| 00:01 | All right. Thank you. That's nice. Mhm huh. Don't |
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| 00:17 | That's still a performer from the Alright, alright. The old |
|
| 00:25 | Mm hmm. So that starts to reach up a little bit of finished |
|
| 00:30 | the last time and mention just very something about another type of hold no |
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| 00:37 | and other than 15 years of the in terms of different ways of working |
|
| 00:41 | insulation and hopefully get to talk a bit about differentiation. Mhm. |
|
| 00:48 | that's it. Okay, so this what So last time as a general |
|
| 00:59 | that when it's not limited just to polynomial terms in most applications that actually |
|
| 01:07 | other functions Suffice in this case instead polynomial is two interplay or get them |
|
| 01:14 | to accept the data and then take to them to use there is an |
|
| 01:21 | expression or the functional expression you get using these functions to do interpolation. |
|
| 01:27 | you can get an estimated values might been between the points. They used |
|
| 01:33 | be defined the feeding and then the that the a bunch of points that |
|
| 01:41 | wanted to the fifth thing that is by values and yeah, basis functions |
|
| 01:49 | that the various notes that you use the sitting and then to get a |
|
| 01:54 | of equations and then we get the for the difference basis function preparation and |
|
| 02:04 | on. Yeah. Also pointed out it's different kind of basis functions being |
|
| 02:12 | so far without falling the basis functions this cardinal polynomial from using the grunge |
|
| 02:20 | um Modelo 50 and then there was news number one just has a different |
|
| 02:26 | and these terms of the price as function of I quite different behaviors compared |
|
| 02:35 | for instance the cardinal polynomial. Because difference one can see here there's also |
|
| 02:41 | denominator that tends to preserve the stale this kind of uh as a physical |
|
| 02:47 | roll so there conceptually and behavior It doesn't mean that they never work |
|
| 02:56 | it's different and the same thing that more known and also it is a |
|
| 03:01 | way be right for an armistice various of X what they independent variable and |
|
| 03:09 | the proper coefficients. So that's kind where they ended up last time. |
|
| 03:15 | I said I just mentioned this is good point because in the book they |
|
| 03:18 | back but these things much pleasure in of looking more sophisticated ways of doing |
|
| 03:28 | . So this temperature polynomial as they and a very nice and that's the |
|
| 03:38 | for them being quite popular and they're in the sense that there are always |
|
| 03:44 | and Between plus and -1. So that sense nice being normalized regardless what |
|
| 03:53 | degree or the order of the polynomial . It's still Has this bond between |
|
| 03:59 | and this one. There is the of your personal construction that you see |
|
| 04:05 | um that they don't sleep like when kind of simple things but. And |
|
| 04:12 | they said in the book and that guess tends to be true that uh |
|
| 04:19 | there are also alternate between The two points plus to -1. So ah |
|
| 04:29 | not sort of hanging on to one the extreme points that go up and |
|
| 04:33 | and you can look the first one a constant, so that doesn't then |
|
| 04:39 | , but then the next polynomial is X. And that of course between |
|
| 04:44 | one plus one. And then in of the second degree it was such |
|
| 04:47 | order polynomial and things were going from one Down -1 and come back and |
|
| 04:55 | they can follow them. They are have risen very orderly and that's uh |
|
| 05:04 | the reason in addition to having use the construct that they used football a |
|
| 05:10 | approximation. So give it one That's it. Let's see. Uh |
|
| 05:22 | . And you worked sing along with anyway, so this is pretty much |
|
| 05:29 | um except it also says one more that I did say last chance, |
|
| 05:36 | come back to that little bit more that yeah, sort of natural or |
|
| 05:45 | least that's the starting point to um the nodes for I really want to |
|
| 05:53 | the fitting equally in the interval of . Otherwise one obviously needs some kind |
|
| 06:03 | rules. And how did you choose ? Uh huh Because if you're not |
|
| 06:09 | the straightforward version of treatment and equal , that was just a good way |
|
| 06:15 | picking them. And I'll talk a bit about that today. And that's |
|
| 06:19 | to the ships of enormous. In the way of picking into police the |
|
| 06:23 | when they're supposed to. That's it this space, correct. And so |
|
| 06:33 | come back to that in a So back down here. So makes |
|
| 06:40 | think in the roof was what they inverse insulation. So it's nothing magic |
|
| 06:47 | the club is sophisticated this bus simply changing the roles Text and one which |
|
| 06:54 | is dependent that we transfer the independent . So, so this is just |
|
| 07:01 | chemical newton forum in this case in the violent values as the independence and |
|
| 07:10 | figure out drafted in that part of it corresponded. It smells like songs |
|
| 07:16 | that. You get the conditions based X values instead of there's my |
|
| 07:25 | Well, so I think they that's another is the slide show for |
|
| 07:35 | . But this is the same So I mean obviously demographic and flipping |
|
| 07:40 | for this. I get from this to this one. But basically rotating |
|
| 07:45 | or flipping things so, oh Instead of X. Being independent because |
|
| 07:52 | tend to view them up then X be the next year. That disappearance |
|
| 07:58 | um then the dependent funding used to the independent. I want to see |
|
| 08:03 | you change. But this is kind the same thing over and over in |
|
| 08:08 | in that case for potential funding. . The exemption for Which the function |
|
| 08:16 | zero if you want to try to their rooms For the function. So |
|
| 08:20 | was just 15 things. Nothing in magic. So there is a concrete |
|
