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COSC 4364 Numerical Methods - Lecture28_Review
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03:06 took off your mask. I did recognize you for one second. What
03:10 that? What? Oh, I . I took off my mask.
03:14 . Yes, I did. So say, you know, I noticed
03:18 everyone here was without masks. I it's okay. I'm on without massive
03:22 what what what what? Right. , so let's see what they're professor
03:39 you be? Uh boston. We do that. Yeah, I haven't
03:48 it, but so well you can't you another. And then also,
03:59 , can you do an exception for more? My consultant? Yeah,
04:05 no problem with no more homeworks. don't know why it does not move
04:23 a little bit. It doesn't What's the problem? Mm hmm use
04:33 this, this bottom. Yes. . Yeah, it's not bad.
04:45 . It's all right. Okay. guess zoom is showing a different
05:02 Mm hmm. Yeah. Is this one and the other? Yes.
05:22 . I think the whole Okay. mhm. Projects. Mhm. Mm
05:48 . That's right. Really? Mm hmm. That's true. President
05:59 for our I'm just asking. He yes, that's fine. Yes.
06:06 just asking. Yes. Yes. what's the number? What's the day
06:13 to? Okay. So any comments questions about? No. Yeah.
06:33 a number of factors that and just it apart from, sorry,
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06:46 , most of them are teaching 100% of teaching backwards what most of us
06:56 they are combined supporting research and All right. And they have a
07:03 high teaching role then on top of . Thank you. Thank you so
07:11 . So, All right. So you're basically going over what we have
07:18 about since victor and remind your but hopefully about it right on friday
07:28 , um, here in the So this is kind of the topics
07:35 covered after the they turned more after spring break. So I'll talk quickly
07:43 the highland, but I think it's that I hope they will remember.
07:48 enforcement and I guess for, so , so in terms of ordinary differential
07:55 , it's basically things you get something in a differential form. So things
08:03 best play models and whatever things are to model in terms of the rate
08:09 change. So there's the things are in terms of their images um and
08:15 whatever the function is that drives So then we have this kind of two
08:24 but they all kind of follow this and best to go back to basically
08:33 turned this differential equation into kind of integral the question instead. And then
08:41 can use whatever they talked about in of how to do integration. Um
08:49 there are some of these where the rolls up, roll all kinds of
08:55 rules to do integration and the two um in addition to this model or
09:04 the differential equation into the kind of problems is how this integration is supposed
09:13 be carried out. So one set methods they use also the endpoints basically
09:24 values is of the integral at the plus H and that makes it that
09:31 trying to predict whatever the outcome is X. At the passage. So
09:37 of 19 building here on the right side, that means you have to
09:42 the system of the question drags and it out. And that's what these
09:46 methods are being cool. So at end point in this instagram, whereas
09:53 explicit methods only use dollars that already or do some other tricks in terms
10:00 all, if I have anything this . so point is understand the difference
10:08 explicit um there are examples and to known this is that's good to
10:17 But practically this is what you should for sure that explicit methods it doesn't
10:27 So that means we don't depend on the system arbitration to find out what
10:33 next predictive value is at some future in time or whatever if it's face
10:40 time. Ah So and these were on that kind of sampling what the
10:50 might be or or how X. involve between T. And T.
10:54 H. So the kind of guests sample what it might be and then
11:00 it was suggested doing away for some these guess work and trying to
11:05 Yeah. XT passage The different presume was to was this sort of splash
11:14 where am on not just instead of guesswork things may evolve. You actually
11:21 what you have learned so far so different times or whatever T. Represents
11:29 earlier points. So then about being polynomial approximation of the history and then
11:36 used that to predict what it might our people are safe. So it
11:40 different design principles and then there was similar method to this and especially the
11:49 included history to try to make it prediction but also including the future
11:57 So that's why I became an anticipate . And so again the design principles
12:06 they are, it's good to even if you use packages because if
12:12 have this method that depends on the , obviously when you started for your
12:17 condition, you don't have a We need something else to get something
12:23 to have this type and then you use system. And the other thing
12:29 that one determines the approximation order. coming back to the same way we
12:35 about throughout the course. It's just out how many terms in this.
12:40 a serious expansion that's covered. But that's something amusing that determines the
12:46 So that's very consistent with a So next time you hear it was
12:54 Basically how you sample the space and upon seven points etc. And how
13:00 weight the difference sample points together. got going to cut the methods of
13:06 orbits. Um, so one thing I think a good thing to think
13:16 is why use one of these instead it. If everything is the government
13:21 terms of goodness or order, why just use the very serious expansion.
