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COSC 4364 Numerical Methods - Lecture15_Review
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00:53 Alright, let's see if it So yes. And I'll cover just
01:01 . Yes. Most cooking all of . Uh, when somebody tells them
01:05 the first couple of sides, Destiny you walks in talked about basically number
01:15 . And when it comes to America tends to be floating point most of
01:20 diamonds. Part of the reason. , where things becomes a little bit
01:25 difficult and living within five years and different months and come back to talk
01:32 each of these aspects. No, monitor seven. No, please.
01:46 feeling so much. Yeah. So would seem before about the specs players
01:54 you orders will be okay okay. side should be on the platform.
02:19 as I remember that being uploaded to um, I haven't talked about any
02:27 solvers and finding and we'll come back talk about this a little bit more
02:35 just probably the first. So, then as being with these parts,
02:41 means that's pain in chapter six. . So basically the certain in terms
02:53 the internal one problem for each one the chapters in the book. And
03:01 if it's pretty simple problem then make than one otherwise trying to cover each
03:07 chapters And it was something that had them I think to myself some of
03:15 or may not have been actually on , that was from oh Lord Charles
03:24 now it's in class this year. , and as I mentioned last time
03:31 you have to work things your So and if we happen to need
03:41 to pay your bills or something more formula that will provide and um So
03:49 be here any glass at the normal . So it's like with assignments important
03:58 is for you. Disclaimer tell us you want at least what's the they're
04:04 what we intended to do. That's more important part than if you get
04:10 calculation right or wrong. That's for . I understand the principle. That's
04:17 what I'm saying is from uh and days electronic means that the default and
04:23 and let me know. So those yes depends on dance. Now this
04:35 like that emphasized the essence of what trying to convey in the course.
04:41 so that these questions were attracted, on that and make sure that nothing
04:49 concepts right? And a little bit particular sport. Not so much the
04:59 of. Sometimes the first examples have done data computations. So that's kind
05:07 the gist of it. I think said in the previous five.
05:15 So um so that was and so call the retail for what? Hi
05:25 concepts that are important to understand this of welkos versus impulse. I didn't
05:32 on it a lot so far. since the concepts oh public main stock
05:39 is a challenge. And it was from the some of the polynomial approximations
05:48 will come back to that in the half of the course and also this
05:56 , significant digits. Our position is too understand what it is. Um
06:05 back again and talk about loss of and how many digits do we need
06:09 that's important for choosing your data types boats and so on. Like a
06:17 in that they don't have percentage but think that's sensible. So in particular
06:24 important to understand that I believe the important standard. No and then the
06:32 of Manchester and exponents a little But the trigger here hard things are
06:37 on this. So the excess representation . This was centered the value of
06:44 experiments fields and Negative brain to consider 70 for it all what are explicitly
06:53 assigned bits. And there are certain values in the opponent that are reserved
07:02 things that things that are not than . So understanding again that there are
07:07 reserve fields um his support and then very different running modes um In real
07:21 there are the most sometimes support considering synthetic that things are imposed. You
07:28 get very different results. Damn. a function of what has been used
07:34 it's cold so understanding that with a you know around the nearest distance and
07:42 just less than a half. Around in it about a half then it's
07:47 up you think someone that is in middle of them you can do different
07:55 either up or down or and a choices but those are important because it
08:03 better feedback from results sometimes. And the particular system. I was wondering
08:13 there was an assignment. You should to remember what the different formats
08:21 but they're actually pretty standard again because kind of fundamental end. Many of
08:27 you use that level or anything else that subject by the airport uses type
08:33 decisions are the kind of reasonably Yes, good value of that.
08:37 work with something else or um Um that has been popular in recent
08:48 . The machine learning and because machine is computation very expensive on my head
08:54 it's also very energy and power So the big guys, they're google's
08:59 amazon's they worry laughed about utility So part of the reason for not
09:07 we're specifying or used using data That has kind of more bits in
09:12 than you need it. Important, line, not only costs of
09:20 time to do the computation, but a large degree also in terms of
09:26 utility bills. So, mm So it's uh and as I said
09:36 a very small part of success amended . This is since we tend to
09:42 mostly in terms of the small Um this gives an idea of the
09:48 yet. And representations. So that's that not only should we have an
09:57 remember it for the midterm but it's after the courses were trying to
10:02 good morning. And then there was default that was invented by google.
10:10 . Yeah. So that's again for learning that they discovered that's what they
10:18 dynamic range. So the exponents in number of bits you have, the
10:24 is larger than the corresponding younger than of it in the Tripoli standards.
10:31 that's the trade off the random numbers for the position in the industry that
10:37 were bits for. So that was important trade off. It started off
10:45 island now twitter and some of them the other hardware vendors support, they
10:54 have the apartment and it's again for applications that now it tends to use
11:00 on the computing. That's the benefit using this. Both in terms of
11:06 , energy consumption. So This has 1%. So in terms of holding
11:13 representation in terms of highlighting concepts and they stick performance, it is obviously
11:19 the most pathetic correct results but you necessarily you don't want to use.