| 08:31 | . So on this guest um, was done. We don't go through |
|
| 08:34 | the steps to figure out what the are. But it's best that one |
|
| 08:40 | use the Y values to define the and the X values that becomes the |
|
| 08:50 | variable. What? That then goes the coefficients. And in this case |
|
| 08:59 | to try to find basically what say bye value in syria. What |
|
| 09:07 | the X values that causes that? gonna put this zero and then some |
|
| 09:14 | them you can sort of see It's got to be somewhere. Do |
|
| 09:18 | think four and 5 and probably family little bit? Uh, both of |
|
| 09:27 | . All right then minus 0.3 versus 0.9. Depending on the hardware |
|
| 09:35 | And so the Haitians at the So in this case on that they're |
|
| 09:40 | part of that. But it's just of using they're all a good |
|
| 09:46 | X. And the function value two different coefficients people the other way |
|
| 09:51 | . They use the virus and used systems. Okay. So any questions |
|
| 10:01 | that because it's it's your choice. one to choose how to use those |
|
| 10:09 | and responsible. You start values you to approximate. Uh, so, |
|
| 10:20 | far investigators, that's one independent one variable. But obviously wanted the stipulation |
|
| 10:30 | higher dimensional spaces as well. So standard way is best to you look |
|
| 10:36 | this kind of a cross products in space. You have the X and |
|
| 10:40 | Y axis. Maybe the independent defended that is for the heart of the |
|
| 10:46 | for instance. And this one space message means that what it does. |
|
| 10:56 | can use leverage or Newton or any of the methods to construct from normal |
|
| 11:01 | respect each one of them ah independent . So in this case you can |
|
| 11:08 | it for instance, in terms of find the card sample known as with |
|
| 11:14 | to the notes you have Along the axis. And then another one that |
|
| 11:21 | the mm hmm notes, you choose the Y axis. So that's for |
|
| 11:28 | independent of the cross product. And to get the actual for normal |
|
| 11:34 | then you need to have the function the different noble points in the |
|
| 11:39 | xy X and Y with the same . Yeah. Following, basically mapping |
|
| 11:49 | to step into the to space and and Y. For each for the |
|
| 11:53 | in the plane you have a corresponding value. And because of the properties |
|
| 12:00 | the cardinal called normals picks up the function value. Both of these guys |
|
| 12:08 | to be done in order for this to count. So if any one |
|
| 12:12 | them is zero then obviously you don't that. So, so all that |
|
| 12:21 | . Okay. So it's simple to the most of the match and |
|
| 12:31 | this is what I said on Both of them happen to be wined |
|
| 12:36 | they only have 51 at this The only thing, wow. |
|
| 12:52 | Somebody so this is the property is times same thing with the formulation Violet |
|
| 13:04 | for the distance the computer professionals and main difference between was down there, |
|
| 13:13 | grants and the newton is the proficiency as we see they are dependent on |
|
| 13:20 | function values or that they wanted to to fit the following whereas the cardinals |
|
| 13:28 | independent that can be constructed just from sports. Okay. And this was |
|
| 13:38 | . Let's step forward and I guess talking about that. That's him and |
|
| 13:43 | questions so far above um their scientific how good is the approximation? |
|
| 13:59 | it's important aspect of course. Um it's obvious and choose to do |
|
| 14:09 | fitting such that there is no error . Uh The chosen known points principle |
|
| 14:18 | from the back to America. LTD decision and on those that makes |
|
| 14:24 | public sector um to do the best can do so except for the invitation |
|
| 14:30 | terms of the americans, It's supposed be zero. Yeah, chosen |
|
| 14:38 | Cool. Ah So this is something activity. Mhm. All right. |
|
| 14:52 | the more points. Yeah. Trying do the same thing to me expect |
|
| 15:01 | things will get better. I remember he said that that's not necessarily the |
|
| 15:09 | . I'll come back to that stuff And also try to understand what |
|
| 15:16 | Why is that not picking? We'll about that today. This is from |
|
| 15:24 | boat. And clearly this is also have a function that isn't continuous that |
|
| 15:35 | jumps. And then kind of in of as long as there's continuity and |
|
| 15:44 | has something at least the behavior that can estimate good and approximations if you |
|
| 15:52 | them. That's what I'm talking So, I mean it wants to |
|
| 15:59 | out like an example taking more points extreme silly example but makes the |
|
| 16:08 | So there's this terrorist the respect Anyone heard of that before. Strange |
|
| 16:14 | . No, there's something in that likes as to uh kind of point |
|
| 16:25 | um behavior is that um is worth about saying that the conclusions when you |
|
| 16:38 | a nice neighborhood functions that are not true if they're not so nicely debates |
|
| 16:44 | . So the students the function is that has his bed it's defined to |
|
| 16:50 | zero. All rational value points. know, not all numbers are rational |
|
| 16:57 | . So and all the other points one. So it's jumps between zero |
|
| 17:03 | dependent if you go through everything else which ones are rational. Which ones |
|
| 17:08 | not. So what it says here So you did the same thing |
|
| 17:17 | This terrorist function for all the rational . That means all the coloration, |
|
| 17:25 | function value at all the rational So they're place for her daughters and |
|
| 17:30 | takes more and more points and whatever the function value zero. So that |
|
| 17:36 | the polynomial ends up being the values the zero regardless of the number of |
|
| 17:43 | to zero. But um if then look at the value of the polynomial |
|
| 17:51 | an arbitrary value nor specifically irrational value the polynomial is zero everywhere than terror |
|
| 18:01 | one. Because the actual function is it's an irrational and it doesn't matter |
|
| 18:07 | many points. It's almost comforting the case. It doesn't help Um to |