13:28 my answer is that the thing is you use taylor series expansion, that
13:34 you need to know we'll be able evaluate derivatives because that's the taylor series
13:40 . The function was an increasing number of derivatives and sometimes you may not
13:48 been added to formulas, you can in and statistically evaluates derivative. Whereas
13:54 roman total methods on reduced function and from the first part of it before
14:02 break, numerical differentiation is subject to of amplifying errors. So numerical differentiation
14:11 hard on your body avoided. So one of the reasons why you want
14:15 use methods and only more fun evaluation function and outfits. Okay, something
14:23 just. And the other thing like talked about the Fortuyn in terms of
14:27 method, it's the same thing. higher order methods gives you sort of
14:32 bang for the buck in the sense we get higher order accuracy as for
14:37 same number of function evaluations. So we went through also below the carcass
14:44 we talked about I guess the most methods but the point higher order method
14:49 preferred because you got higher wonder higher for even the number of function
14:58 So that's the point my mom wants do. Right. So whatever in
15:02 after class. And in general the order methods are not very much used
15:08 practice unused is higher than other And then it's up to you how
15:14 in order you want because they get complex that I ordered this. But
15:19 the 4th and 5th order I was common practice and after methods is prettiest
15:28 forward but in order to make them , you're always in in some form
15:32 estimating the error but it should and were two different ways of finding estimates
15:38 the enter and based on that you decide whether you can and when you
15:45 something you like in the integration whether can some of the in this case
15:51 step size that is called from you further in time. So instead of
15:59 d plus age For whatever age is maybe keep us 23458. So the
16:05 time you can go and younger brother have fewer evaluations and accuracy. So
16:11 that what? But you need some of estimating your parents error in order
16:17 figure out whether you can increase step or you need to reduce the step
16:22 because they are animals too. So that's what I focus on the design
16:30 except remembering exact about methods of what's basic idea wow and we never
16:38 So then there was this other kind Class or better consumptions or the different
16:47 combined two things. So I said wrong Dakota and oil and all these
16:54 not because they are acceptable predicting the and I believe that your prediction is
17:02 , don't worry about it. Um or as this at us as for
17:11 kind of cheat a little bit but molten was this implicit ones so it
17:16 use of the thing on the right side, so to speak. What
17:22 supposed to compute so you don't know your guess. Um and then you
17:26 to solve the system on the So in this correct predictor corrector methods
17:33 you can combine the prediction by and using the prediction as it was.
17:39 fact the action Um final solution at so you know kind of iterated to
17:47 her for the thing but instead of the old value X. R.
17:52 . You now use the predicted value hopefully is very good estimation about the
17:58 . It's so So this is in sense one iteration. Well and that's
18:03 we kind of call that correction. have this predictive value and then you
18:08 it if it was actually refining outcome get them gold. So so that's
18:13 notion and predict the correct let's see and then there are other things that
18:19 the quick so wants to know what do and it's simple single first order
18:26 the lead on this expanding possess then you have four variables so you have
18:35 system in this case then you can turn it into basically renaming variable
18:42 collection of them. And then you the system of first community differential equations
18:48 be fooled by the symbol sex or . Just make it kind of a
18:55 component but for all the variables that have and then it just turns into
19:00 major explosion of what they have. and that was that and the same
19:06 one goes to higher order, uh . Police permits to get value
19:12 derivatives. Yeah. Then you basically new variables again and then so you
19:22 this to turn it back into the of first of all the differential
19:26 And so kind of the workforce is figure out how the first of the
19:31 equations for a single variable. Um it into a vector or matrix versions
19:36 it and then you can use similar if you have by your own
19:43 So this that's going to basically be hard with marriage these things. one
19:50 that is our difference quality, nature constant over the east that uh you
19:57 be familiar with them because the severely are what the teams you can use
20:05 how they behave and in this case what it is means to have a
20:11 range of time scales for space scales the same problem. So this is
20:17 to illustrate about these things. You certain things that happened fairly rapid.
20:21 kale and we have something else That's longer. So that's fine
20:31 So the two large step sizes for , miss all this finding of rapidly
20:36 things. Um and on the other , if you take very fine step
20:43 system, make sure you don't lose fellows, then the solution times depends
20:48 large because the overall solution also depends the other underlying gardens don't dictate
20:54 So that's why this kind of difficult manage efficiency competition early because of
21:01 Big range of time scales and that's stiffness referred to. The big range
21:06 plank scales or space scales. And there are special methods that we did
21:11 talk about to try to figure I have to manage these things without
21:16 exceedingly small time steps to cover that funds correct and saying that they were
21:22 solution for it. So just now stiff means I'm very nice. If
21:32 ever happen to have such a go and find their proper matters to
21:41 . So any problems you just interrupted questions. So the other thing talked
21:46 the most it talks about calcium elimination is principal and unstable solver unless your
21:58 that stabilizes the scent elimination. So are others that are not data
22:10 So in this case doesn't elimination means need to go look and figure out
22:15 which war column or elements to choose traveling. Um So otherwise you may
22:23 up divided by zero, divided by small number of things goes up.