11:25 the bigger data or more bits to this stuff from me. So
11:36 I don't know this idea. But did the study of the energy consumed
11:42 machine learning and some of the most actually learning algorithms um for the training
11:51 that consumes as much energy as the time. Yeah. Last time all
11:57 cards. Just one track. So serious. It's not marginal and then
12:06 thing loss of significance is another practical that's um it's important to know how
12:15 kind of um There was so we this example of the vegetable once it's
12:23 things that happens is when the subtracting are almost the same and since you
12:28 a limited number of digits division wants ends up being zero and then you
12:36 just a few LTD floating point representation of the normalization that gets sort of
12:47 zeros but they're really not knowing what should be then yes get shifted him
12:52 they are not particularly. And you that when you talked about just for
12:59 approximation if I remember right we have example that increased to decrease the spacing
13:07 interpolation points And for a while when ira got smaller and all of a
13:12 they started to grow so I was their practical things and then it will
13:19 significance is important and disappointed. Not there is no standard formula. So
13:25 have to be clever and trying to out what tricks you can clean in
13:30 to not be so exposed to also and in this case as you may
13:39 the used and tobago series a bunch sin X. Where the leading term
13:45 solved the big number. Yes removed the things and exposed. The difference
13:52 much more than that. Yeah and there was this other one where we
13:59 also play around with ah you send an underground properties of expressions too exposed
14:09 potential is small and extra small for . So it's still under standard
14:16 you just have to the conscientious about do you want to come up with
14:21 trick and then there was a lot potatoes serious, not only knowing what
14:29 is. Yeah, so hopefully the that was kind of standards will be
14:36 to figure out stop afterwards of what approximation so and I think yes,
14:44 we use taylor series expansion and then goes on forever in both cases and
14:52 they're coming upon so usually amendment of approximation and want to understand how good
15:00 is. one looks at the tailor's and figure out how many terms have
15:06 covered but the approximation and then there's rest that is not coming and that's
15:13 problem. There was another episode that kind of flying that for the first
15:20 here, that is not included in approximation part. Yes, then their
15:27 , that's what we used to figure the order of the normal approximation for
15:32 derivative approximation with the order of integration what was bad and trying to figure
15:39 how many of these stated seizures terms covered but whatever question you used and
15:45 that, the less overt terms, that was conversions and then there was
15:56 special case where the signs and the here expansion here as they are calculating
16:06 , negative. Get it down You do not need to worry about
16:11 to estimate and all the rest of serious you can approximately behavior or everything
16:17 with the first time that's beyond but was this whole alternating series and then
16:28 was this main value there and that convenient. Any context for the
16:35 A function is continuous and as derivatives an interval from a today, somewhere
16:41 that interval the derivatives. This is same as the so for second between
16:49 straight line between A and B. that gives you kind of the slope
16:55 somewhere the function delivered in equal start stuff and that happened and um maybe
17:04 more than one Point to see whether is true but at least there's
17:10 So this is just me about your two cases to properties for stuff.
17:20 knowing again what I mean by that what it is. That's a good
17:26 . So and then there was this individual that was just it was synthetic
17:33 that especially in this the separate Yes. So when we did,
17:39 we have a problem with that, it okay if we use that since
17:42 position limitation. So we have like swine and the farm and then we
17:50 run the numbers or I mean if to calculate the, can we use
17:57 like a little notation. Yeah, I understand. Right, explain.
18:08 So I think I think that one the assignments or certainly one of the
18:12 details. Line for the lecture. . There was just showing the savings
18:16 arithmetic operations learning they do this form opposed to having some of Yeah multipliers
18:27 mono meals instead of us doing something constant plus whatever. Eight times X
18:33 B squared. So in that place point as in this savings event carry
18:41 operation. I want to thank all time and that's exporting the book when
18:48 commented on what's in the lectures for . Yes it's always good. Just
18:55 very offensive operations but in real life a memory system. So depending upon
19:01 the variables are, family moral concerns but looking for the member system mentally
19:12 and reduction of many many accesses and . They're still in many situations you
19:19 to minimums so we'll see if and there was this inflation that's used and
19:33 what it means is that it's basically arts um Inspector. So you have
19:44 polynomial and um back there or something if you're lucky they are here efforts
19:52 be such that there's no remainder in space it's basically it's a true factor
19:57 the polynomial because there's no remember and we use that in a number of
20:08 . Not necessarily been route finding in southern procedures. Just understanding this principle
20:15 want to do too. Yes. I don't know if this is the
20:24 scheme where you have in mind and the question. Yes. So that's
20:32 fine. And that's again based on idea of what and here I think
20:40 is as a notion in this, it multiplication and the scheme of us
20:47 basically that you go column by column left to right. And then the
20:55 course the highest orient contrition is also historian coefficient on the there were the
21:02 that is on this online. And you get the next term of basically
21:07 factoring whatever is the result of the column comes uh, the factory trying
21:15 remove and you put that in, know, against this proficient. And
21:23 we have colonists as a procedure. it's a very simple schema to come
21:30 with the tube coming over. So how this deflation process works or
21:41 they should be coming here. and this is one more example where
21:46 can continue. Then that's fine. to you again. And that may
21:50 really interesting. They want to find wherever there was something that was than