|
| 18:16 | more and more points because in the maximum one. So it's stuff that's |
|
| 18:20 | bad. But in principle it doesn't different. Yes. And as you |
|
| 18:25 | , there's a lot more irrational values rationals argues basically there is one almost |
|
| 18:33 | . So. So so this is this is kind of. And also |
|
| 18:41 | I don't want to talk a little to try to figure out how to |
|
| 18:48 | and then obviously it has to do what they're trying to approximate the |
|
| 18:57 | Oh yes. And then the example couple aside from it talk to this |
|
| 19:04 | and then you will see most of coming back from what I was going |
|
| 19:06 | say. But it's like documented. and I can think of this as |
|
| 19:15 | doing this kind of a newton way the sense that what we see |
|
| 19:20 | Sarah Newton polynomial restarted best of some of points. And they are the |
|
| 19:26 | points and they want more time to terrific polynomial. So in this case |
|
| 19:33 | can say it in this First we 1.0 and eight. So and the |
|
| 19:41 | this this constructive is first take the extreme points 07, 8, 8 |
|
| 19:50 | zero. Um the next one. let's start with this one. It's |
|
| 19:54 | a constant. You have too Then we have two points that affect |
|
| 19:59 | . The polynomial is a straight time between them. So you get this |
|
| 20:03 | car that is key. 1. two points. 1st degree polynomial. |
|
| 20:09 | then it's okay, let's have one then add the point in this |
|
| 20:16 | 3.2. So this cr 3 So that means that the next fall |
|
| 20:20 | I'm gonna say to them it's not bad. So and then you keep |
|
| 20:27 | points. So in this case the degree, you have five points to |
|
| 20:31 | fourth degree polynomial. That as you as a higher degree. They don't |
|
| 20:36 | start to perhaps makes sense. Who what the real value is already |
|
| 20:42 | We don't have any knowledge. But know that it's very likely that the |
|
| 20:48 | thing underlying these five points behavior like . Probably. So in that case |
|
| 20:54 | can see the no, the polynomial started more and more extreme in this |
|
| 21:01 | the more points you have remember again was constructed to just show that more |
|
| 21:10 | . They make it worse. It Matter in which order and principles because |
|
| 21:14 | your table five points that looked at think that is pretty much the |
|
| 21:29 | That that's as it gets more and extreme. So now um right. |
|
| 21:39 | want to talk more about that Marilyn actually they hate. So I don't |
|
| 21:48 | we're back to this function and short early the last lecture but there was |
|
| 21:58 | function is 1/1 plus six square um bell curve and I'll show them on |
|
| 22:06 | exercise. And that's one where increasing number of points is not helping. |
|
| 22:17 | here was their only function. And I am the signs here. I |
|
| 22:25 | the interval -1-1 instead of might start find that for some context slides of |
|
| 22:31 | book. Somebody said to draw and can get that but they're still putting |
|
| 22:38 | . Scale expert finds that strange. intervals connects me most deeply. Ah |
|
| 22:45 | run this function between plus and It's the same as moving to take |
|
| 22:51 | as long as the scale brush, is the function. Okay. And |
|
| 22:58 | In this case takes six Points. is the 5th order polynomial immigrants. |
|
| 23:05 | type of approximation. It's not 30 . It's pretty big errors in many |
|
| 23:10 | . So my one really wants to points and started getting better. So |
|
| 23:18 | is now doing in the tent over for a moment and yes on that |
|
| 23:24 | in many other places here. But the Other it is or towards the |
|
| 23:30 | point plus and -1 things got I got a lot worse than |
|
| 23:35 | Just six points. And then I tomorrow. So here is when you |
|
| 23:43 | more points it's scaled. So it's the bump was to get things in |
|
| 23:48 | what that looks like this but compressed it's not that's the same function. |
|
| 23:54 | then one more here, those things much more than sweet. So they're |
|
| 24:00 | a shorty dog compressed on one side the beginning. You have seen this |
|
| 24:06 | , but it just shows having more more points in terms of the maximum |
|
| 24:12 | it gets worse so much for your . So, so it's important to |
|
| 24:24 | to figure out ah after I got idea that in the era B but |
|
| 24:33 | defends you about the function is and you happen to no only able to |
|
| 24:39 | about the function here. So here the fact um I want to see |
|
| 24:50 | be or years actually. So the and the that's not her in the |
|
| 25:00 | were considered is proportional to the m first derivative. And then the product |
|
| 25:09 | um the distance between the point X on the double points that you |
|
| 25:18 | So it's very reminiscent and some can do it. And you also remember |
|
| 25:29 | the tape of civics expansion and they here polynomial then as the right set |
|
| 25:37 | coefficients. And this has um first turn in the tendency is expression so |
|
| 25:49 | it depends whether this is strictly to this polynomial politicians, that's exactly |
|
| 25:57 | The relative dividing by the factorial There are serious questions. So in |
|
| 26:03 | book ah so to prove that this in fact true to another way that |
|
| 26:12 | on the next slide. So it's little bit, that's what I |
|
| 26:19 | I'll try to talk to you And plus it's another one this aspect |
|
| 26:26 | churches perhaps I used to know so this simple. Just the most on |
|
| 26:34 | other side. And to manipulate and some the person to prove that it's |
|
| 26:40 | . And then we're going to use from the next couple of space to |
|
| 26:45 | around with it. A couple of . So so to try to show |
|
| 26:52 | this is true, do a little particularly here and some salt looking at |
|
| 26:59 | product part ah a little bit of variable when it's going to be |
|
| 27:05 | So instead of having X up there t. Um and it will become |
|
| 27:11 | little bit bigger perhaps what this trickery sense. Now also if look at |
|