22:28 that's the problem with Justin elimination. these others are called stable methods don't
22:33 that problem. And talked about householder and givens rotations. And the householder
22:43 work this way and go back and at but especially zero is out and
22:51 elements of a column. So I I should have a column before we
22:55 without a is the vector that has Lecter and the point is to them
23:03 transformation on that vector such that only element is non zero. So this
23:12 of does the same job as one in the Gaussian elimination and zeroing after
23:18 taken all the diagonal. But the is that this householder transformations that kind
23:24 formed this way where the is related a down here is that this matrix
23:34 guarantees that it's kind of unitary or . So it's scale preserving so things
23:40 never going to blow up. That's of one of the benefits from this
23:45 the transformation And come back to that remember that was also used 10 years
23:52 and I am value solvers and singular decomposition and because of its nice property
24:01 it may not be the dominating thing solving linear systems of equations but if
24:05 being used the reason why it's not much used and so on and then
24:12 just some stuff in question is it's choir is about twice the amount of
24:16 work compared to boston elimination but um particular is the work. Um Alright
24:25 computers, clusters and the problem is . The fact that householder transformation has
24:34 data independent control flow. You don't to search through the whole bus to
24:38 out what to do next. Ah may not matter too much in a
24:43 note because it's just local memory references if you have large problems that requires
24:49 of the nodes these methods are Are you sure about that moment?
24:56 generally use. So it was just of the same thing, how it
25:01 . The end result is that to what's known as and you are factory
25:07 and the Q comes from the general . The symbol for MH 67 is
25:15 normal. So that means the columns the matrix is proportional to each other
25:23 each one is kind of unit language and r is enough for triangle
25:30 So this thing also works if it not the square matrix for any matrix
25:37 that was this used for instance and of the least squares method we talked
25:43 they can solve the first problem. by doing the normal questions to the
25:48 characterization of the majors itself and I'll back again. So what And then
25:54 was giving his petitions that is best them doing one element at the time
25:59 supposed to call actor at the So then it's an answer questions and
26:04 what he sees the length of the sign and sign. Ah let's use
26:11 let's provide you this house some of circle. What kind of course and
26:20 general is that? What it You don't have to buy two problems
26:24 stick the call sign and sign in right places to syria what the particular
26:29 are using and it just works on them to roll this month and
26:34 a second Role here and this one the 4th round. I wanted to
26:40 the rest of it is unaffected and same works on the same set of
26:44 . So you're only change things in worlds. So that's what's also used
26:51 somebody arguing about methods later on and to make that case load things onto
26:58 diagonal from off the um, direct that are not talked that much
27:09 Well, I want to at least remember what the goodness and a little
27:12 what they want. It does have . Um, the thing that they're
27:22 and then it looks not just America and computer science books. Trying to
27:25 you something about compilers, something about enough to tend to use the hallway
27:32 as a model, Um, how operate on the race of more than
27:37 dimensions or one dimension. So you see the covid situation in many different
27:44 . It's not the picture of the , it would be a method for
27:48 systems of equations, but structure is simple and if you're not all that
27:56 about computer time, that's fine. easier to golf. The other one
28:02 basically, it is kind of bad step method. They have a complete
28:10 interests or whatever it is and To the new one, using a formula
28:15 this. Um Now, since Many you don't compute them all at
28:23 You compute them 1x1. And that when we go through the process of
28:28 components of the vector here, you know some um The asylum method was
28:35 taking advantage of the components or that confused and then I will try to
28:41 what's known as successful relaxation and that they kind of do the weighting of
28:48 and I would estimate in computing. the next step that suggested that the
28:55 of the whole thing when these methods as starts actually has come out the
29:01 with two anniversary that this iteration, G on the right hand side and
29:07 properties of the chip, the situation , it's the things that governs the
29:13 rate. Um If you have, that means expect radio something called this
29:23 . When we talk about ideas, the best of the expected value,
29:26 maximum value of this Iteration Matrix is than one. Then the method converges
29:33 of the stopping factor. If that's true, it may seem convert but
29:38 not counting. So that was that test computing island values is kind of
29:50 . So simple ways of finding out your method may work is to use
29:56 proper bag on the dominance. So means if the absolute value of the
30:01 element is larger than some of the values of the off diagonal elements also
30:08 wise, then they started that dominance then the convergence is fine. It
30:14 the same thing kind of before the elimination or their composition. If this
30:23 true I wouldn't need to worry about this is just a little bit more
30:30 this together in terms of formal to uses es war methods than the legislation
30:37 the the weighting of old versus new to be in this trend to guarantee
30:47 . But the dominating method is what's as safe for solving systems today is
30:54 the contrary, greater method that has to have I would say a zillion
31:00 versions um and the point is on steepest descent that thing. And I
31:08 seen when it's kind of tends to popular in many areas of the steepest
31:14 and methods. Um The cardigan very method instead of using thanks fewer maximum
31:25 , trying to be clever and get the point a little bit faster and
31:31 doing that by incorporating the properties or matrix. One is underlying the questions
31:39 you're trying to solve. Um So factors in it and reforms this kind
31:47 a in a product that is So when you look at them in
31:51 products that is being used to help directions, it's based on the property
31:59 system you're trying to solve, it's kind of fine but the standards graduate
32:07 by forming making these support a congregate ensures that they are basically orthogonal to
32:13 other in the A in a product , that means the form a basis
32:20 anything known at some certain point Okay. And this is for you
32:27 through all the different situations, kind . But then you can basically express
32:35 director in terms of this continent So that's one in this case.