21:55 fungus is often then that's the least . So you can repeat this And
22:02 number of times to get successfully lower for the documents. And that
22:09 that's what we talked about the horse the whole news schemer and restoration was
22:16 much and how it works and what meaning of the terms are in your
22:22 something hopefully mostly reputation for most of . I'm sorry before coming to us
22:30 much about the past and the elimination solving systems of equations. But I
22:35 hopeful in nothing to take that Maybe the concepts of and then using
22:43 . What happens in the process of this in elimination. What's what's the
22:48 to lower triangular? Yes. The of destiny was like a fact
22:56 Right. I remember when I when did OK and we only usually do
23:04 the ford. We did the hard but then we did like but then
23:08 eliminated all those other values. So gonna try to get like like some
23:14 of identity electric. I see. , you can. Yes. You
23:19 what they're talking about? Yes, can do it that way. But
23:23 it's not. So I have to for instance. Yeah. Not necessarily
23:31 . But math library is out front no, I don't know that you
23:36 know, can you? Or it's not because they have all kinds of
23:42 things. But um it comes in of the routine that does this thing
23:50 just work on the matrix A and produces the L. And the use
23:56 then you have other routines that are of triangular system solvers and so the
24:02 systems over one part of it myself the right hand side of the And
24:09 is sort of the form of elimination the right hand side and then there's
24:13 triangle is over the, that applies kind of you in some ways to
24:19 right hand side. Just unravel things the bottom up. That's what's going
24:25 . So usually you will find the mouth flavors factor between separate trump will
24:33 having resolves. And the reason is the medicine areas you have one matrix
24:42 . And you may want to yes what access for very different balance of
24:51 So I can be they may represent thickness or weight of the structure of
25:00 cell tower or something. And so it is what it is. But
25:05 you may want to look at it the various wind conditions or something else
25:11 pressure on the structure and you want evaluate but it's the same powers.
25:16 the same A. But then you at different scenarios so that's why I
25:22 to separate out in most applications dealing A. And then figure out access
25:31 respect to be. And the reason think that next side. So the
25:36 is that dealing with that is expensive dealing with the triangle of sorts dealing
25:42 A. Is it has been an operation. So you know, if
25:47 can use the same or F. . Is the same in many
25:52 You just want to be the ones they used the result of that to
25:58 at the projected and that's what caused structure. Mm hmm. one set
26:05 routines for during the fact sensation and than finding two sort of right hand
26:16 . What? Mhm. So and the problem with the gossip elimination
26:28 The way we tend to do the supplier multiply one have a role
26:36 the matrix for the proper packages. wants something then in a given column
26:44 no, we compute the multiplier but kind of normalized. They're old.
26:51 using as a pivotal also. But I am agonal element is one.
26:57 I mean sometimes that they can divide the number that's close to zero and
27:02 goes into the nation doesn't. So that sense there was big news.
27:08 partial pivoting to when you're going to what they're going to do next.
27:15 if you have got to hear a since this part and now is comfortable
27:20 this column, you don't just take top row and whatever is left to
27:25 done. But you look for the that has an artist absolute grand in
27:29 bag and also want to normalize All the things will be the most
27:35 but that preserves the scale and then elimination But it's not cannot avoid.
27:43 the most. 15 snipers out They have to do partial pivoting in
27:50 . If you don't need to search another role or the search for another
27:53 but you don't necessarily need to fix . So but there seems to be
27:59 is to write your own little thing have to be aware of. I'm
28:02 you don't divide by a small number and then there's the big uppercase.
28:15 did you use this? And we about supplying there that I've used in
28:19 have systems to find proficient store um sprites. It's also used many different
28:29 for the song partial differential equations. So it's a special case also using
28:38 elimination. But Since most of the cities are zero practically you never store
28:44 . You just store these three non diagonals in terms of the and then
28:49 just operate them. And it's pretty since Most of the parliamentary zero just
28:56 below the diagonal. So in order them and the only thing you need
29:00 do is modify the and that goes the way down. So it's very
29:06 cheap to be the china national And it was just showing also that
29:14 it is what's known as the that's you don't have to worry about
29:18 and those information and preserves this diet . So that's one thing to be
29:26 of it. They are going to the systems that are nice but it's
29:30 good to know that the operations you want to do that and information preserves
29:36 property. So this kind of race or not. So that was I
29:44 just chosen computational just chose gratitude to few operations. So in this case
29:54 dealing with tractor ization of china in system it's not enduring the still the
30:00 or whatever it can. Just a operations for you. So that's part
30:08 the thing to and things are doing right substitution. So okay, that's
30:18 other questions on so understanding and just relationship way back and try that on
30:27 . Maybe not, but they're It's good to know what is property
30:33 are going to dominoes it preserved. is preserved for them systems. It's
30:40 necessarily what they do reserves. and understanding that Yeah, absolutely.
30:53 divers out there. That's the tribunal and they don't use 10 square storage
30:59 this store and everything is a married number and the system Okay. That
31:09 finding um, the important part overall it's first to remember which ones what
31:17 of approximation order that or convergence That's proportional experience. Hello, the
31:27 the intermodal researching this case is So, so the dissection is simply
31:35 dividing. They evolved in half whatever in order to try to figure office
31:41 the group was and that means it's converges linearly in terms of parents all
31:47 time. Whereas other methods I could that's an in particular that's convert is
31:55 faster in terms of the error as function of the distance is working.