| 27:20 | expression what things? It's useful if remove the product onto the left hand |
|
| 27:28 | then we get this part. So my speedy X divided. But this |
|
| 27:35 | now with the X. Itself. so in that case um the sea |
|
| 27:45 | is not dependent on X. Right that plastic seen it. It's |
|
| 27:57 | So then the form is function here is but there left hand side of |
|
| 28:04 | model T. P. O. . And then. No this expression |
|
| 28:11 | if you use the argument T. instead of X. Um the product |
|
| 28:18 | systems. So this is essentially ah everything side to the left hand side |
|
| 28:25 | they are empty. So that's what going to buy your onward in order |
|
| 28:30 | improve that. In fact the error acceptable. It's just something here. |
|
| 28:35 | that's going to be in the upcoming order to get to that argument. |
|
| 28:39 | they look at this function. Um know because he Interpol it's the function |
|
| 28:49 | exactly have the notes. So we that this function is zero. All |
|
| 28:59 | notes in the interpolation. But it has the curious property that if you |
|
| 29:06 | in X. In this case, it's also simple but it's also for |
|
| 29:11 | study because therefore X minus P. X. That's the left hand |
|
| 29:16 | Mind you see that is this property W. T. So it's basically |
|
| 29:25 | . Tina Republican acceptable to express And then left and C. Is |
|
| 29:32 | to this expressions on self destruct zero for the experts. So the point |
|
| 29:39 | that is then that inspired here has plus two um notes or methodology Arguments |
|
| 29:50 | which is zero. And I mean that is being used to come up |
|
| 29:59 | conclusion that this system too is that kind of like them from the mid |
|
| 30:08 | . Therien rates we have a function It's continuous between two points a. |
|
| 30:15 | B. And somewhere in this Did you take in a pair of |
|
| 30:25 | values or arguments of the function fire it's zero? We know that there |
|
| 30:31 | these two points. And if this function would be the gazelle can cut |
|
| 30:36 | or go down and comes up. somewhere in there that derivative has to |
|
| 30:42 | zero because the continuous function that that's same level of the beginning. |
|
| 30:50 | What's up, man? And now have basically M plus two Points where |
|
| 30:59 | are zero. So that's for the . And if there are points um |
|
| 31:07 | is zero at least between any pair points because it's all zero leave for |
|
| 31:13 | lives. So that means the derivative emphasized one rules 4.10. And then |
|
| 31:20 | can keep doing this argument. So means that the implants first derivative As |
|
| 31:24 | least one location for a zero. then we just plug it in |
|
| 31:31 | But first at some point Somebody argument zero. Again all the expressions. |
|
| 31:39 | the emperor's first derivative of an advanced polynomial zero. So this time is |
|
| 31:45 | to drop out and what they have that it was these two terms. |
|
| 31:51 | the best process essentially than that. The fear of X is that it |
|
| 32:00 | left the same plus one term divided N. Factorial. That's what you're |
|
| 32:05 | is. And you have to see up there. There's the questions of |
|
| 32:12 | . Ah The formula is true because in front of the product, it's |
|
| 32:18 | sea which is the emperor's first derivative is still the best. So this |
|
| 32:27 | the seeking an arbitrary thing and and excess. Another point where the expression |
|
| 32:36 | zero and figure out September 60, ? Yeah error. So that someone |
|
| 32:49 | probably do the same using the taylor . As long as someone has to |
|
| 32:54 | that the polynomial proficiency itself. But is a ah in the taylor series |
|
| 33:03 | . So they went around about the . Alright so now we're going to |
|
| 33:15 | around with this expression a little bit kind of messy. It's quickly. |
|
| 33:24 | so the thing is to try to something boundaries um There so one part |
|
| 33:35 | independent of the function. So the and other parties we talked about on |
|
| 33:42 | product of the distance between the orbiter X. And all this note that |
|
| 33:47 | for the definition. So this is deal with the product that's just down |
|
| 33:54 | and it's just manipulating this expression and what this kind of the side song |
|
| 34:06 | uh huh. Looking at X wherever is and the closest to points and |
|
| 34:20 | this kind of consideration comes appears many and it's slightly different forms. Um |
|
| 34:30 | so we have the product here, we have X with respect to all |
|
| 34:33 | difference Knowles but the single out the closest nodes and then we have actually |
|
| 34:43 | all the other. So this is this product and they're splitting it |
|
| 34:51 | So yes, so the one part the knows that are not the closest |
|
| 35:00 | it coming to the left and this the product for the points that are |
|
| 35:06 | So the right but not including the response. So then we have to |
|
| 35:11 | then estimates the two terms for One is xJ times X minus |
|
| 35:18 | Tape was one. Ah Well, intuitively, we can see if X |
|
| 35:27 | very close to extend my plus then it's going to be very close |
|
| 35:31 | zero. Similar effects is very close extinction than one of the two factors |
|
| 35:38 | the product between X and this and . And that His forms are being |
|
| 35:43 | zero. And it turns out that midpoint is where I fished that things |
|
| 35:50 | the Lord is the turkey many times you want to throw something and it |
|
| 35:55 | out that and maybe the best of worst so much. So that means |
|
| 36:03 | look at the product that state between . And these two points. They |
|
| 36:07 | it's basically less than it for you texas in the middle. So it's |
|
| 36:14 | it's so if it's in the middle servitude, each one of them is |
|
| 36:20 | . So then you plug it in and manipulate it formulas and then you |
|
| 36:25 | get an expression that looks it? . A. But it is you |
|
| 36:31 | something to or age. Um the between the points is to the Plus |
|
| 36:40 | . You have two factorial and to it enough then the best I |
|
