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COSC 4364 Numerical Methods - Lecture18_Ordinary_Differential_Equations
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00:00 kind of like you're reading or writing part of the cancer is different parts
00:04 on whatever it's kind of this padding something. You'll see when we get
00:09 I think goes through it like step step way even I think I knew
00:27 to like but like you're probably gonna an easier time understanding going through
00:31 I have to like you might research square Okay. We don't know what's
00:44 on. Okay so we'll continue to talking about all these or initial value
00:53 . There are other problems and I they're also mentioned in the book but
00:58 won't cover them. There is no valuable problems. The initial value problems
01:02 the ones we talked about. So kind of know the storing values for
01:08 the questions doctor and the boundary value you kind of no the point values
01:16 a mixture of it of starting well glass has discovered the initial. Let's
01:26 . Yes. So um we'll continue to talk about where we left off
01:32 time and looking at methods for solving these with based on our experience from
01:42 integration. So come on. Nothing to something you already covered and generalize
01:49 not just to deal with a single but more equations and higher order in
01:54 O. D. E. Not just the first order runs and
01:58 some particularly problematic guys known as all and if there is a non stop
02:05 the next so where we left off first reformulating the first order potential questions
02:20 kind of an integration problem where we the solution of the given time and
02:26 I want is to find the solution the next time step and then you
02:32 do it by integrating the right hand of being differential. Which and then
02:39 you apply whatever integration method, one to evaluate best for this in Japan
02:48 this time we're doing it this way pointing out that there is a difference
02:56 to the integration problem looked at before when you saw so ordinary differential equation
03:04 for that matter fortune but we won't about that. But differential equation then
03:09 kind of march along in time is sitting there could be space but whatever
03:15 variable you march along with that. most of the time. So I
03:21 you do, you have a history that we don't have when we look
03:27 the freshman just know uh we can with the integral between senior people sage
03:33 whatever integration and the molecules But in case we have a bunch of prior
03:41 evaluations or estimates of accepting. So behind this quarter here. That is
03:50 from applying quality ship for all these for depression. So then there was
03:58 what to do. Um, so have access to a bunch of fire
04:04 and when we looked at during integration the whole idea was that they found
04:13 integration formula that made a linear combination the integral values at a bunch of
04:22 . Captain's orders. All. There to Simpson's alone, there was three
04:26 two endpoints and the main point demonstrates Simpson rules. But now we also
04:32 the history that we have done these for. So now. So basically
04:39 is still kind of integration formula but the X values can include ones prior
04:47 a detective and the one that we to figure out how to determine this
04:55 or the weighting of the different historic in this case. It is
05:03 When we did the integration of the rule here that it won't produce the
05:09 . That's beginning integration of this function this interval. Now for the linear
05:15 of the old stuff is such that right hand side here equals the left
05:24 side for any maloney allow up to in order that is less than one
05:30 than the number of points in this there are endpoints, investment and integration
05:37 in mind response. So they stopped and that's always something that's everything.
05:45 in terms of mono meals on order less the number of points should be
05:52 . Exactly. And that gives us sufficient number of equations here to address
06:01 computer scene. The left hand side this analytics. So basically we integrate
06:07 guy and this is the and the for the invitation results basically our to
06:17 ice divided by the first If you derivative about to be I divided by
06:22 . And then they get the same we know the left hand side or
06:29 bunch of ice and on the right side this is what it sees.
06:36 kind of the referent times another vector this submission of the day for a
06:41 time. So basically it's kind of roll over that and then you have
06:48 for the different parties. So then got this matrix anywhere interest in the
06:55 for the second, that is all . So you have no left hand
07:01 and you have the matrix. so then it's good for so that's
07:06 way point guards what in this case formula E is for doing kind of
07:15 in here. But it's sort of the combination of No one values or
07:23 let's consult this guy then for this for it and it's four then one
07:27 up on this particular proficiency. But procedures general so that we can and
07:35 it turns out which I think is mentioned in the book that the accuracy
07:41 this type of procedure that gives you entire order weapons. If you do
07:47 points, you get financial From this is four points. That means This
07:54 in the 4th order method actor like the ah so that was where we
08:04 off last time. Then one can be a little bit braver. So
08:10 this case what this impression expression does looks at the careful treatment. So
08:20 you're plugged in. Yeah, equals The first time. That thing is
08:25 argument of gear is one which is what we're trying to get. So
08:31 is an implicit formula because it makes on the right hand side something we
08:36 know, it makes use of the . They're trying to find some principle
08:42 had to be solved no iteration but still go through the same exercise in
08:50 of what polynomial should be integrated correctly find out the weights for the values
08:57 the dates of the values in this . The problem is that no one
09:01 not known. So you can't just the club again and so that is
09:08 to the iterations And that's the same . This we make an estimate of
09:14 and plug it in and get the out and then um so that's why
09:24 itself this formula is not used very . But what is making you is
09:35 combination which results is most known as predictor, corrector method. I use
09:42 first one that doesn't make use of values of T plus H. But
09:49 the known values Up until the 19 they used that to predict gas
09:59 30 passed H is supposed to be then you pretend that that is the
10:03 value and stick it into the small formula. And then again, in
10:08 group estimate of what The rather than plus h. So that once this
10:16 kind of known as a predictor. I guess what it should be and
10:22 one just has one iteration for the small town with the predicted that gets
10:28 in that should be corrected system. that's the notional and predicted correct
10:38 Oh, so here is against the . Obviously the guy's name In this
10:46 , the 4th order four points and predictor and I'm sticking in this
10:52 They haven't smoked on formula and yet a group about predicted. So at
11:03 point for methods using quarter should be on. Mhm several prior points.
