| 00:03 | Yeah. Something people want to. , so continue that um inspirational |
|
| 00:20 | And I started at they're almost simplest . There's a simpler version of it's |
|
| 00:30 | . Mm hmm. So we'll talk little bit more about that then how |
|
| 00:36 | can do so chapters or go to something more accurate but using sensual |
|
| 00:43 | Sunday extrapolation. We're trapped. And I talked about another matter. Eventually |
|
| 00:51 | is same sort of there's always something . Alright. So here about Yes |
|
| 01:00 | remind her. But the tropical rule was 6 3 approximate the integral or |
|
| 01:07 | area under the curve area under or by chapter swords. Or there is |
|
| 01:17 | the straight plane piece by speed. approximation of the functions affect a bunch |
|
| 01:24 | notes. And you evaluate the function those notes and then rule as it |
|
| 01:31 | things correctly. There's the function between two or two neighboring points behave. |
|
| 01:39 | a straight cut and I'll come back that thing. So Sandy's going from |
|
| 01:47 | formulas were fairly simple in the end the best behalf of the function bodies |
|
| 01:54 | the and then just some of the . So that was 16 and a |
|
| 02:07 | version of it. And the reason that's the person with respect to the |
|
| 02:14 | of the interval. And the reason talking about it is I think we'll |
|
| 02:19 | it later essentially. It's an extrapolation is kind of based on mr cursed |
|
| 02:28 | of the tractors are so so in case the gun like when we talk |
|
| 02:39 | division they sort of refined somehow. or make it smaller and smaller as |
|
| 02:44 | function in this case and and has divided very attractive too forever stepped in |
|
| 02:51 | the final process. So H and former is in fact depending on them |
|
| 02:58 | on the given subject about being his and his father something. So this |
|
| 03:08 | that half of the 10 point values then in the summer of internal values |
|
| 03:14 | by the winds. Okay. Um as always. Well, that's that's |
|
| 03:26 | , yep. Mhm. So what now in this kind of notion of |
|
| 03:32 | intervals, manufacturers through and looking at finance, finance of the intervals? |
|
| 03:40 | you can see basically uh some of points, what they're saying us in |
|
| 03:48 | immediate course of of division. So only thing that happens is that in |
|
| 03:55 | you add some new functional evaluations and we change the weights of the functions |
|
| 04:01 | are actually and then we'll do Then we'll get this occurs the |
|
| 04:08 | So the first one was easy, the two endpoints and the width of |
|
| 04:14 | interval from the start finish from A B And just one track. Thanks |
|
| 04:26 | . 5 10 minutes. Some divided this can use the big point, |
|
| 04:35 | the intervals of april life simple. then they got the endpoint and the |
|
| 04:42 | points and that's the first step is The life does not demand as a |
|
| 04:48 | by two. And then there was of the influence with the board and |
|
| 04:51 | there's the other guy, the other once a month and rearrange the |
|
| 04:59 | So wow this thing I'll have the endpoints divided by two manipulating this survey |
|
| 05:18 | . This is half of this kind the new volume, that was the |
|
| 05:26 | point. That was interesting. Thank . So because again age is kind |
|
| 05:35 | one way of thinking about it. you split the interval between A and |
|
| 05:41 | . Into two intervals instead of Um The length just um as |
|
| 05:46 | so that's kind of the scaling factor that half a song. So and |
|
| 05:56 | there is the value for them so and then ah okay so do this |
|
| 06:09 | . It's like this formula and red the previous slide kind of related a |
|
| 06:16 | bit so manipulated from an exercise and what and also mostly invisible come |
|
| 06:23 | It's not worry too much about the endpoints porch and it's called it see |
|
| 06:27 | now and then we take a look what happens when some something in the |
|
| 06:37 | That is not true to the N to have Someplace to go from Say |
|
| 06:41 | from to this one. So too that age is kind of different normally |
|
| 06:52 | you use advice one because that would the interval divided by tweet today and |
|
| 06:58 | instead of the end. So to the same meaning of age and both |
|