32:41 . The solution can only be expressed it's conjugal factors as well as in
32:46 standard partition. That's when they And so then it ends up being
32:53 of an iteration formula varas you've got thirds or estimates of X based on
33:02 direction of this next can you get ? So this one has all the
33:08 conduct that factors in it. So is just adding sort of part of
33:13 space that wasn't previously covered. Bye and waiting between the old and the
33:23 inspiring at least. That's the way a product face petition. And then
33:30 is the updating how we find the community factor here based on what's known
33:35 the residual. And the residual is how far what is kind of the
33:41 And supposedly you're going to solve X B. So there's a deal
33:48 It tends to something about the So the area forms the basis to
33:52 out what the next and their action should be. And in the end
34:00 not that complicated that confidence is So the whole thing is one matrix
34:06 product, which is this formula A. And that's related to the
34:12 . In the product and then there's rest of it was just assembled in
34:16 product. So and then there is scaling on an additional director is something
34:22 . So the computations are quite mostly in a problem or breakfast
34:30 And this is the most expensive. is a major part. And that's
34:34 have in the simple Colby and both down, it's just um, the
34:41 is how you handle the major spectral . So the additional work is very
34:46 compared to the it's very simple and methods. And that's why again,
34:53 has become the dominating methods for solving by objective methods. And yes,
35:05 didn't put it on the side. the problem from here. So the
35:11 we're using entity methods is I would mostly, yeah, when we have
35:18 matrices, all the storage requirements are from related sahar. When you use
35:27 direct method, whether you use housing or householder or give us methods,
35:37 if they were starting matrix, I to work with, remember your industrial
35:43 , you form mnu by doing linear , rolls to get new things And
35:50 means the non-0 solution in the same . So things that used to be
35:56 when you live in the combination of roads comes down zeroes. So that
36:01 the major cities tend to fill up if you have big problems, people
36:09 major cities, there are hundreds of of rows and columns and initially I
36:15 we have tens of elements in each or column. So it's a huge
36:24 between heaven. I said £10 100,000 zeros versus have 100,000 square zero the
36:33 . So that's why in terms of , 8 30 methods are very much
36:39 when it comes to problems of enemies escape. The other part is before
36:45 that they do provide approximations to the . You don't have to go through
36:50 you so um for two or 300,000 and columns, but you have to
36:57 in Gaussian elimination. So a large of steps to find a solution And
37:04 interesting that you might do, you , even for a very large,
37:08 just maybe 200 or 1000 steps. good actress before we get to
37:15 So the number of steps to get good approximation and the amount of memory
37:20 the texas kind of the best using acceleration. All right. And then
37:29 something you probably already knew that before class of different kinds of norms.
37:34 again, we should be familiar with vector norms, what they are and
37:39 their company Constructed in the 17th. the L one norm ist just sum
37:46 all the absolute values. So we kind of traveling statesman problems if they
37:54 them extra. The second person is to Yes, one should change
38:00 Not travel in the same small. , alright. So let's just sum
38:07 all the next and whatever it from A to point B. That's some
38:11 the other one. And I'm not . The Euclidean norm that is specially
38:18 . Look at the sum of And this is the dominating, I
38:24 say the one that is being used it and certainly the one for this
38:30 and some sort of squares, we to make the most of someone
38:35 So that's where this is where I used in this class. But there
38:40 also other ones where we'll make potential focus on their macs and it's
38:48 You want to minimize the maximum possibly. So, there are different
38:53 about understanding That's between now and one 2. No, and then one
39:02 of based matrix norms and a little harder perhaps to get intuition about how
39:08 work, but they're kind of based the veteran norms. And it's kind
39:12 that's to say the normal way is on matrix right there, brother of
39:17 matrix A with the vector X. you put constraints on the vector vector
39:23 . That you're used for finding the . So they that Director X.
39:30 hard, unit length in whatever normally from the other one that's the
39:37 Ah and then it carries over and we get something related to the
39:42 Matrix. Otherwise it has similar properties So the vector normal center.