11:13 , I mean, you can't really this technique when you start the process
11:18 you don't have an entire force. that's why it takes a different starting
11:24 to get a sufficient number of time that you can then start to respond
11:30 this quality based techniques. So that's is what is known as in this
11:37 , using the method. The 4th method just started. So that means
11:43 least it's a portable limit in terms the accuracy in the beginning and once
11:48 have enough points you switch to this corrector type approach and using these four
11:58 . This is also for older So the whole thing then ends up
12:04 accurate to the 4th order in terms approximation. So the order is consistent
12:12 this case of using and of course want to cut down. Plus these
12:16 points, there is something the dick collect so has continued findings on the
12:28 slide but I'll ask you right So yes, Even though this is
12:35 order, this is fourth quarter. so in terms of accuracy they are
12:41 of equivalent. So why not just when equipment all the way? What's
12:47 benefit for switching to this type of the solution record instead of continuing?
12:55 sees an event? Yeah. So look at this step here.
13:25 this is already known so for any this is already known of goodness.
13:32 . So what I'm going to do , it's essentially this one plus the
13:36 combination of this. So to do , I only need one function evaluation
13:44 you put that all this requires. for function as I really. So
13:51 this case the four functional evaluation for temperature. Remember this picture of sampling
13:58 placed ahead to the next point. is for functional radiation. So and
14:04 of course there is another one functional . So this one is one the
14:09 is through functional regulation of diversity for year. So you save in terms
14:13 functional aberrations when you switch parents in confirmation. Oh, this is what
14:24 of track to summarize on this line . Yeah, no, according to
14:29 order for the method, they remember order of method as long as the
14:35 local translation. 4th order. That It's not just up to the 4th
14:43 chairman or delivered in a very serious local translation areas the next time let's
14:49 get forward. So I have been since persons a number of function evaluations
14:59 this is an implicit methods and We need to iterate and who knows
15:04 many iterations before. So that's why didn't say against but I can do
15:14 predicted. Perhaps the version you kind just one incubation on this fact by
15:19 the predicted. So that's the best to function and I wish so that's
15:26 of nothing helpful to keep them on relative weren't involved. And again,
15:35 evaluations then your combinations of stuff is comparative during the Oh, so that
15:46 that any questions on when this Uh so this is kind of a
15:54 but for us it's best to go that everything back to how we did
15:59 this basic single system. Um place the three categories, a method that
16:07 talked about, taylor series from the and the others. So morning.
16:18 , you are Oh, so now guess first a couple of points about
16:24 than one equation. Things don't look this example here. The only point
16:31 this example is we need to introduce concept of couples the french and the
16:39 . So what's different, you could a collection of equations but supposing this
16:46 did not involved why? And this did not involve acts. Then you
16:52 two totally independent systems or questions and on each one individually. You don't
16:59 to solve them together. So the system is you can't sell them one
17:05 1. We really have to so kind of both at the same
17:10 So that's the notion of couple system is Yeah man, this example,
17:18 we'll only practically a couple of systems here in this example, two equations
17:25 does, you can solve these from . I don't need to know about
17:29 . Why are you trying to solve X And if you want to solve
17:35 this is kind of tribunal, it's of in terms of methods,
17:40 you know, for a single question you can apply twice in this case
17:45 for a couple of you need, I said, the song live
17:47 So that's what we want to be . So the first thing was done
17:54 and you're serious method and in this We have two variable system administration and
18:02 you need to be terribly serious expansion each one of them and then use
18:06 and plug it into economies to make prediction for each one to solve
18:17 Yeah. Okay. So basically We in this case for this Sarah
18:25 This one just take the derivative with to this one was given the equations
18:32 then you have 2nd and 3rd and assist depending upon what the order of
18:38 you want but that's previously that means you need to have access to you
18:44 function and enable ideally to do what's here to do an analytic differentiation.