| 07:04 | , they have to plug in the two here because Divided by 2 to |
|
| 07:08 | N -1 instead of two. So can so this is basically says that |
|
| 07:14 | and this former starts for the boats this form. So then we got |
|
| 07:19 | here that's also that replaced age but reach whether it's just a minute here |
|
| 07:27 | that she wasn't provided and I want play with that. That was from |
|
| 07:33 | previous slide 1st 1. So now you look at this difference that was |
|
| 07:40 | , is this one that comes from deal our end. Come on |
|
| 07:47 | And this is the one that comes this expression but we multiply this by |
|
| 07:55 | they have so they can see me see so that disappears an obvious to |
|
| 07:59 | about ah those three expressions now. since now it's just the same |
|
| 08:12 | right? So we can notice for the even um ice and this formula |
|
| 08:19 | corresponds best with 2, 1 Of function values and this one. So |
|
| 08:27 | it's only the odd ones. That's . So they aren't ones essentially the |
|
| 08:35 | But I knew that was not in state to to the to but only |
|
| 08:39 | the two For the three invigoration. space beside as well and this proportion |
|
| 08:45 | you're looking at it. So if have already evaluated this one you get |
|
| 08:50 | next one but there's a different function these options for the new points, |
|
| 08:55 | of the proper, so that's So, so this is based on |
|
| 09:06 | personal formula and wow. So summarizing , ah and the reason for this |
|
| 09:23 | we're going to use this formula You don't have to do it this |
|
| 09:27 | . Obviously you can start from scratch whatever function values Savant or on your |
|
| 09:32 | and the proceeds directly with us, you can also go through this the |
|
| 09:41 | of what you're doing stan The previous with a factor of two for a |
|
| 09:49 | divine gratitude. And then you have you function in it. Okay, |
|
| 09:58 | little bit of discourse before coming to book. So doing it for multi |
|
| 10:10 | is sticking straight forward and for that , but it's the first Look at |
|
| 10:16 | standard interval from 0 to 1 and it's just the scale and arbitrary intervals |
|
| 10:23 | standard interval And sometimes this community 0-1 sometimes it's convenient to use the symmetric |
|
| 10:31 | around zero. So from one is , the look of the scaling a |
|
| 10:36 | shots accounts later. So, so especially you can sort of recognize the |
|
| 10:47 | of rule because yeah, these are two endpoints, There are scale events |
|
| 10:52 | 1/2 and then you have all the points on the two and the two |
|
| 10:58 | also this basically the sum of all internal points, everything may be |
|
| 11:04 | but it's expected and then make it simpler notation, just look at |
|
| 11:11 | Then the coefficients here, various It's basically the same for all the |
|
| 11:15 | points. It's best for them one here And then we got the |
|
| 11:25 | And so in order to do that thing, it can be real taking |
|
| 11:32 | dimension at the time which is done this thing. And the chapters are |
|
| 11:36 | with respect to X. And that's . In principle, I do it |
|
| 11:42 | all the wise. It's not done . Is to keep the integral and |
|
| 11:48 | two guys commute so you can sign and now the best to have waited |
|
| 11:56 | more while his wife is determined for chapters overall. In this case for |
|
| 12:02 | extraction. And then you can still the integral in the Y direction |
|
| 12:09 | you know something his own rules also the wine. And they get a |
|
| 12:15 | way to some respect to the white . So basically you get in this |
|
| 12:22 | intervention, you got some kind of great yeah xy notes. And then |
|
| 12:31 | function values those notes. And then have the weighting factors with respect to |
|
| 12:40 | um X integration. So that's all internal points added together. And then |
|
| 12:46 | have to be a little bit careful the boundaries will lead us to ask |
|
| 12:50 | for the border, depending upon where points are, but it's diseases are |
|
| 13:00 | the end, points in exile. the after boundaries for X. It's |
|
| 13:05 | heart and then in between it's wow, and then the same thing |
|
| 13:11 | the Y direction in the corner. what happens? Did you use that |
|
| 13:18 | of formula? But on something that satisfy the, well 12 seconds. |