39:52 I don't think that's more of a . And this is kind of what
39:55 get in terms of the military. make that turn on some of the
40:02 . In some cases it gets for . Well some uh right from some
40:08 in some cases together ah call themselves and we have to know and then
40:15 you can expect on spectrum and also then remember this important part in condition
40:23 whenever that is something you should definitely because that's important when you solve whatever
40:30 of equations because it has gives you idea of errors that you make and
40:39 expect. So if it's a large number here they lose a large number
40:46 digits. And that was statement early I think in the first part of
40:51 course and I think you also showed example in the past, Simple thing
40:57 did in the first very first class looking at doing things for three or
41:01 significant digits or accuracy. And then we talked about, I am banished
41:09 could relate to the fact that things totally inaccurate to the fact that this
41:15 that particular example had a very large essentially the ratio between the maximum and
41:23 IBM value that tells you something. ? So this is what I like
41:32 sent. So condition numbers important properly unfortunately places of computing. Canadian
41:42 That's kind of an expensive way of things out. Um So for that
41:48 have Oh right, I'll come to . This cash foreign. Let us
41:55 one simple way of getting some idea they're hanging back off. So maybe
42:02 to that and they basically properly, know, what is an argument of
42:07 problem. And it's defined is that vectors that basically gets mapped back onto
42:17 and even make expected modifications. We this matrix factor thing basically we got
42:24 scaling the Russian director back now the factor is this set of values and
42:36 are your values are very useful I showed him the bridge example that
42:45 , it relates to systems. If models this the system that may not
42:52 perceived as a dynamical system, you probably think of the richest dynamical system
42:58 pretty much all structures have some They're not totally rigid. So then
43:04 tells you a little bit about what I get most basically Restaurant three Places
43:11 the structure and as I mentioned, don't want to put energy into restaurants
43:18 our um well, definite modes of them then they kind of they don't
43:30 out so they get to amplify their energy into that that made historic destruction
43:36 what was happening. So understanding values. This one example for dynamical
43:45 . It's understanding how your values. also important for what it would be
43:52 reduction. So we're trying to figure how much of them. Big problem
43:58 actually into deal with that are getting risk from the good answer. So
44:05 that's um sometimes we put falling in of India algebra once you figure out
44:13 lambdas here or singular values they call rank revealing. So values or singular
44:25 and when the smallest sufficiently much smaller the largest design there. Well just
44:35 order the rest of it. And make a smaller problem that corresponds to
44:39 number of. I do not accept . So that's just kind of the
44:46 thing about understanding why did he design mind investors and then the spectrum of
44:52 is just to set the argument values the spectrum rages. What's the maximum
44:57 value? And again you just show that's in the 14th quantity in terms
45:01 convergence but encrypted manifest for instance And then there was bunch of the
45:12 of violence related to the matrix. power of the metric. The item
45:18 of the power of the matrix is as I invited the part of the
45:24 of the original and carries over to normal constructs. And and then there's
45:34 that are useful to know how the of the island values relates to the
45:39 of the natures that is real symmetric general not the real. Even if
45:46 matrix is real. Um And this I meet him then because of the
45:53 , even if it's a complex, have been real and one can also
46:00 it's positive definite and that means stronger for so these are young understanding a
46:08 bit what to expect. So if know something of your metrics and use
46:14 routine and this turns out not to true or something wrong. And the
46:19 thing that was mostly used in the was talked a lot about there's some
46:24 of characteristic polynomial which one can use find the argument values because once again
46:33 find a nontrivial vector X. So equation is true. That makes sure
46:39 found this is true and excess So basically that means that this matrix
46:47 to be singular and then you can try to figure out what the numbers
46:52 . That makes that true, forming determinant to put it in determinant zero
46:59 . And that's not very useful. for small problems, it's also true
47:09 the real major cities if there are violence, the commerce kind of
47:16 somebody is strong. So it's as as it that's that's not a good
47:21 to do it practically. So I'm coming to this thing. How do
47:27 get says if you were to use system impression for this matrix A and
47:36 things do computing the Iranian values is . So many other things you try
47:43 find shortcuts to understand something about saying spectrum radios or mhm And then there
47:53 just very simple methods that to the the the so called discs,
48:00 1st amounts of this and the complex centered. Um but today I'm gonna
48:08 and reform one step for this based Vivian, some of the diagonal elements
48:17 a row or it can form another in terms of um All right,
48:24 do this. Someone's eyes for this gone. Right. And so the
48:33 . So we have this, that's two sets of this one. Both
48:40 road disc is centered for the diagonal is and then the radius depends on
48:46 in the drawer. So you can that and then um you know,
48:53 still in the main oh yeah, unit of all the circles where are
48:59 embodiments maybe? So you can find for instance if there are separate the
49:04 from zero. You know that things not sinking. Perfect. Yeah.
49:09 this example image, mostly everybody's are or less near the like center of
49:14 disk. Is that more likely than or is it? No, I
49:19 think that highly dependent on your matrix and some of them may nice trip
49:26 that. Mhm. So if you for instance something that is very difficult
49:34 and this is a large spread on values. So that means I cannot
49:39 me. So, I collected All right. But its computational very
49:49 right? You have your matrix and up the absolute bank withdrawal and balances
49:56 . It doesn't take much time to . So we got to go now
49:59 that is good enough picture and kind for range and for you to decide
50:06 things are well conditioned or unconditioned or the spectral radius all kind of less
50:13 one. So it's kind of in . Um Yeah, so that's part
50:19 this interesting methods is um to choose let's go back to the simple ones
50:29 Kobe and outside Del however if there's minors or whatever. So the iteration
50:38 bit comes close to the identity That's kind of the game. So
50:42 has to try to comment compressing like in this and compress them so that
50:48 want to do, you know, business. The situation matrix that should
50:54 of be can be seen that it be less than months or somewhere.
51:01 there should be a way other things know in terms of what the vocabulary
51:11 and that's the orthogonal unitary symmetric everybody is just equivalent. And for complex
51:19 and transports, its transports and So these two are finally intimidated and
51:27 . So yes, you have to . And unitary is in the complex
51:35 the corresponding to orthogonal, that means kind of the product of the matrix
51:41 itself. Either whether there's some of left side product or right side product
51:47 need to transport because they are generally square, so we need to make
51:52 that it rolls in column matches up that's why you have to transport.
51:57 then the prevalence in the complex case the transports, cardigans And it's these
52:05 then turns out to be an then even 1000 all of you And
52:11 one can relax the strength of the and then you get what's wrong.