18:51 the questions that you can actually different this you have formally inspector derivative of
19:01 and then they need to plug in expression for extra anywhere from that we
19:05 from here. Yeah, that's So um yes, this is just
19:17 it ends up being um in this it's cold example and plug it all
19:23 and the native serious formula. 7005 years validation of this place down
19:32 steps and then for convenience for these of equations. Uh huh But will
19:43 use the documentation and I will do with a bunch of companies fight so
19:49 this case the kind of um either or later if you have more of
19:58 , your best with introduce new variable . Investment subscript according to the number
20:04 variables that you have any system of or in this case what was maintained
20:10 the X and the Y. Besides company will see less than excellent.
20:16 , 2, 3, 4. mean dependent variables you have and just
20:23 hopefully we'll be straightforward too. okay. The doctor comprehend what's going
20:35 . So then general a larger not just the variable subscript variables according
20:45 . So previously had two unknowns. dependent variables that someone do stuff for
20:52 for the future. Systems of equations general form right on the clock.
21:02 on that stuff about yes, formal terms of the save the serious then
21:11 them excellent money and whatever. Not dependent variables you have and formally still
21:18 same. That's not dramatic, thinks that. But in general that's saying
21:27 seniors methods again we start use because means we need to have access to
21:33 derivatives preferable to genetically. That's why mentioned last time in the numerical differentiation
21:42 usually not something you want to do it's very sensitive to errors and then
21:49 amplify errors. So that's why the of equations like the singular equations you
21:55 to use the methods that that's not . The valuation of derivatives. It's
22:03 of step forward that to Also general only equipment methods to be more than
22:10 dependent variable. Okay, so it's square with the price. Everything that
22:17 to be single Moschella vera husband actors agencies formula is pretty much the same
22:27 versions when it means more than a variable but otherwise it's exactly the same
22:35 in terms of function evaluations at different and that the Entomology and she can
22:46 us reach. I can do this function evaluations that are now that's the
22:52 instead of Australia. What's this? is nothing otherwise remarkable. Mhm def
23:05 insist the list of right concerns. . T and whatever they are
23:27 So I think has a bunch of examples, strength that works um and
23:37 not to differentiate the system now differential given it is not protective about it
23:43 derivatives of the dependent variables and the right hand side and then the initial
23:52 and there's a whole lot of there a couple of things except from
23:58 XP system and maybe next time this basically evaluating the function of these different
24:06 returning the corresponding values and then successively uh evaluates the a combination of the
24:20 function evaluation at the core sample and what you see. It's one of
24:25 The four times to evaluate the case then they will be defending a combination
24:34 remote performing an expert system. Something here is here is the right hand
24:42 . So the extra system for this example what it does essentially the function
24:49 some of the points that we have mark a little about that For example
25:07 are using 4th order better or an solution in this case that came out
25:15 the I just think nothing and in case themselves for these sorts of operations
25:23 instantly expression so And it's plugging T so questions so far, That's what
25:41 talking about the procession of prevention methods systems of equations and it's pretty much
25:49 same. That was before. Mm . So now it's just making things
25:57 of vectorized vector values instead of scalar . The second thing. Yeah.
26:02 if that's the um the electoral function to be evaluated at the news but
26:12 we are, this is from. , but we need to compute this
26:19 . You have already computed. So still it doesn't change the weight.
26:24 still the same because the winds are or function. Remember how they computed
26:33 ? It has nothing to do with that was basically the mono measles was
26:37 to find out the weights that was intended under the function itself. So
26:44 just fairly straightforward. It's just that becomes electra valued instead of skating.
26:52 , sir. What I tried to , I would like to stress that
27:00 special molten. It's a family of and they're not just this particular
27:09 Um this particular formula is the 4th of, you know, read too
27:18 into this depression that it is in for reform. Yeah. It's just
27:26 of the difference between this function. function evaluation of course. System
27:36 Yeah. You know, the combination the predictor corrector and it's the same
27:41 . Same rates is independent of their for the scale or effects.