32:01 that's good and we give some but the first order of business is
32:06 know which was essentially then there's a difference between them in terms of which
32:12 might come much faster than not. practically there's no jewish difference between
32:18 But these methods are in particular Newton this version that was effectively Newton where
32:27 approximate the but that they are basically where versus order H. And that's
32:35 when you try to select the method find the road that you may want
32:43 consider something that the work is much . And there are private simple method
32:51 this was I guess what you that's the dividing into our criteria and
32:55 figure out on which side it will hold on to that particular something.
33:03 from secretary anybody simple and you discovered there's been the signs of the function
33:10 the endpoints of the interval should be . If it's not, then you're
33:17 of out of luck with this subsection in the sense it could be ruled
33:23 it could be many routes or the way when you're sure that there is
33:28 least one rule that is from the side sometimes. And as I
33:37 there is best way is kind of knowledge or adjusting game. Yes,
33:47 and B. Trying to figure out just and we seem to go that
33:53 but it was stolen. Stand on picks up understanding something without the function
34:05 times and we'll find out I don't anything. So this was just simply
34:14 to figure out. So it's very to figure out if you want to
34:20 intolerance but that's how small the names all needs to be. That's it
34:27 you don't know where they are also simple just um error tells you the
34:35 of the And then so that's pretty the bottom line of these things and
34:42 that for you to figure out if do you really sort of dissection process
34:47 start with some sort of big ones you can tell. And the customers
34:52 organization to figure out in the constitution best for you. There is determined
35:01 the small intervals so we can build much directly if you want to three
35:06 of accuracy in order the length of Yeah and that's a little bit just
35:17 the difference and the refinements in some on the straightforward by section. That
35:24 instead of yeah splitting things in half is trying to look for in the
35:32 or the straight line between the two and then look at that time process
35:42 well that's equal zero about the function the state's 90 0 and then figure
35:48 what the function value is at that . And then she was whether the
35:52 is to the left. So instead just picking the midpoint, look at
35:59 point and that's this regular fancy. it's kind of fourth position of
36:06 Whether it works better or not depends the function. So what's the point
36:11 it? People think that works off often enough that it's out there.
36:21 then there was once of trying to more clever and trying to find instead
36:26 going to during the regular dialysis step step, in this case, the
36:32 step would have been the next step them. So it kind of straight
36:38 between the new um 10 points over interval and this one, but then
36:45 modified instead of going just wide, do a basket follow, find where
36:51 line parallel to the one we just , we're in the process dex
36:58 And we picked up and in some , again, that might be working
37:04 . But um so this is you're a complicated at least understanding what
37:13 differences are. Yes. So in case that all both times it's less
37:19 less than zero is their case where , or I'm saying like let's say
37:29 say you're doing the bisexual and then and then you find that that's a
37:35 let's say season midpoint of mp. then you find out that of a
37:44 mind. Never mind. I feel feel like I feel like I think
37:48 . Okay. Okay, so, then the Newton's method is the most
37:57 method um to remember of them all other ones are simple. Newton's method
38:03 not particularly complicated either in terms conceptually you know practically. But the difference
38:10 that in this case very basically it's marathon point. So I guess the
38:21 approximation of where the routers it's the of the attendants of the function at
38:29 here And of course then you need look at the functional element was not
38:35 but then investing all success single but that the different points that okay and
38:41 if you do that, if it out that the convergence here is apologetic
38:47 this is the reason for the All of them is transparent.
38:55 It's a little bit more involved in sense that not conceptually as far as
39:00 concerned but it requires two to both have access and can evaluate the function
39:06 spanish. So and as I pointed under the approximations pulling on over or
39:15 many times the expensive part is this about women now, you know the
39:23 or this subtract operation that's cheap compared you guys in that sense, it's
39:30 expensive than the other method because you to evaluate not only the function but
39:35 its derivative. So in that sense computational more in terms of most of
39:44 time. It's well worth it in center it converts was quadratic and they're
39:49 to be. So that's the reason the popular. So again, the
39:57 that works and what the error term , but also, but the difference
40:03 computation between the, it's just, . Then there is not too hard
40:11 figure out hold the list and quadratic . And I know sometimes that I
40:18 that question at least, I don't , don't remember him because it's simple
40:27 good to be able to convince myself the government property. But it's also
40:34 case of simpler reception related letters. not food growth. So the time
40:44 . So this is part of a of the failure cases in this case
40:49 derivative is such such a That's a . Just move for further away from
40:53 zero directions for this. Um This case is the site because it's
41:04 happened to that, you know, one point finding derivatives here and then
41:10 go down find the function about me get the new derivative and you end
41:13 where we started. So you get pointed out it really extends needs a
41:22 slow convergence. But because of round in practical situations, yes.