| 36:49 | Yes it grows with it victoria but age also did for a fixed interval |
|
| 36:56 | smaller at the end. So so I understand that this is what they |
|
| 37:06 | up with for the products maximize that the short term and wow and it |
|
| 37:15 | a double area part. And so what it says. Yeah. So |
|
| 37:20 | I knows um something in, come we have an equally spaced points. |
|
| 37:29 | ? So then um we have this a derivative. So this one over |
|
| 37:42 | plus one factorial. That comes from sea in the previous one. And |
|
| 37:48 | had the in factorial here that was of the product estimation. So then |
|
| 37:56 | left is M plus one and the . Um Yeah this is from the |
|
| 38:04 | and um that's fun. This is proficient in front of to show them |
|
| 38:16 | that's what they have something done. where now the error maximum is proportionate |
|
| 38:23 | the emperor's first power on the space between the points and divided by basically |
|
| 38:30 | the funds and then times the first of the function in that. So |
|
| 38:38 | one happens to have about, it's nice to be hit function and |
|
| 38:44 | well it's maximized some number M don't that's a pretty simple. They were |
|
| 38:50 | up the yeah, the maximized first then as a function as the number |
|
| 38:59 | the end, the number of points . Uh this obviously usually ah h |
|
| 39:08 | smaller fixed intervals. So it goes and also on the next couple of |
|
| 39:15 | . But we got the formula I want to estimate the yeah, |
|
| 39:25 | you know something about the function of , I think it's on the next |
|
| 39:30 | . So on this just kind of a simple example. So Sine |
|
| 39:38 | And we know take functions. You the sign becomes first, the sign |
|
| 39:46 | said yourself, they're all nice look . I mean all that, whichever |
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| 39:51 | rivers is the but it's a sign go sign, it's always less than |
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| 39:56 | in magnitude. So it's all in case. M is very simple. |
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| 40:01 | just the one so that depending upon M is one and the depending on |
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| 40:08 | interval how many points you are and it's yes, following that. So |
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| 40:15 | this case, in this case is wasn't a slide, I think for |
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| 40:20 | two lectures packs an approximation of the and In a 10.49 intervals then once |
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| 40:30 | and figure out what the introduction, I want to come back to |
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| 40:39 | So, um, we're talking about guy. Why doesn't it work for |
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| 40:46 | ? You're on your phone? So problem is that the derivatives are not |
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| 40:53 | in that case putting abound The sign band was one. But it turns |
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| 40:59 | that this is it's an ugly expressions , but basically as the it's an |
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| 41:08 | function. So you can take whatever of derivatives you want? Mm |
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| 41:15 | The magnitude basically girls on Barbara. the harder it is, the number |
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| 41:20 | points ah the larger, well bound the and our employees first to hit |
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| 41:31 | B. And it grows sufficiently So the fact the intervals get smaller |
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| 41:37 | make up. So that's why you this highly or structural things happening when |
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| 41:44 | increases because this grows so rapidly. pretty much, yeah, there's certain |
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| 41:53 | , all the other factors in the disk question. So, it's kind |
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| 42:00 | just the way of assorted having an of course, is there an expression |
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| 42:11 | the approximation and it comes back to the function. Do you have a |
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| 42:21 | as long as one have nice functions took functions are financials and others. |
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| 42:26 | okay. But many times you may more complicated functions that function The name |
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| 42:31 | science? Mhm wow. All So the next point is, so |
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| 42:47 | back to this thing. So equally points is well, if you don't |
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| 42:51 | any insights, what else should you ? Oh and there are particular other |
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| 43:00 | that was said to that turns out be more useful and get your smaller |
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| 43:06 | if you don't get an equal respect that is uh, well we'll see |
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| 43:12 | this. Um, so this is they unequally spaced notes for 20. |
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| 43:25 | . What stop. Well. so they should have put them side |
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| 43:32 | side for equal and Monica police Um, so this was Still a |
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| 43:39 | better. The 20 points was second last and maybe you have one more |
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| 43:44 | and I remember and earlier on it highly participatory and extreme points. So |
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| 43:54 | this choices in terms of No points doing the 5th thing, obviously it |