27:49 So more sensible for, you starting it up with someone in nothing
27:57 . It's just the thing is now of dr value and stand up stand
28:03 what and I think the next thing that it turns out for the predictor
28:12 method, uh volcanic facts the delegation picking up the local For step one
28:22 face error is sending us to make . Is this where we started and
28:28 then is the outcomes especially for people so we'll see what the difference is
28:34 terms of what correctness predicted left value be corrected. That comes out of
28:44 one. Um I guess the relative and mistakes and that's obviously a function
28:51 the step size. H someone can that also developed and the doctor procedure
28:59 the local error and the prediction from formula here is large compared to what
29:07 is comfortable with then on come redo for so half the step size and
29:14 considerably smaller there than they said. we can increase our step size and
29:20 that's what's being said on. Well was I guess a suitable of course
29:26 not remarkable. I'm sorry investment analyst for but in the book some of
29:36 effects to skip this thing but it's both of them in combination. Not
29:43 the interactive piece related to perhaps the and this suspect starting and then switch
29:54 yeah this quarter debased the reporters that's evaluation and things and like the thing
30:05 function evaluation on the right hand science the expert system this is kind of
30:15 then there's a part of it here you can see the A.
30:18 System. It has this special part predictor part And then they have the
30:24 part. That is the correct part the combination of the two and that's
30:32 book and then it shows your evaluation and eventually get the area estimate that
30:42 from and a little busy right pointed one of its so um I really
30:52 I think that's everything is uh oh was just trying to be conservative management
31:00 so often it's a personal and important you I'm just trying to be patient
31:08 wait in the cold. That's nothing do with the americans. And then
31:16 don't really talk about the adaptive methods reducing terrorist innit thing in order to
31:22 whether you should decrease step size more increased exercise without buying your acceptable error
31:32 results. So what uh what I let's see but any questions on this
31:46 interpreted the insistence about the is gonna and have a good understanding of a
31:55 system. Single unknown very well. in this case from ordinary differential questions
32:04 one independent variable. Mhm. This of industry or to generalize to other
32:11 first order depends on the question questions so on. Obviously the nature and
32:22 problems require that but also understands have solve higher order differential equations when the
32:33 is formulated using Oh yeah, the order. So what problem is you
32:44 you have to think of something that in the form of a higher order
32:52 first I'm quite sure that economic Many about one. The streets are not
33:02 question. Yeah, Thanks for the when I have a simple example but
33:06 think you were going to the acceleration the function oh, part of the
33:17 over the thing that needs to be and the force and then acceleration as
33:24 order is changing but possibly unit and is basically changing position as a function
33:33 time. So with that there's the that's come out here. The second
33:39 instead of the first order differential acceleration is the second derivative of position
33:48 that's a functional force divided by the and escapes. That's the very accumulate
33:55 of what things natural when they come formative in terms of that higher order
34:01 . A second order thanks to So let's just get alright in town
34:11 represents. So that's interesting. what's this, can I want to
34:17 this is compared to the previous line introduced subscript and variables where extra ones
34:25 fact the position and its derivatives Is velocity that has X two electrical
34:37 Yes, that's two is velocity which under derivative of next one. So
34:45 happened here now is instead of having second order differential equation, it was
34:52 into a system of first order So now we're coming back phenomenon several
35:00 and just went through, you know to deal with the system of first
35:04 differential questions. So we have listed the high order for the problem into
35:11 system of first order for leaving So this is basically the technique used
35:18 highlighting media for these going back to of personal for these, which is
35:25 to follow the realization of a single one. So I don't think that
35:32 from the general question here. How doing this morning? The victim's things
35:37 us to now ah and and so into everything. Uh huh. The
35:45 is falling the derivative dr it's dependent okay, potentially all the lower order
35:53 , improvement itself and on your best do the same substitution as I did
35:59 the site for Newton that subscript the and introduce move over for the various
36:07 until we get to Yeah, on one told the derivative and then you
36:14 your expression just use the vector of . You know where the electric components
36:22 the various very much including the variable . So from this forget something.
36:28 analogy with the new terms example. that the whole collection assistant versions depending
36:38 and the number of variables for the of the system depends on the
36:43 Wow, differential the question and they them back to sort of make it
36:50 notation using their implementation so that the number of components of productive the
36:59 inside the function boundary where most of are. Mm hmm. The equation
37:04 introduced the system of first over all and the last one is another election
37:10 given life and site of the original . So this is a concrete example
37:25 I got to worry about extra That's the first component of the vector
37:31 the success of derivatives as additional components the vector. Um, And the
37:39 order and then well then you have the things you need for the right
37:45 side in terms of components of And then you have the river think
37:49 the highest story introduced us in the three crimes into the last question.