41:29 not stuck in some cycle. And of course all of a sudden the
41:34 happens to be zero. That's kind our blood and it's also that,
41:41 , and sometimes it's always like to section, you need to find A
41:45 B. That breakfast. Ah, this method is kind of important too
41:53 not too far away from where the is the guarantees don't have sex
42:01 you know, you have to have that will be the guest again,
42:05 what you eat. I hope that will come. And the other thing
42:15 you also kicked off ah have the . Um or it's too expensive to
42:25 , you can obviously approximate the derivative this and talking about the soap between
42:30 successful forms. So this is the thing that is kind of, there
42:35 this method with working with them and american estimate of the instead of correct
42:43 violation of the and it slows down convergence a little bit. So it's
42:50 quite the square was kind of close because it's a product. two successive
42:57 are supposed to the square. So both things. And then there was
43:08 other principle that was six point Yeah. What's the castings and try
43:18 solve an equation of this form? investing six point is you have a
43:24 and plug into an argument X. I'm working on expediency exit sucks,
43:31 . So this is the the the breakfast. So we could write something
43:35 this eventually come back to the single . Same next from the less than
43:40 rights and stuff. Kind of illustrated this picture, understanding that the fixed
43:46 as his property is that if ah of contours of 31 and we have
43:53 examples, I guess this one that the G says they want to find
44:01 rules as well functions and then finding function G. That has the property
44:09 that fund to get the solution to fix point iteration type scene texas is
44:15 fact the same As the root or zero or the original function in the
44:23 . It's not unique. You can many good performance of x in many
44:28 . But again, the function. it's again creativity a little bit.
44:32 do you choose the function G? least you can? Quick convergence and
44:38 this point, thank you. Just , understanding that was this notion of
44:45 or is it supposed to work? , that was that mm hmm.
44:56 pricing uh to emphasize then had the approximation Then it talks about 62.
45:10 been ways of doing it. one the most calm and informed is a
45:17 polynomial. And so one of the things about the background formulation was this
45:25 apartment owners are only dependent on the or the points where they want to
45:31 the installation. The only dependent actually depend on and reduce that. Apart
45:36 developed some of the integration happens that separate out finding in that case the
45:45 projector function values independent or the function whereas in this formulation that this takes
45:57 successful coefficients here as well, increase order of the polynomial approximation. They
46:05 dependent on the function line. So then they have this um divided difference
46:15 of computing this case of them the values I think that's going to the
46:22 slide. Right so we have this where yes, thank you doing an
46:33 or some function. The first thing the best thing please for Austin.
46:39 there is a constant. So that's the picking the function of the point
46:45 the next improvement is what happened his methods, integration is straight line
46:54 So then best when you have in case first the concept and then you
47:00 the slope that comes with the territory then when they want to have a
47:07 order approximation it's kind of a just the term the the the curvature.
47:16 I said there's a difference in the derivative between two successive intervals. So
47:23 so as we can see the function it just gets into these coefficients from
47:29 case we don't have a nice property occupiers and polynomial as can be dr
47:37 perception is just based on interpolation that's not true And the neutral formulation
47:44 um the functional values gets built into profession and then we have this just
47:55 schema for how to the environment and the function evaluations our best for the
48:04 column. And that comes back when also talked about derivatives and integration methods
48:14 functions evaluation in those cases still happens the first parliament is Amos and then
48:21 problems against the combinations of function evaluations the function values have been the first
48:33 . So that was part of the . If that's together many terms
48:39 let's sit down something. Yeah. so part of the things again that
48:45 were both in the garage and the item that is kind of lose the
48:55 of powers of X as um pretty in expressing. Mhm fallen order then
49:06 didn't talk much about it once. want to just point it out.
49:15 frequently use much more so than the formulations because they use Paula normals.
49:24 nice properties. So In the sense they have this equal calculation, they
49:29 them between plus and -1. So never it will protect the large derivatives
49:36 sort of modest so that someone to a linear combination of this basic championship
49:46 that is an approximation of the And the same way they are doing
49:54 finding polynomial approximation. Not just based monogamous but on a paranormal various degrees
50:06 in some other way is what was also that these parts of these plants
50:11 the set of functions. They were just you know the power of X
50:17 powers. So that's in each one them. No these parts. So
50:23 that case it's the same thing when is approximation defects What's known as his
50:29 functions and Romano meals are one. . You can also point other sets
50:36 basis functions that may be in some better to approximate the function and that
50:44 maybe that if you do use. that's basically behind did you compression and
50:53 to find an alternate representation of the set that requires. Mhm Let's um
51:03 information on that case we have the function. So the basic functions that
51:10 variations and they just sort of skated how much you know, so that's
51:17 you have just as an example against the chip conscience. We haven't talked
51:23 about it but we will talk about second half of the course they are
51:30 basis for thanks compression and this is same instead of just doing inside the
51:40 way point point point. Yeah the is one of the basis functions.
51:49 just need to have the scale in face and then so that's two numbers
51:54 then we felt that this is So the only thing you need is
52:00 damage in your face. No wonder are supposed to be used in the
52:05 of the sine function instead of all values along the space. So that's
52:12 this notional basis functions and their behavior important if you have um but no
52:19 else said they tried and then we expired for trying to design expert.
52:25 . Just put it on us as of X X squared and experience that
52:29 both And they get to the data and you may need the long
52:34 I gotta go in there bus stations have used that assigned itself as a
52:40 function is just okay. So this something for it, you know.