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| 44:02 | up giving a much better results in of they are being much smaller than |
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| 44:09 | you take them equally and it might be obvious but funds mm hmm. |
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| 44:17 | I should have gone straight lines down the X axis. It's pretty obvious |
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| 44:22 | intuitively at least that it looks like points are much closer. Much more |
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| 44:28 | spaced towards the extreme values in the and there are a little bit further |
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| 44:35 | . So in this case during this spread on the nodal point ended up |
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| 44:43 | much better polynomial approximations. Okay, how was that done? So obviously |
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| 44:52 | equal space versus an increase in, sorry. So Uh huh. |
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| 45:03 | wow. So so it turns out what was used for the unequal these |
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| 45:15 | where the so called Championship notes, are the roads of this particular |
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| 45:24 | So or the production is down on several lines that so in this case |
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| 45:29 | can see as to increase the number points Jag ar obviously sort of clustering |
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| 45:37 | , part of points for the Those these functions are the roots of |
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| 45:44 | Championship Colin obeans. So that's So when it comes back to the |
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| 45:50 | and Championship for the normal seven Chef normals are very good functions um to |
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| 45:56 | the basis functions, but also they're good for picking interpolation points as if |
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| 46:02 | pick them as rules. So at least. So when you see |
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| 46:12 | many times in an arbitrary interval There's over 1-1 center point and then just |
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| 46:21 | of straining the doctor in order. right, so this is what I |
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| 46:37 | and we'll come back to the pulling over in a later section but |
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| 46:41 | now No one of them and also that they stood forward where you're taking |
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| 46:48 | the points and trying to be it's not terrible, the best choice |
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| 46:56 | this is a very simple way. kinds of work very well in many |
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| 47:01 | charge up choosing in preparation for um is a little bit joining what in |
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| 47:11 | case we have this their formulas before now. Seven plugging in and values |
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| 47:21 | . I'll do it completely on another for next slide. Ah So this |
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| 47:30 | one kind of the next question for Championship. It looks like this. |
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| 47:36 | look after equally spaced points. We have the part that depends on the |
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| 47:44 | itself, the derivative. But the in front is quite different. So |
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| 47:51 | you look at equally spaced points, 1, 2 to the four of |
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| 47:57 | on top and then yes, you an exponential amendment, the denominator. |
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| 48:04 | if they look at this term then have been Bar of two in the |
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| 48:09 | . Annual Factory. Yeah denominator. this is the coefficients um that bounce |
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| 48:19 | owner is generally and sometimes much They can use the championship points from |
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| 48:30 | the derivative part to that too much what the scaling of the maximum |
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| 48:37 | But this is very different. So am but I didn't expect this max |
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| 48:46 | on the previous side and just plug uh changes with them. So this |
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| 48:52 | temperature and the equal spacing, what proficient in front of the max |
|
| 48:59 | what they are and the numbers and can go counter racial. So for |
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| 49:05 | smaller values of man. Yes, some difference but it's not that |
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| 49:12 | That's to take more and more You can see that the difference historically |
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| 49:16 | queen significance on this case Is pretty almost a factor of 1000. That's |
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| 49:22 | it points up. The championship scaling of the derivative 2000 times more or |
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| 49:29 | less then for equal respectable. Mm . Right. So this is unless |
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| 49:44 | somewhere outside the city. So that it association. Thanks. So I'll |
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| 49:52 | questions on this. Interpolation hitting stuff on that. Okay. So I |