37:56 is that together equation Now when you in this is next to and this
38:03 into the text. Three. So . Mm hmm. Initial conditions also
38:13 this case um, these things through all the derivatives are up until the
38:20 that we're trying to find for countries in the form of So that kind
38:27 stuff right here. Come on, . Itself is still limited and secondary
38:35 defined system. Oh more. Mm hmm hmm hmm hmm hmm hmm.
38:44 is again it can be it also is another example where they spend on
38:47 question is this a couple of Mm hmm. There's something tragic about
38:54 that just being careful and exercise in case they're showing that um you just
39:03 the variable list and in this case dr components to include other dependent variables
39:10 these two variables X. And and just keep the components of the
39:15 It's nothing unique. Forgot the number variables in the number of equations.
39:21 just make sure that to get to form for his only the highest derivative
39:29 copy. It's one less. That's derivative of the breast cancer. And
39:38 you get the derivative of the vector then protected over there. Well that's
39:45 in this case that we can triple of Why did you have any questions
39:53 ? So this is kind of watching here and yes, hopefully not conflict
40:00 the central position following the prevalence. see some kind of questions. The
40:13 operations. So once again higher order map onto the systems of first
40:19 The system is the first order institutionalization the central first of all.
40:29 if only is as a part of different. So um and so the
40:43 of if this is not to have hmm. Instead of me equation,
40:52 are differential equations. So they're not in some sense the normal change and
41:00 have stiff differential equation. What it . Mm hmm model phenomena, that
41:12 or solutions of very different landscapes, . So you have things that changed
41:21 quickly and change that. Uh I have a picture but I think And
41:29 to get one you can use in of understanding understanding that they're both kind
41:35 small and large scales. Thank Think of family for c your house
41:45 some wind ripples. There are various . One small what? Perturbation on
41:53 surface and then underlying have very big that have making extent where it's just
42:01 . But the windows is fairly So your friends about controls pictures like
42:08 one here are you having in this some components of the solution that changes
42:15 quickly and there is another one that changing and this particular house administration here
42:27 relevant for this political problem here. in this case one has an elliptic
42:34 . So it looks like their analytics here. Um I can see that
42:40 It involves two expungement chills and one is Christy and another one that is
42:48 30 19. So this is the of this Tara decreases very rapidly as
42:58 function of team. And this one more slowly as a function of
43:05 So it doesn't take a very large before one more or less can
43:10 Ah mr The problem is um now these differential equations is that both lessons
43:24 you to choose the step size according the fastest became component. Otherwise you
43:32 to be there. So in this one would need a total solution time
43:38 be determined by the slow decaying solution the step size determined by the fast
43:46 a solution six years compared to something has more of a homogeneous timescale
43:55 These differential equations are very hard and and if you try to go straight
44:01 where you're solving, There was one to remember. The notion was fifth
44:07 equations and they are not so Um the behavior of the solution is
44:15 important in terms of recognizing whether one to resort to something that is suitable
44:21 stiff differential operations. I suppose sometimes is more in publishing but don't go
44:29 a whole lot of techniques was different functional equations in the book but I
44:36 to be aware of this notion because then there's a little bit of going
44:45 what I said in terms of for the requirements for choosing the steps sites
44:52 in this case no one can figure in this case there is another solution
44:57 I can't figure out what the maximum is for this to converge to the
45:06 and in this case it defines as can see there was a time constant
45:10 I call it, you know what this tends to be known as time
45:17 after it in front of the team South african things are and I can
45:22 both to my go back to my guess first time for talking about only
45:28 this simple circuit, the capacitor and resistance. The resistance and the
45:35 There are cds that time comes The of the two that tells you how
45:41 things will change. So um no can as it starts here um in
45:54 case you have good expressions or if use all the equations that was and
46:02 the existing value to predict the next . So you can plug it in
46:06 get an expression and then you've got questions that looks like this. Um
46:12 now it's a linear equation that's nice easy. So the parts of bihar
46:19 the unknown on both the left and hand side and that's where manipulate these
46:26 only got something like this. Is So in that case you can hold
46:33 it. So in this case even it's an implicit form, you have
46:40 expression where they can do the We don't need to hit directly district
46:46 to investigate converted to the solution solving system and question whether you actually form
46:53 universe or not. You probably wouldn't back to my comments to be any
46:59 you don't want to the soldiers. can say Gaussian elimination or something.
47:06 but it means that then the final , it takes this form that you
47:14 the starting value, its initial value . So the solution that time and
47:19 best for iterating ah this reverse. you and how good this is,
47:28 on the properties this kid's here. , alright, so so always place
47:44 you take powers of this guy the , right? This is the end
47:51 of this fellow. So what it that um in some value, in
47:59 way the value of this expression has be less than one now, this
48:06 the matrix. So it's not clear that means. We haven't talked about
48:14 up with a single number for the soon, but basically it has.
