into L.A. Pack,

371
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which is has ever since been the standard scaler pack for parallel computing air pack, which I mentioned for eigenvalues.

372
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Here are the templates books. In ordinary difference differential equations.

373
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I don't have much to mention, although cab fun is very nice in pdes.

374
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That's a huge field of course. And different packages tend to be good at different things.

375
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So one that's very popular for conservation laws is claw pack or for geo physical conservation or geo claw and so on.

376
00:43:48,170 --> 00:43:59,240
So that's a nice example of a system on GitHub. Open Source APL Htmg is an ancient but still evolving educational tool for finite elements.

377
00:44:00,320 --> 00:44:04,430
What else comes on? Multi physics is commercial and though in principle we would all be using it.

378
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I don't know many people who seem to use it who uses comsol. That's interesting.

379
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It's a big company that does a lot of good stuff, but not in our world, I guess.

380
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The MATLAB PD toolbox hasn't been touched in 50 years, so it's a piece of ancient code that still some people like because it's available.

381
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If they've got MATLAB going on to optimisation, we're going to be talking about that more.

382
00:44:26,160 --> 00:44:32,580
So I won't linger on this, but that's a very important, well-developed field with a wonderful online interface.

383
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So nice. For example, the server in the Midwest in the U.S. is a very important resource for optimisation.

384
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Other numerical topics?

385
00:44:42,910 --> 00:44:49,479
Well, I've certainly mentioned the top 500 supercomputer site, which is interesting, the digital library of mathematical functions.

386
00:44:49,480 --> 00:44:53,230
So every few years the US government shuts down or nearly shuts down.

387
00:44:53,230 --> 00:44:56,470
It's about to happen again. And a couple of years ago it did shut down.

388
00:44:56,830 --> 00:45:02,620
And you might think if the US government shuts down, how does that affect us here in the Numerical Analysis Group?

389
00:45:02,650 --> 00:45:09,820
Well, the answer was two things. One is that one of us had planned a holiday in the Grand Canyon and had to go to Utah instead.

390
00:45:10,510 --> 00:45:12,480
The other is and this was the killer,

391
00:45:12,490 --> 00:45:19,150
the online digital library of mathematical functions was no longer accessible for eight or ten days, and that really was annoying.

392
00:45:19,510 --> 00:45:22,720
So the US government does do something useful and there it is.

393
00:45:28,660 --> 00:45:34,330
In CFD. Traditionally, a lot of the best stuff is commercial and you have to pay big money for it.

394
00:45:34,690 --> 00:45:40,630
I guess fluent, for example, is in that category, but there is some very interesting open source stuff.

395
00:45:40,840 --> 00:45:45,190
Phoenix is a very well developed one, and then lately one hears more and more about open phone.

396
00:45:45,340 --> 00:45:48,490
Who's used Phoenix? Who's used open phone.

397
00:45:49,370 --> 00:45:55,130
Interesting. Okay. The Dolphin Joint Project is in the Phoenix setting.

398
00:45:56,270 --> 00:46:00,229
Statistics is, of course, a huge, huge, important subject, which tends,

399
00:46:00,230 --> 00:46:04,130
for whatever reason, to be a little separate from the rest of numerical computing.

400
00:46:04,310 --> 00:46:07,219
But Oxford is very much involved in this through R in particular.

401
00:46:07,220 --> 00:46:12,950
Oxford is one of the big are developing places are as an open source statistics package.

402
00:46:13,130 --> 00:46:16,520
Most of the ones listed there are expensive, not open source.

403
00:46:16,670 --> 00:46:21,230
So who's use are? Oh, that's impressive. And even though who's in statistics?

404
00:46:21,620 --> 00:46:24,949
Yeah, I thought so. So many of you have used our. Even though you're not in the statistics.

405
00:46:24,950 --> 00:46:30,080
That's good. Certain things can be done on the web, I think.

406
00:46:32,120 --> 00:46:38,299
Predicting the future is always difficult. We now live in an era where everything is stored on the web, the cloud.

407
00:46:38,300 --> 00:46:41,540
You know, nobody saw that coming. But suddenly everything we do is stored on the web.

408
00:46:42,140 --> 00:46:46,010
Whereas 15 years ago, people predicted a lot of computing over the web.

409
00:46:46,010 --> 00:46:52,340
And of course, that is also there, obviously. But somehow it's the storage that has more changed our lives than the computing.

410
00:46:52,580 --> 00:46:57,250
Especially at the high end numerical level. But there are some interesting things.

411
00:46:57,260 --> 00:47:04,280
Of course, Wolfram Alpha is one that we all know. Just to mention some others, more mathematical.

412
00:47:04,490 --> 00:47:10,850
If you have a sequence and you want to find what is that sequence, go to the online encyclopaedia of integer sequences.

413
00:47:11,360 --> 00:47:18,439
If you have a number and you want to know what is that number, you know, 3.141592653.

414
00:47:18,440 --> 00:47:22,700
What is that number? Then try the inverse symbolic calculator.

415
00:47:29,820 --> 00:47:35,400
I'm going to talk about Chevy Fun at some other point if you're interested in Schwartz, because tarsal mapping, there's a beautiful package for that.

416
00:47:35,910 --> 00:47:38,879
High performance computing is, of course, rapidly evolving.

417
00:47:38,880 --> 00:47:45,570
And I see here I don't even have anything mentioning GPUs, but of course GPUs are a very important part of that these days.

418
00:47:45,990 --> 00:47:51,000
Traditionally, they're the two paradigms of high performance, which is shared memory and distributed memory,

419
00:47:51,000 --> 00:47:57,030
and there's open MP and MP for those graphics and visualisation I don't know much about.

420
00:47:57,030 --> 00:48:00,150
There are a few commercial things, some of which are then gone. Open source.

421
00:48:00,840 --> 00:48:06,300
Many people end up using MATLAB in practice, but of course for high end stuff, there are other packages out there,

422
00:48:06,480 --> 00:48:10,670
so please take a few moments and play around and click on some of these links.

423
00:48:11,040 --> 00:48:12,210
I guess that's it for today.