52:48 then we used also and other properties turned out normal approximation. That's important
52:55 say that um, to remember that straightforward things is to basically divide an
53:04 for which you want approximation. Think intervals is based on the incorporation points
53:11 those and equality in the in the but that's not always a good
53:18 So it turns out that many insurances . It is better to take the
53:25 or the points for the good Important to use the contribution those zeros
53:32 this particular function. So just knowing folks routes such as for normals there
53:44 a better choice than equal in space above all the equal space notes and
53:50 all has been resource and again, don't think that's focused in this but
53:57 to run their functions and nice permanently how observation and the observation is much
54:03 smaller when they choose this point. when they saw the difference in behavior
54:14 basis functions. Okay then they have . Oh, so the differentiation was
54:27 of things to women and so we're to approximate derivatives and what to do
54:32 too many of those places. But and 2nd order the relatives was what
54:37 discovered in the book and it's one to remember tend to be generally
54:48 So called one sided approximation. You the soaps that are going forward or
54:54 backwards mm hmm. And that gives approximation or whatever that is proportional to
55:04 distance between the two function radios. okay. So that means to get
55:12 accuracy in this kind of ah approximation the derivative has taken the soap.
55:21 , no, very good. You see quite small age before we get
55:26 . Perhaps they decided accuracy. That's . The center of differences then it
55:33 be do a much better job because it's these cases proportional to the square
55:40 the distance. It's not just half the square. So that means if
55:47 get good accuracy for a given level accuracy, you can use much on
55:53 age. And then the other thing the, which is an extrapolation that
56:05 just looking at this case from the difference what various terms of the Davis
56:11 is substituted knocks account for and then start The original one and then having
56:23 H. Showing again another expression here using H over two and actually to
56:29 then within your combination of these two get rid of The eight sq
56:38 So now they use this function values now they have something that converges to
56:45 fourth power of age. So it's trust again. So what it means
56:50 you need to work evaluation all in bottom here to get something that is
56:58 to the four compared to using just age. Yeah, the straightforward expression
57:10 So once again the game Is that fewer function evaluations? They got a
57:19 error charming compared to the rest using straight forward. So yes, another
57:30 , Yes. So you can keep it in a number of times.
57:34 that was the principle which is an , quite simple to look at their
57:40 and then using H and H over a number of times successively illuminates the
57:48 that is ignored in this area. hmm. Mhm. Yes. So
57:57 is what I'm saying. So. then, yeah, come back
58:02 There's lots of significance that they had in this case the ages small,
58:09 luxury values are almost the same for States current function and then whatever condo
58:15 errors. Again, in one of , can we go to the difference
58:19 the function value. So that means get something that has potentially very high
58:26 . So that's also something she you're wearing one of those plastic doing
58:34 is very aerial drones practical. That's to have evaluate numerically. Not just
58:45 derivative, but I mean a I would recommend to use some software
58:52 out there mathematical or about 51 right the best. They tend to do
59:00 good job and taking precautions. errors don't blow up and integration and
59:16 also the integration is much more forgiving in that case they don't get to
59:23 to find that their derivatives or it's of smoothing because it has things up
59:36 a good way. So areas are and down eventually to summon up.
59:42 tend to cancel each other out much than in the text derivatives and looking
59:47 challenges. Thanks. So, and simple methods was just to know that
59:55 tractors or method in the investigation doing straight line approximation to the function and
60:05 clear counting the area for each one these chapters. Oid and someone
60:09 So this is kind of being he's your approximation of and then look something
60:20 there areas for each one of these off and it's a very simple schema
60:27 there might be another expression of the 8 to 11 points and then your
60:36 all the function values in between if have the same interval length. Otherwise
60:42 can wait individually with the distance between , but it's the given fixed in
60:47 world and it's just there for the silence. But some of it's a
60:53 simple formula two ways for me to and this. So it's not too
61:05 . It's over. It's square, the next month that we talked about
61:09 was the Simpsons, you'll have basically age forum was Not much more.