48:20 if you multiply this with itself, important that somehow in magnitude is less
48:25 one because it multiply something by That is less than one? It
48:30 smaller and smaller. On the other , if it happens to be larger
48:33 one then things blocks up. So important to understand this matrix theory on
48:42 you do with this all your method is implicit, but as long as
48:48 has a reason, variability, you ? Um So there is the one
48:54 that I'm not talking about, so the next topic. So yes,
49:00 something in order to I understand that is about the head, that means
49:06 the argument values that's the next Oh, this has to be greater
49:12 among that means happy. Um Are in matters of the inverse is less
49:19 one of the of the inverse is among this method. That's one.
49:29 the questions of sticks or use another happens. Okay, that's right.
49:43 Oh, less than more of the dancers. Something in general about the
49:49 of the questions never able to play . Some best summary. The system
50:01 have the essential equation experimentation questions Higher order systems american systems the first
50:17 and then monty generalized. It's also systems of higher order that X and
50:23 . And an example memorized. So was when I was coming to cover
50:33 terms of what the definition of operations now something that is problems they are
50:44 or and so forth. There are useful so but either about hiding from
50:57 problems or sending about with the competition been diagnosed realization. Yes, that's
51:10 when you're dealing with that. So books the book talks about about the
51:20 talks about the mathematical aspects. Smart talking about talking about algorithms and
51:30 procedures finding first being central available. I will have things sometimes of procedures
51:40 how. So it's kind of But the first thing on give one
51:50 and then we'll talk a little bit analytic ways of leading them. But
51:59 gives you a sense for this. problems were? Oh, mhm.
52:09 ? Oh, mhm. And I everyone likes. Let's see. Make
52:15 thing work also. Mm hmm. hmm. So what? No
52:28 Mm hmm. So very short Thanks. Mhm. Mm hmm.
52:49 is the real thing. People standing 1st. Yeah. I think this
53:00 an example of sorry for the quality the whole. Is that a movie
53:08 ? And right. one part for most the money. I just got
53:24 on the bridge. And somehow because didn't want to go home four.
53:33 a windy day. Huh? Is this film like this? But
53:41 watch this another minute or two and see what happens here. There's a
53:46 coming off the bridge at least is . That's nothing. Yeah, I
53:54 seen it before. Sure. He's through like you know, he's like
54:04 walking. So Oh chia was I am 40. I think
54:25 Let's let's get them out. That's of the end of it.
54:32 Mm hmm. So what does this to do? Mm hmm. So
54:53 whole thing was doing in decades. on a mobile. So that was
55:10 , ordinary differential equations. Modern Um, so that's a long
55:18 Including dynamics. Yeah. Responded to discovered the distance. So,
55:35 dynamics then um, mr moon. . So, um, yeah,
55:52 is Yeah, 13. If the is ordinary differential equations terms of the
56:03 that was explained equals or if it's inner system that it was from this
56:10 methods. It eight times. So major disease. And since then when
56:18 talked about the stiff. All the . We talked about onion values.
56:24 , Uh, system for Simplicity's 100 matrix is done. The villa haven't
56:35 talking about what that means. But will start kind of capturing restaurant frequencies
56:43 structures. So if you yeah, it's science and his friends will win
56:54 has energies very close to or Where restaurant 3.6 are the kinetic amplifier.
57:07 that's my understanding. Iron values yourself that's one or more industrial jobs.
57:20 current stops for a company in the hmm runs in that case it was
57:30 , yep, the indians, but productions do scheme and still got through
57:39 turbine and spins generated. Mm And what happens the way these are
57:51 is between sensitive or resting equipment and has decided to use it. What's
58:02 between the operating 3000, I hope things coming out of here zero.
58:11 are restaurants difficulties. So what do need to drink this breathtaking machinery across
58:19 critical frequencies very quickly. In other , the chloroform, but it's very
58:28 , important. So that's one of things we're gonna practice not just in
58:33 of bridges, all kinds of equipment devices defending the hall. Even with
58:41 parts, you know, that certain a little bit more than others.