|
| 50:03 | the whole message of all this inspiration that. Mm hmm. We have |
|
| 50:10 | garage. Again, you can construct corner polynomial without knowing functionality inside of |
|
| 50:17 | parents and getting so um mm mm hmm. There's some forms and |
|
| 50:24 | more convenient. Someone else's scope in future. Ah, this one has |
|
| 50:33 | . Trying to approximate functions, The straightforward equal spacing of inspiration, |
|
| 50:41 | situation and not what you want to . Oh and without taking terribly sophisticated |
|
| 50:49 | service endpoints. It's a good strategy of well, mhm one time again |
|
| 51:02 | formulas the number. But basically what interesting today they're about is probably the |
|
| 51:12 | of the functions of the border. one higher and the order of the |
|
| 51:18 | . It's and also that cardinal behavior point a normal also got scales |
|
| 51:33 | Okay. Um so therefore having an expression. It's always advisable to |
|
| 51:48 | Um The other six elevation or differentiation finding the derivatives and Biblically compared to |
|
| 51:58 | in America. So has another olympic in with this message trying to do |
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| 52:05 | . But sometimes it's important manipulation their health, their findings and I |
|
| 52:11 | that this question for the community um differentiation is not necessarily tricky but it |
|
| 52:28 | to emphasis or increase so they wow some ripples and waves or any single |
|
| 52:46 | goes up and down. And I see if I'm trying to take the |
|
| 52:50 | of it. It's probably much worse the single except in terms of this |
|
| 52:59 | , you know another contrived example but is going to function as one example |
|
| 53:03 | the function itself looks nice but if look at the derivatives that have to |
|
| 53:10 | so that's part of the reason why american differentiation is 40 50 And that |
|
| 53:19 | that class. Of course. Um it ever happens they need to do |
|
| 53:26 | if there are. So that's all there. Our device would be because |
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| 53:33 | attempt to take all kinds of precautions prevent things of growing up for all |
|
| 53:42 | . That that's the preamble. Um are confident about the standard procedures are |
|
| 53:51 | approximations of their objectives and so this what the expression has seen many times |
|
| 53:59 | . It's best to serve. Well look at the slope of the |
|
| 54:04 | Think of that as an approximation of attendants or political reports. I wanted |
|
| 54:12 | find the derivative of X One can the function by there's Tuesdays and points |
|
| 54:20 | that one distance, X plus H then draw a straight line between these |
|
| 54:28 | points and divided by the distance and picture. And in general this is |
|
| 54:34 | take away your doing things. It's less time forward derivative. You think |
|
| 54:39 | consequences, you can also use the one but taking minor states instead of |
|
| 54:47 | , and we'll talk about that Well done trying to figure out how |
|
| 54:53 | is this in terms of the error you have. And then I want |
|
| 54:59 | that I am Davis serious again, the whole point of the first three |
|
| 55:06 | in the series of pension function Its first delivered there. And then |
|
| 55:12 | the only interested in the first But they have they can stop there |
|
| 55:15 | they said and our translation terms that uh, following that. So from |
|
| 55:22 | expression we can solve for. That's . And so on. I |
|
| 55:28 | please forget the expression on the previous . So that is, you |
|
| 55:33 | a reasonable approximation of derivative. And start mm hmm. And so this |
|
| 55:46 | what it says here. And it's . I have this uh, than |
|
| 55:56 | of the derivative is proportional to is distance between the two points that they |
|
| 56:03 | for. That's amazing. So this also known as first order approximation in |
|
| 56:13 | sense that the area's football to the between. Thanks. The two points |
|
| 56:18 | using training an approximation of the and the illusion of particularly grapes at first |
|
| 56:26 | is it principal? They wanted to higher reported approximations the derivative evaluation more |
|
| 56:36 | and we'll get, I don't So the simple way that might often |
|
| 56:42 | this is this first order, this just an example. Cool max |
|
| 56:56 | So this so there is nowadays um value at the exposition and export from |
|
| 57:07 | function design they wanted by the So that's just an approximation of the |
|
| 57:13 | . Now the analytical expected there is sign that gets closer. So here |
|
| 57:18 | the actual value of the derivative FX here's the estimates of this is now |
|
| 57:23 | effort in terms of this way of the first to review them when it |
|
| 57:30 | to the side. So I think this line, the bunch reformers that's |
|
| 57:37 | best player look at as a function age, what the true value is |
|
| 57:46 | what the pair is. So now let you guys comment on this people |
|
| 57:53 | sight what do you see what happens and she gets smaller. Yeah, |
|
| 58:02 | bigger While some areas starts at the of -2 something say that's kind of |
|
| 58:12 | but then all of a sudden his starts to build so what happened? |
|
| 58:21 | , yes, it has a minimum life. Absolutely. Really. |
|
| 58:35 | well, okay. Yeah but it smaller. Everyone now it comes back |
|
| 58:48 | the I think back to the first second lecture. Alright so at some |
|
| 59:04 | that X. Is small. This it's not me. It's almost the |
|
| 59:15 | . So at some point you do yes returns from these guys gets to |
|
| 59:23 | So the difference between our way down the number string of bits string and |
|
| 59:29 | I'm gonna do the subtraction most of . That is what happens here is |
|
| 59:36 | some point um you don't have enough in your representation that some of them |
|
| 59:46 | garbage. So that's one thing to about when you have do this kind |
|
| 59:53 | thing that you just project takes a and smaller. Um Yes if you |