61:16 I almost have to come in The one necessarily integration and don't bob is
61:26 right that I'm called to do something . The Simpson's rule instead of the
61:32 , it's not practical expensive and there's much actresses on the end. He
61:39 saying computational expense budget um I guess that but they're also the recursive algorithm
61:53 that was the news too. Supply extrapolation to the trap is on
62:04 So they but the recursive thing here that when you add got an interval
62:10 half from one to the next. have already evaluated all the function by
62:16 at all these points. And then new is the function values of the
62:22 points that they didn't do before. then invest the halves of previous integration
62:31 And now that they've been divided by different it's now half. So they
62:38 somewhere the red triangle is half so really something and number two here is
62:48 and I just need to fight in , results from the function values at
62:54 point with the proper, with they the world was just using after having
63:04 the british state forest in performance. when the companies wanted up to only
63:15 half the weight of what it used be the first time in Georgetown and
63:25 following the same sort of on bad I want and then find the schemes
63:30 there on the functional evaluations are in first columns and then one is a
63:36 of two of these um again the car, then when it came to
63:45 the Simpson's rule, It was only square. So so it then in
63:53 long boring it's the same thing, this one an extrapolation because some other
63:58 came up with it. So algorithm it may be that some extra operation
64:04 systematically. It's all down here. to the next column where they're gonna
64:09 too for doing this combination of values then you guess what it's for so
64:17 . So as you move from one to the next, we do not
64:23 the function of our relations. So you probably don't think on this
64:29 And it was done golden sample and lecture was basically you look at what
64:38 do you want for them and in and you know what column you need
64:42 go to. So that's the best only don't need to cover the rules
64:49 gives you the diagonal elements moving here has the property. So that was
64:59 cold example. I was founded in book Monroe because something. Mhm
65:11 And then I got something to talk Simpson Simpson. So then we took
65:16 other view of it. Like I said, what happens in the
65:22 It was um mhm fiscal linear combination function values. So for the tractor
65:31 there were arrested two endpoints there was them that was on the road with
65:38 perhaps investment. Oh the kind of height set the context for a straight
65:46 between the once. So then in more general way of looking at the
65:53 methods and said okay they want to an integration method. The integrates polynomial
66:02 something that you need correctly. So why they got and so then one
66:10 was then so okay instead of doing in your approximation let's try we're doing
66:15 approximations and now we need one more evaluations to do that. So and
66:20 kind of the arbitrary that we just the midpoint between men and be so
66:28 find what are the ways or is function values? In order to make
66:33 the quadratic polynomial is integrated where and you saw them so was very
66:45 Think of it in this way that have is cardinal polynomial in that
66:51 The integration of this cardinal polynomial then should have waited for this. One
66:55 the functions. That's fairly simple principal to think of it. What you're
67:02 to get this conflict looks like an combination all the functional lines in the
67:11 and then it's time to produce the . And so I guess I can
67:20 out but that's um So in this in order to find the eggs and
67:28 seas what they did. So it integrate um in this case, not
67:37 a linear approximation like chapters or goal didn't if the function is indeed,
67:46 a bunch of straight line segments. exactly correct. So now I wanted
67:51 talk to correct for mm hmm. that means the integration are respectable.
68:01 constant and text and X squared. to be correct. And that's the
68:08 . So then you have three conditions to integrate X and integrate X squared
68:12 integrated for X and X squared. you Stephenie Immigrations. And that was
68:18 us to get the politicians, c questions, the unknowns. Then it
68:24 out that the white ends up being actually it's so well. And then
68:31 one could also then on what to to traffic sides and not give them
68:36 example. But it turns out that the difference in that case, it's
68:40 the different weights of the function values the end. Still using the function
68:45 . But the way these three function are used ah Is better than the
68:53 since method in terms of X to 4th approximations instead of. And then
69:02 have the script and councils that we this notion that was sort of close
69:10 interval and that in this integration intermodal of formulation and like but it doesn't
69:24 to be including the enforced with them choose uh other interpolation points. It
69:33 meeting to the end points A and . Um and if that's being done
69:39 what one guest was known as the intervals. The open interval is respect
69:45 father notes are chosen chosen for the of the hello And the reasons in
69:58 cases let's see what I have next that. So examples, well guess
70:05 one of them instagram with employing around a few examples of sin X over
70:12 To integrate that from 0 to The dynamics of that is the principle
70:18 the time. I'm actually zeroing the that's zero divided by zero. So
70:24 stick that into it will change the that they've been trying. But it
70:30 mean in the case of the shoulder has a well defined integral and the
70:37 more directly be replaced in annex with overseas expansion to start with the next
70:44 then it gets That was X&X to next to the 5th power failure series
70:50 Cynics so then becomes the one and experiential in that case When X0 them
70:58 show that they're very strong equals one has to put that part what
71:06 instagram or a good interval but you not want to do in a normal
71:13 where they function by the museum or that interpolation points in the interval of
71:21 and B by the voice. So was the notion that it's open amount
71:27 they don't include 10 points. During the approximation. And then there
71:38 this girl squad richer. And the for including that is it's similar to
71:45 we talked about the polynomial approximation. do you choose your roles? And
71:54 is the argument was if you choose known as the points where his championship
72:02 has its roots then those are using as notes from the interpolation history in
72:09 areas. Now in terms of integration kind of what discussed products. It's
72:15 clever choice of the interpolation points in polynomial that has integrated correct. And
72:27 find those points by finding phenomena excuse has the property that integrates some power
72:37 X. For this polynomial to over . Sorry to interrupt here It is
72:44 zero and that should be true then a wonderful meal so far as the
72:50 . Up to one or 1 or so and versus team as long as
72:57 the order. There is one destiny this point. And then you know
73:02 the find their roots supposed to find accused then you find they are also
73:07 . Then you have the points of . And then so you can get
73:13 move on to try to figure out you wait those function values from the
73:18 war. And in that case the is it's basically yes. Ah The
73:27 is respect upon an umbrella twice degree the step forward choice of notes instead
73:35 them best free and equal to the of polynomial. That is now almost
73:42 delivery. The number of points from point here is kind of the steps
73:50 find that you find the roots. well done. It is no you
73:57 the loads and then they can move to what they do before to find
74:01 weights. We'll have to wait to brothers at those notes. The what
74:08 expression for an integration formula. So is the, that was quite popular
74:14 it's it's so um do you so order accuracy reports And then there was
74:24 genre that figures out good ways for these two point moments. We don't
74:28 to remember them. But just that's principles. So there's basically tables those
74:36 for abnormals. The founder of paranormal gives you, they finally not going
74:44 ask for my number. Yes. know that there are so oh some
74:51 in almost have a right way that defined the personal formula too.