58:55 as a way of motivating them. ? It's important. Unfortunately, it's
59:01 is very important. A bit more . Morning. Yes, it is
59:09 an engineer, then it turns out what I said, even if you're
59:14 engineering scientists and dealing with physics and , it turns out there is,
59:21 know, everybody does searches in the of Patron. It's something that depends
59:26 understanding how to do develop efficient methods doing. It depends on the properties
59:34 . Thank you. Don't you understand they're more keen on Arizona a lot
59:40 the internet computing stuff and not so in physics. Understanding your values or
59:48 version. I would call that a by the decomposition, which is kind
59:54 development of life in Ireland letters. very important. Lot of reduction means
60:04 you have maybe in our systems of questions of some flavor. Of
60:10 an organized system is the most computational . It gets to deal with
60:15 So if you can replace it with small system that captures the essence of
60:20 you need, then that's what And , so that's what model reduction means
60:24 have some things that opened correctly captures the features you want. But then
60:31 can try to also figure out which are particularly important and which ones are
60:35 of in the north. So on considerable reduction, I can use this
60:44 to come up with a smaller problems chapter 17 The one with the times
60:50 the actress. Okay, they're not of moderation, you're trying to get
60:59 the light and values is another one one that corresponds. Yeah. So
61:05 it turns out and also really um a million methods with the greater structural
61:14 . So these days computed everything is ago or science multi core. There
61:23 also the clusters and all kinds of systems and that's because she typically it's
61:31 normal food building but we try to of money large problem then we need
61:38 divide up the data that you have the data structures among the and one
61:46 you. So I hear about the you're a professional and there's one example
61:53 actually the book cover designing a building by colleague the like of the needs
62:02 our administration's cover his books. But again shows. So that's another thing
62:09 remember your techniques that comes up and somewhere deserved it. It covers a
62:16 of money and a little part for computation. So what is now a
62:23 problem? So when I get what problem is the problem that Um one
62:32 to just define lifters such when you a major complication, your best bet
62:44 to scale back. So this is essence of other medical problems in this
62:52 . The funding of the investments are gained factors here is the onion values
63:00 directors are having so you know, the vector and the welcome to matrix
63:10 command rotation with the matrix, whatever is. So it's best if it
63:19 and you can think of it. it wasn't a computational geometry, right
63:24 investing in my teammates have rotated the space and then it also tends to
63:31 the vectors. It gets longer. ? Um So this is for any
63:37 way to tell you not just in please suffer enemy joint to find vectors
63:44 that when you do the matrix like , that's the best. That's what
63:50 state factor. So finding these landlords are the values that talked about the
63:58 understanding and the structures and then also reputation. So for the additional problem
64:06 actually focus on the other electric So other things that to keep their
64:17 . Us these investors are not entirely the sense that if you scale this
64:26 Tuesday, that means also the left side. The scale factor too.
64:32 he has to be about I never they have the same idea scaling and
64:38 get back dr doesn't change the nature the problem. So these are not
64:47 within um it's getting faster. I'll give examples coming up. What
64:55 do you want? Yeah. The thing that is important to understand so
65:00 times you have a real nature but doesn't mean that find your vectors and
65:08 values are we tested the complex examples Islam does complex obviously has in the
65:18 otherwise you cannot ah this is a and this is real, then it
65:25 work if it is real because the hand side is real and the right
65:28 of the complex. That can be . This is real man. Also
65:38 not complex. I have one in and the other one not one this
65:49 behind. Yeah, separately it's not quick. So we'll start with the
65:59 examples of trying to the industry. very simple um things. So This
66:09 eight times 1 director 1 1 gets time, But just five turns one
66:14 . So Same vector scaling factor I can value five vector of the
66:25 . Um and here is another 2007 same tricks. Then it's these
66:34 , which is why it's important to self control. This case is a
66:38 x 2 matrix. So it has this being Tigger's score and the corresponding
66:52 right, there's another car. a real man On one end of
67:02 case you can see it just comes and just it's an eye director doing
67:09 value. Multiply this example right hand and do this on the left hand
67:16 and that's the truth. So this an example of a million matrix complex
67:23 value complex sector. There is the carpets again. And and the corresponding
67:34 . Okay, so and experience this a nice the animal rights drips on
67:47 case things are trivia and his hands . Both one and 2 are are
67:56 serious? And the corresponding on So when you have the diagonal matrix
68:05 diagonal interests are the final rounds. a number of the methods for computing
68:13 get values and corresponding I am factors to diagonal matrix and having original matrix
68:21 some form and then you try to transformations on the matrix. So you
68:26 to something that is No no, me. It's also the case that
68:33 was given here which is an upper matrix in this case it's also the
68:40 that whatever is on the diagonal and upper triangle reform um are the Eigen
68:47 . And then from that figure for again that pressure that corresponds to try
68:57 . Mr there's a little bit more than that. It's not given you
69:02 take matrix audience directly this over any of the values in the matrix represents
69:08 embodies from the previous example. So this case is the population for now
69:15 we can check it that it turns it was too on this connector for
69:22 night tricks and verify and it's just see just for example under this same
69:36 then I'll tell you how you find what are your values and that's what
69:40 class This science and in the next does not. And here's another one
69:47 in real matrix and mixed case complex came back here. Mm hmm.