52:18 it's just to know yes, I of visually and hopefully Eastern furstenberg for
52:27 it was essentially triangular matrix plus I don't know. So they can
52:33 upper or lower. So it's this for me. In fact, the
52:37 triangle is kind of not guaranteed to down zero. The lower triangle,
52:42 guaranteed to be zero except for the right with all the main diet.
52:48 the rest of it is guaranteed to zero. This is just the transport
52:52 for that. So understanding the system because that was one of the forms
53:02 and computational e efficient argued that singular decomposition. So the other thing was
53:15 back in the form then I get resulted that you know that so that's
53:19 of if you have things matrix and diagonal formula, they know they argued
53:24 this, any of them at the , try to generate the diagonal reform
53:30 trying to generate something close to it that's not always possible to get.
53:35 in the same way that I got the fourth, that was up and
53:41 a number of methods were based on similarity transformation as it said, that
53:47 the original made to extend you too and post multiplication, one is the
53:55 and you can put them on either . Um and the point of this
54:01 as then the Eigen values of the A are easily related and so are
54:12 ideal lectures. So, but if is not particularly simple matrix in a
54:21 , it's kind of a general matrix we don't know much about it.
54:25 you may have, you know something finding transformation majors. Is that like
54:31 Eigen values and Eigen breakfast or B simple. So in particular,
54:37 if we can transform a diagonal matrix you already done mm hmm. So
54:47 , A and B have the same writing values But they don't have the
54:51 vinyl records public season two basically recovered Eigen vectors from a from finding the
54:59 breakfast for weeks. So that's so bunch of the methods kind of
55:05 Um this idea ah a particular tell us about this A has optimized
55:16 . You're No Nice Transformation Agency 6 , don't tell the other party and
55:28 is just more or less in this if they're square medics and then you
55:35 get it into triangle electrics or Yes it's sufficiently if it's her
55:42 So that means the Sequels when you transposed. Um this is true.
55:49 the triangular matrix and it's only the is the same. That is the
55:54 chance is yes. Just making sure that star operator is the same as
56:02 . Yes. Yes. Thank I should fix that from the
56:10 Yes. Yes. And this is wrong and if you have it's to
56:24 19 arrested for for the matrix that the correspondent thing and then we can
56:32 then worked. So this is Once we got the triangular form the
56:37 and then and the land there is one of it it's just for a
56:43 form. What is some of diagonals the argument is the same as if
56:49 mean just have been diagonal. So easy to pick up one of I
56:55 understand that here and then what the is something like this and then you
57:01 um partitioning into how many more minutes wasn't. Sorry but this pleasure.
57:08 can then easily Mariah the correspondent. am elected for. Yes and then
57:18 can go back and it's on the of the class that are in the
57:24 and figuring out how to get will ? I am a message for the
57:30 Ameritech's um other things that were used terms of designing methods for computing.
57:40 demand answers. Were these methods on shifting? That that's huge. Mm
57:48 introduce a shift. So there's the scale and subtracts in the same value
57:54 all diagonal elements of a. Then equivalent to this subject from the
58:01 But I will find the any of in that sense. And sometimes cases
58:05 simpler the work with this matrix and original matrix and we have examples of
58:12 in class with us but just understand if you choose the shift then shifts
58:21 since you know the shift to introduce can recover and then we have property
58:30 there I can values of the university a texas thing. Conversion of the
58:36 is an atlantic. So and with efforts like the bar mats are subject
58:40 the largest in magnitude. I imagine perfect distance. I have no
58:47 So it's just one strict in order magnitude gives that the land. We
58:55 then also work with a the inverse a instead and then we get yes
59:01 the smallest Eigen value already before one of the smallest value is standard
59:08 . For used in some of the other methods were kind of using the
59:15 of our method. Yeah smoking pot effect of the mentors by itself.
59:24 then there was a trick to show on the side basically activations or everything
59:31 gets suppressed fixing except the ones that to. So I think this is
59:40 , reminder of what we're thinking about terms of the argument values, nah
59:46 actually need all the onion values and only need a few or the one
59:52 our efforts do you need it and which on the whole lot what kind
59:57 metric and that's what they want to . So again, it's for life
60:01 class. This is probably the most that you should try to remember if
60:05 need to solve an Eigen value declare um what the needs actually are
60:11 then select method based on what the are. Um As well as well
60:17 the 70s hits small or large That's not enough. So it's best
60:25 this department electrically wasn't the form is ratio where are the best so chris
60:38 they components in someone's of the island that corresponds to smaller island value in
60:47 iteration process because I got invited to raised to the power for your situation
60:54 here. So this is less than and then all of these things eventually
61:00 suppressed and what you're left for something this and the check was sent to
61:07 and then your function that gets out dominant time of wealth that's about the
61:15 sort of the norm and the maximum out the largest magnitude. Yes.