74:56 successive for a moment and then there tabulated. This is fairly simple that
75:05 know the formulas are developed or standard zero. The one or sometimes It's
75:12 to have a semantic into one and and then we can scale the variable
75:18 the expression So it's one part is translation and one is stretching so how
75:35 translate things from running towards another one something that it is very good.
75:42 also what the launches try to remember to do those in the world changes
75:51 that there's a translation parting the midpoint where it should be instead of being
75:58 or And that the interval 1917 1 two the main challenge and then there
76:11 a little bit of trickery to figure how to do the work potential singularities
76:17 sometimes it can be a terrible Yeah, that's also and finally it's
76:26 working on slides, slides spines so separation. So so basically the things
76:37 remember the difference between lots and they also interpolation points and not our
76:45 for smoothness by the while functions themselves or where derivatives are considered as
76:55 That's great. And then you can stuff the degree of the spine tells
77:01 that's how many um derivatives that we the function self zero. The relative
77:09 continuous number of derivatives that can be is up to one order less than
77:17 every other polynomial. And I'm sure it's trying to force the, the
77:27 from the same order as a So we continue. It's going to
77:32 up with a single response. So and the government example in this
77:45 the other thing that was example where population points were chosen to the different
77:53 . Oh thank you. No. where are the conditions of our
77:59 That's a warning for a normal. it was also that for the first
78:07 first degree and everything was define so to speak by the population
78:14 But switching to higher order spines. there were for the 1st 2nd degree
78:21 was one order and one degree of . And All right, paul until
78:28 expands on the industry. 2° of left that you find the continuity conditions
78:36 the function. So when I guess can use this extra three conditions to
78:43 properties today. What? Mm And I guess that principle that the
78:53 was started with things that are the order delimited that was supposed to be
79:02 . And then we assume whatever their of that was at the house,
79:09 were unknowns. But based on those , we can write down a bunch
79:13 migrations and integrated things up from the of the derivative. That was
79:24 Mm hmm. This fire is linear across center and once we have integrated
79:35 so we have the equations pretty much all the spines and is applied interpolation
79:43 and conditions of the other. During , not the highest order because that
79:48 satisfied by construction. And then we a bunch of equations. And eventually
79:54 for 4% before that for a second . There was one recent conditions that
80:00 needed this for the child and decided and for the To the spinal needle
80:06 two things. But basically we have machinery in place to figure out then
80:12 is assumed during which the gardens were . Good morning. So then we
80:20 up with this one case the diagonal is kind of a degenerate version
80:26 Let's try that. I go Um, for this opponent was the
80:30 system and productivity. Few respondents also systems, but the disease are our
80:40 , the relatives that were being sold very good construct. Sorry, very
80:47 for normals. And then, but these points we have formula and like
80:55 said in this case in different shapes the championship Kuala normal. But in
80:59 cases there are and the basis functions we produce thanks wow. Or
81:10 The spines are various degrees and for second degrees planks there was a bit
81:17 accomplishment machinery. Let them also was others fights, not just for these
81:24 . I think it's time we talked last time. Populists and question
81:32 And this is why you want the in the first place is that's what
81:37 repeated a few times. It's better many the waterfall, a normal person
81:41 the world. And if you hire have brought enormous possibly the order polynomial
81:48 not get lost. Very much so if there's two things exactly the number
81:54 in preparation for instance what happens in And that's why the other one is
82:00 . So many of those. So think that's why that's the opposite.
82:11 was a quick one. Just just higher order things that hopefully it's not
82:17 hard to run them. Okay the final exam, final final,
82:35 good. You know? So well in principle, yes, but in
82:41 not so much so that most of questions that McConnell will be on the
82:45 board from there. Yeah, it be that there is one or two
82:51 , but in general it's best. supposed to be what hasn't been assess
83:13 . Alright, so we can send really? Okay, so you want
83:23 then one day? Yeah, Thank you. I just need a
83:28 bit more because I'm like overwhelmed with bunch of like understanding friday.
83:41 so monday, it's okay. I'll you Tuesday doesn't matter. Okay,
83:55 you say we just focused on the and homework problems something? Mhm.
84:03 think it should be enough if you familiar with what I covered. Obviously
84:10 is texts that may be easier to from and they won't know what's coming
84:17 I have not talked about that is the book. Yeah, if you
84:21 need to worry. Okay, no . Okay, thank you. Alright
84:27 , thank you. All right, mm hmm it's going to run.
84:51 , not quite Okay. Oh um , that's fine. The weight of
85:12 object. Okay. Yes, you'll be back for the yes.
85:27 , yeah. 6 45. There just, mm hmm. Oh,
85:44 . Okay, 65. I The Okay, implementation of problems go
86:16 . Okay. Oh, okay. the author of the I've got
86:34 Oh, okay. Yeah. Okay. Mhm. Yeah.
92:08 hello. 1.5 minutes. Oh, . All right, mm hmm.
92:35 . Thank you. Mhm. good. All right. Mhm,
94:39 hmm. Yeah. Right, mm hmm, mm hmm.
96:33 Sure. Mhm. Thank you. . Okay, mm hmm.
98:31 Right. Right. Mhm. Mhm. Right.
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