69:57 . The next thing has done a in the book. Has to use
70:07 can talk about that next. But not the practical procedures again, except
70:16 very small systems and we have a . Okay. But they do advise
70:23 this transition zone there 1000s, hundreds thousands of operations that have so 100
70:32 this topic. But this illustration is as a way of understand the kind
70:39 approach that either. So what we're to do is print screen to have
70:47 thing that was supposed to exit access in vector in lambda isn't buying
70:53 This relationship is true. Which you best rewrite as moving everything to the
71:01 hand side. And then when you zero, that's not the right hand
71:04 . So that means for this to interesting. Obviously if X zero then
71:09 tool is zero. So that's part the trivial and not just so the
71:14 is to find a combination of number . Our Texas Non tribute to zero
71:20 that this equation is true. Mm . And then someone needs to look
71:29 the properties of this matrix here. remember something from the old things.
71:35 it's the determinant of this guy. . So this is not just focused
71:39 the matrix. Then the system is . So then we can find nontrivial
71:46 such that this product is to what this now? So this is not
71:56 equation, which one can try to Landers such that this equation is
72:04 And that's so good. We'll talk procedures have to do it now.
72:08 dealing with the characteristic elements. For . Ah So that means when we
72:14 figure out how to give up the . Sorry, I can't do that
72:22 get a polynomial in terms of Yeah. This pee is then known
72:33 the current list for No. Well is the arguing about this for which
72:38 determinant of a minus on the Mm hmm. So and here's some
72:48 . If one doesn't remember how to the determinant to this, This guy
72:53 crossed out 1st row and first Another determinant of whatever the remainder of
73:00 . And then you can go down row and the next to this column
73:05 second row and then we have the So what happens is that?
73:11 the determiner. And then you personally this summer. Thank you determinants.
73:19 then you know in this case if of a from direct a. It's
73:26 is online. Then we have numbers all these questions and they have some
73:31 of numbers. So this is kind potentially construct the characteristic polynomial. Just
73:44 few uh couple of examples um I'll back to that later but there's arguing
73:54 may have multiplicity if factors can be out in this form. Yes,
74:00 are somehow. Okay. And other that it's a real major still come
74:06 to that too. Then there are properties between pairs of our values.
74:13 subjective. Use the notion of protective . If multiple system this large in
74:19 monument is very supportive policy and this things that will come back to trump
74:24 competition procedures stuff. As I said on an H. S.
74:30 Such depth it becomes diagonal and then have the diagonal. The argument values
74:38 of this matrix straightforward and then we relate ideas, values and vectors of
74:45 two originally and that's what we're talking next time. But for now let's
74:51 this comment um characteristic polynomial. He naked say they wanted to doing such
74:59 determine finding Eigen values for this matrix you determinant or indiana assam dies with
75:06 Van die. And then this is the nature same one as some guy
75:11 determinant of it. And the way determined computation is then take this guy
75:20 whatever is left if he crawls up row and first column which is this
75:25 and then the next term is best left before here and then a few
75:32 this one and this one and then have seven left in 1914. So
75:35 is not the determinants. And this now the characteristic polynomial that you can
75:41 about this way And putting this equal zero, then you can solve all
75:46 states of this tattoo argument is. that -4. That's a straightforward um
75:51 get the values of characteristic for If you have something nice and easy
76:00 , it doesn't give the argument respecting . Um, so um another
76:11 All right. Ah this is just so now here. So then the
76:20 needs to solve The next one and zero and then this constellation, it
76:29 tell you how you're actually thinking about . All of the things we can
76:33 that this is correct in this It's not a good procedure. Again
76:43 . Yes, that's just a small event, but it's not. Mhm
76:51 , I think this is a good . Well, however it is
76:59 So the best day it shows you know, so this is all
77:02 diagonal matrix, it's a small entries the diagonal then they are still
77:10 Are they large areas in the mm hmm. I guess the point
77:24 yes, but it is a reason it's not a problem because both Amanda
77:29 next product, nice and non limits both land and unknown. But um
77:39 himself so move on to the next . So there's a whole lot of
77:45 and understanding of matrix properties before But hopefully the importance of and supervisors
78:01 composition. So significant trouble understanding the . Let's go forward. So I'll
78:11 the proper text pragmatism. Probably more to cover. Oh yeah,
78:23 thank you. Okay.
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