61:29 , it's now you need the largest can be the problem as well as
61:36 gasses have a starting vector, you make inspector multiplication and it is a
61:42 matrix and real starting back to and will never find anything complex. Are
61:47 largest because of all the standing on expects um this was never talked about
61:55 the shift plus used to targets other badges and the and putting the shift
62:02 to something else that you want to away. So we've kind of maximized
62:06 distance yourself or minimize statistics. Mm . And then there was sort of
62:15 inflation method. Again we're in the transformation come back on the plate Where
62:21 am on business, zero out columns I'm fine. Again now this is
62:27 least this will end up being on diagonal proceedings. I'm I can value
62:36 since this is a similarity transformation as keep working and still preserve the union
62:43 . So then you work on the problem that is not one. But
62:46 carl um smaller and continue to do transformations and B until you find this
62:53 , are you in value such you and then you can recover the correspondent
62:59 am. That's what song inflation is smaller and smaller problems. And when
63:06 apply the householder transformation to this they don't ruin whatever they gave them
63:12 first column and the first stroke. that's the proper nice property of this
63:19 transformations. And this is a show the outcome of the public. Here
63:25 go. Yeah, I know. . So we'll start finance. And
63:33 there was just a called entertainment situations kind of um no trying to do
63:46 things at once. So that's going have a bunch of different starting
63:53 But then you can correct into matrix . And then instead of doing one
64:00 vector time now eight and 6 for collection starting vectors. But the problem
64:07 trying to point out here, when it converges, it converges fears
64:14 the I am vector from the largest . M. Value regardless of starting
64:19 . So these kinds of actors eventually very much alliance. So this lecture
64:27 increasingly automatic. So in this matrix increasingly in condition so it's not particularly
64:34 method to try to find more ideal . So in order to do
64:42 understand going back to what's not ask our factory section and again we can
64:50 pass all the transformations to find that are characterization. And then there's a
64:58 procedure they're doing. But now if are a factory ization, the Cuban
65:04 orthogonal vectors and even not normal. that kind of the iteration records apart
65:11 make sure that they do span the initiate operation. So they don't aligning
65:18 with each other. And then if do that then it's a more efficient
65:23 than to get multiple lying investors at same time. So that's the cure
65:30 accusation is basically trying to prevent the finest decorations to end up being in
65:39 or Yeah, the correspondent vectors are business. Mm hmm. And then
65:52 just correcting everything else. And eventually get um, they're all similarity transformation
65:59 okay, so long and military from agencies and so these are.
66:08 yeah, similarity transformations deserving not that investor side and I can't breakfast your
66:19 to applying the two and then one also like they can manage his shifts
66:29 that can make things from Argentina And then he has to keep track
66:34 the chips were used to the Are you serious? But in terms
66:39 finding an elector's income doctors Caroline Okay. Um, so I think
66:54 . This is no different from the . Okay. Sorry. So the
66:59 thing was if you remember a picture you take your general matrix finding.
67:06 the transformation is determined the things. some big markets then you buy human
67:13 too. Mm hmm. That's something matrix into the tri diagonal waitress.
67:21 then so for a given by So I'm gonna try that. That
67:25 the procedure getting used. So householder transformations comes back again. I
67:35 understand singular value decomposition. 40 I guess whatever it was. But
67:48 if you are in fact positional Our software. So this is um
67:56 this is just tell me what the complexity was but it is expensive,
68:04 more expensive and sometimes if you can more expensive and so the system of
68:09 question and so there is something more that was covered and I think that's
68:17 of the family now. So when come back convinced and talked about this
68:21 methods and just waste, they work they're not just methods. So these
68:29 the cure iteration, the reductions that something informed this is kind of mhm
68:37 were closest in terms of doing good invaluable solvers and for sports cases from
68:43 support fragile space subspace methods that again on nature expect a modification similar to
68:54 can't get pregnant. Something about P D. And the point is
69:02 . It has this form. So is applicable to that's the energy matrix
69:08 dozen square. And we've got the matrix in the mill which is symmetric
69:16 singular values. And then you have is left and right singular vectors and
69:25 symmetric has a certain properties and they the same but it's not generally or
69:35 . So this is what I said , I want to always start this
69:40 in principle you have as many singular as you have the size of the
69:47 but it is rectangular, then it a little bit different. So in
69:51 space this is what's known as a form of the principal. You kind
69:56 standard and in that case there are many single garages but most of them
70:02 zero. So that's one element. the book depends on his his form
70:08 practically in our services well as And so here's the thing that you look
70:19 the american methods and there many of are labor that's rank revealing methods and
70:26 what they do. They're trying to explicitly or implicitly find singular values.