|
| 59:59 | infinite position that helps you but in it might not. I don't think |
|
| 60:06 | another son. So yes that's what said. So I can also trying |
|
| 60:12 | figure out from the return and that's I think we got it done And |
|
| 60:17 | finding powers this is very useful to out what age needs to be to |
|
| 60:24 | a single position accuracy and there is point in doing it single position. |
|
| 60:31 | can do the same for double And don't try to get you know |
|
| 60:36 | do better than that. So there's places that they want the single |
|
| 60:45 | This is supposed to the inter We'll give you a single position. |
|
| 60:51 | yes, in this case you've got little bit better the smaller the air |
|
| 60:55 | doing stuff like that smaller age but then point lots of positions contaminated |
|
| 61:08 | Ah Okay. Right. So So I said, this was one way |
|
| 61:22 | this was for the 8th. The order. That was the basis for |
|
| 61:27 | form of difference that we used on cynics example. But the devil first |
|
| 61:35 | approximations are not do they not from works when you want to get older |
|
| 61:41 | least the second order like eight So now if you look at the |
|
| 61:47 | shears expression here, Yes, previously kind of stopped and this was the |
|
| 61:57 | term but if they now include a bit more terms so then it's free |
|
| 62:02 | instance make some difference and divide by . That if this term is |
|
| 62:10 | Suzanne vega. First thing this is . Including the error term starts with |
|
| 62:15 | guy, this is included. This part of the error term then things |
|
| 62:20 | going to be order eight square so is that much better. And so |
|
| 62:24 | so probably do things. So now produced. Look at what happens when |
|
| 62:30 | use it the one taylor and X age and another one projects minus age |
|
| 62:37 | then you can take the difference of this type from this card then the |
|
| 62:43 | has gone to disappear at You get times this guy and this is the |
|
| 62:49 | term. So this is also going disappear. So now it's just an |
|
| 62:54 | where you have this promise before and next term has a street in |
|
| 63:02 | So if they're not, they're still to with age, then they're going |
|
| 63:07 | get a survivor too. Um then going to have an expression for if |
|
| 63:13 | where the kind of first a return a square. But this is this |
|
| 63:19 | what's happening on this side. So you can have this expression uh then |
|
| 63:27 | still the first derivative. But so similar to what I mentioned like when |
|
| 63:33 | talked about the divided difference part. you kind of have uh kind of |
|
| 63:39 | soap from extra expressive and the slope X minus H and X. And |
|
| 63:47 | that you use these students and average out. And then you've got an |
|
| 63:51 | that has and every term that is square. So and this is not |
|
| 63:58 | a scented difference if it said on slide or not, but centered differences |
|
| 64:04 | this property that they are difficult against higher order approximation. And if you're |
|
| 64:11 | happy with something that is a you can take more terms and kind |
|
| 64:16 | manipulate his expressions to make sure that get that's prime is left and higher |
|
| 64:23 | terms um disappears. You get an . Prime and but you need to |
|
| 64:29 | different than your combinations are at different . Alright. And that is something |
|
| 64:36 | done. And some of the assignments certainly not many examples in the |
|
| 64:42 | There's no country exercises how to take functions. Then you're a combination of |
|
| 64:49 | to eliminate whatever number of terms that in terms of the terrorist serious expressions |
|
| 64:55 | he's the 1 71. Uh Yeah. So now you can then |
|
| 65:03 | the same thing, you know what a return it and you can use |
|
| 65:09 | to figure out what age needs. in order to get to this example |
|
| 65:15 | position. So all of this because I use it as stem but it's |
|
| 65:24 | interval divided by two departments. So I think I did now with |
|
| 65:33 | center of difference. Oh I can how things behave and for the same |
|
| 65:41 | for a while things get better. you can see now tickets lower much |
|
| 65:48 | quickly in the salt tenants. It's like four here. Uh versus it |
|
| 65:56 | six or seven or whatever it was the previous one. So now that's |
|
| 66:00 | you got single position. So about something is a square then, do |
|
| 66:07 | think that there were the larger age of them and error in the |
|
| 66:17 | So this is the And so I there isn't comparison between for a difference |
|
| 66:28 | they sent the difference and you can that Yeah, almost yeah squared kind |
|
| 66:37 | chance of exercise. That means again less strong too. So I'm saying |
|
| 66:52 | From one best thing At least. square section. Second order approximations. |
|
| 67:02 | you and yes, instance. What? Yeah. Bye. |
|
| 67:13 | That was just next time we're going go through another, you can find |
|
| 67:18 | the other approximations that they prescribed. a rule of thumb thingy. So |
|
| 67:24 | is the entered some extrapolation. That the trickery of trying to make use |
|
| 67:32 | ah the other part of this section addition to being able to use larger |
|
| 67:45 | logic based values. What is Richardson is awesome because sometimes dysfunctional valuations are |
|
| 67:56 | . So about using higher order methods then picked up by using the extrapolation |
|
| 68:00 | we talk about next time. Um can get away and reducing less functional |
|
| 68:06 | as well. So that's two parts it. Using fewer functional evaluations and |
|
| 68:14 | against joseph significant, limited position and stories. Thanks. What? But |
|
| 68:27 | a burden someone can look at and preparing you for the switches and stuff |
|
| 68:34 | that's fun. Um Alright, just on base terms, dropping everything else |
|
| 68:43 | coefficients A's. I can see trickery in which one can within the combination |
|
| 68:53 | get rid of successfully or all the and with just an extrapolation questions |
|