70:35 at some point they cut it So the rank of living means that
70:42 have a kid in the context larger values and the corresponding you and the
70:52 and discard the rest against the smaller problem for computational efficiency. And then
71:00 was also the other distinctive. Sometimes can use it to the inverse
71:06 So when the matrix is singular, in principle the embers of aliens just
71:14 the normal paychecks algebra and then you in the middle here the universal Matrix
71:21 singular values and of course is a value is zero. That means you're
71:26 by zero about this stuff too So in that case I understood the
71:32 force things and instead of Making anything to infinity, one of us pretended
71:38 instead. But that's the way everyone something for observable universe and it is
71:46 use what it turns out to be very good approximation of the induction can
71:52 used computational so Right. So this using the university but several members and
72:06 at least squares fit. Okay. um but then I was best over
72:14 years in the book as has memorized errors in terms of predictive value versus
72:24 observation, whether you're getting from the or measured observation. And then the
72:31 is to find the basis functions that used to form linear combinations of approximately
72:39 value. And the thing is that these basis functions on a diagonal,
72:46 this is the normal matrix, it's diagonal matrix and then on top they
72:54 normalized and this identity matrix. So that's the scent here sort of 19
73:00 championship all in government has expressed an bills. These were kind of good
73:06 they are orthogonal. Um or one use to check functions that was done
73:12 the four years example also both put functions and take functions. As I
73:18 , we've been used in the compression and putting them and whereas these are
73:26 used. One of spaces can be as space dysfunctions about the results in
73:31 condition major system. No. And is the best of what it
73:38 Here's again formulate this minimization criteria and normally it's the other side they're solving
73:47 normal equations. And then and she she were a fact position of the
73:52 . A we don't work with 88 Okay, has bad condition number then
73:59 has to be the worst condition number it's a square on the condition number
74:03 it. So you don't want to that. So one thing was your
74:07 ization and just plugging it in here simplifying it. And eventually the gap
74:13 finding your petitions for that. This functions that you have five taking the
74:22 you have and use this what the natures do that. They'll find something
74:30 . And you're saying the matrix are inverse. That's the trivial operation to
74:36 this complex For some of our c to one. So that's why that's
74:43 method and we are, that's what went through was to use singular value
74:48 and with the same matrix operations and we'll find out speaking in that case
74:54 have your coefficients for the approximation of basis functions ah lowering these singular values
75:04 to singular raptors. It was six up to a couple of minutes to
75:13 guess so four years here is what want to say in the last couple
75:17 minutes. About four years years. very commonly used. And one of
75:24 reasons was for commonly used is that actually compute the fourier transform. One
75:33 the discrete fourier transform and for that is this fast for your transform
75:39 So people do all kinds of tricks convert whatever the representation they have into
75:49 of a tactician space, so they do is for your transforms and for
75:55 past four years transformed the computer. one would think that, you
76:00 thinking about for the climate, Yeah, I live on a sphere
76:05 of so circle coordinates would make They're not quite the same efficiency of
76:15 to compute functions on the sphere as the condition space. So therefore that
76:22 perhaps problems of the sphere, they're trying to convert or do converted to
76:27 in partition space and then use fast transforms us one step in there,
76:34 , it's all over so this is remembering that part of the reason is
76:39 going to be nice recently, nice that is continuous etcetera. And if
76:45 just have enough coefficients then we get good approximation. And then there's now
76:51 tend to abuse this all our coordinate and think of it as basically so
76:58 sing this here and that tells us . And then there's kind of a
77:06 going from the continuous formulation to the formulation, I went through in class
77:11 the end, you get something like and that shows that the discrete fourier
77:16 is nothing but the nature spectrum And then the whole thing is that
77:22 discovered as this particular matrix for major modification has very nice properties. So
77:30 can reduce it from an n squared from Angola n squared enough problem.
77:37 this reduction from this quarter and Logan I would say the reason the popularity
77:43 the that's cool we transform or the transform as such and there's no one
77:50 . And then the other thing I to the mindset without, I guess
77:55 thing is kind of the best part it does from you. I think
78:00 needs to be conscientious of our things as well with the transform and wherever
78:06 find the output. And the other that wants to I guess is this
78:12 to remember that? Yes, for complications, it is not their ratings
78:25 choosing is important. So The use 15 libraries, they tend to be
78:32 of it so they just make sure you use aerobics, that corresponds to
78:39 particular level in the hierarchy that they in security. Kind of small ratings
78:44 data deficits and registers and then there some larger ratings for things that and
78:49 cash and next thing for us to etcetera. And the point is not
78:55 much to the views. There are work but it is to reduce ah
79:02 amount of memory references. So that's the weakest point in all of computing
79:09 to then. So focusing not so and minimizing bureaucratic stuff. The
79:16 Memory is important part and I'm sure time is up. But in terms
79:22 the 50s, I was saying it's still investigating. FST on software out
79:32 is something going on S. T. W. That was created
79:36 a couple of students at MRT years and they had me turn american operated
79:41 you can find it and download this source. And it's still one of
79:47 most competitive the festivities the family to to remember F. F.
79:53 W. Nothing. You got any students the Putin government reverse worst fastest
80:03 still w. And then many commercial and vendors that have been developed and
80:14 versions that they support of for the . Uh huh. Came from or
80:22 original telegram from his M. T. Students and again the
80:31 Okay so for that I will thank again for taking the advice Just for
80:38 final exam. We will meet on at 2:00. Yes in here but
80:46 yes that's what I think. So the university that you know if
80:50 have whatever algorithm technical and I have schedule classes and to to fire since
80:56 also longer the normal civilian example. I have whatever I want to figure
81:04 for Classes don't want to to 30 . They end up you should be
81:17 and we'll put together some if you time to try another example. Thank
81:29 . Yes. Yes I'm gonna do practice exactly and put that up.
81:36 . Right that was just want to of ask for. Right we will
81:39 that thing and I don't know if have time. You will find one
81:43 you can work on. Okay? , did you breathe to mark me
81:56 ? You will not be punished We're trying to do okay? We're
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