|
| 13:27 | let's say integrating 1-1, one over or something. Okay. Yes. |
|
| 13:37 | if um so in that case it's signal point that zero. Um So |
|
| 13:53 | that's a good question. So I thank you. But that's a |
|
| 14:08 | question. Whether so, what's the of the rest? Um So that's |
|
| 14:18 | low content, right? Is the to everything or one overhead? |
|
| 14:24 | So the world function is not, simple zero. So then in that |
|
| 14:31 | it into $20,000 makes sense. So it, hmm. So if |
|
| 14:42 | remember figure out the being dependent on relative some water dependent in that |
|
| 14:58 | I guess the first term that was no one was second or third. |
|
| 15:05 | . But that's so what's the best relative of one over next against, |
|
| 15:11 | know, one over X. Squared stepped up with the science. So |
|
| 15:15 | affects the suspicion is more with the of stuff because for some reason, |
|
| 15:25 | that's why there's no bound up on . So correctness for several magnitudes. |
|
| 15:33 | I think, yeah. Okay, . So then it's just increasing one |
|
| 15:47 | two dimensions just by looking at the thing and so you invested in the |
|
| 15:52 | product from the approximation of respect. . Um So This Sunday's two points |
|
| 16:09 | , the Gander there were is proportional the number of points that have to |
|
| 16:16 | the function evaluation of so much and the most functions that people use or |
|
| 16:22 | end up using various contexts they work evaluation or computing function values. It's |
|
| 16:30 | lot more than they had multiplying. need to be careful. So, |
|
| 16:40 | now it has the same spacing in the different dimensions. So that's new |
|
| 16:45 | in the a number of points in dimension. The means that the work |
|
| 16:56 | the power of the number of So, and you know, functional |
|
| 17:04 | in one point and then you might of an N by N grid or |
|
| 17:08 | squared from evaluations they keep falling in to the number of dimensions. But |
|
| 17:17 | type of work. Whereas the accuracy determined by that edge is facing in |
|
| 17:27 | dimension so that it did not So that's what I'm trying to |
|
| 17:31 | So it's kind of a complicated This is straightforward. Um that so |
|
| 17:37 | this is basically, did you stab in as we will end to |
|
| 17:43 | Okay, powers. So this is one over n squared. That's been |
|
| 17:50 | to the square just to remember. that is in terms of the trip |
|
| 18:01 | over all. A broad separation in of what type of functions 15 cigarettes |
|
| 18:08 | well. It does not account for past that's doing more work. I |
|
| 18:15 | have more dimensions given age, grand and other things may actually get worse |
|
| 18:22 | this. That in respect to an position that this is the case and |
|
| 18:29 | independent. Okay. Dustin stop. Oh right. Um, what about |
|
| 18:48 | multidimensional, I'm not coming back to This recursive formula um, chapters on |
|
| 18:56 | . 20 questions. All right. this is um, um, it's |
|
| 19:04 | first of formula looks like uh, this is the way it looked at |
|
| 19:11 | sports, the person of yes, the chapter is older than the first |
|
| 19:20 | . This was no reversion. I just breaking it down and starting get |
|
| 19:25 | this point where the length of the because side revival but for each |
|
| 19:41 | that's it. Mm hmm. Dr martin, it's not okay for |
|
| 19:49 | , sorry. So now again, what I wanted to say. So |
|
| 19:55 | so called bromberg algorithm depends on The first formulation of the trump resort |
|
| 20:04 | , it's just supplying the switches and idea that they used for derivatives before |
|
| 20:12 | started, when we talked about they have reformed their percentage differences of |
|
| 20:20 | derivative and for that the error and approximation was um mr terms a bunch |
|
| 20:30 | terms of the docs expansion. There powers of X square. So I |
|
| 20:36 | a student. second power is the is 6 and Richardson was down good |
|
| 20:44 | that doing, Looking at H&H of two and producing Those two expressions to |
|
| 20:53 | 1st the X squared term and you something that is accurate to the edge |
|
| 21:00 | the order and then it's complete cold they wanted more, what are your |
|
| 21:05 | ? So it turns out that for trap assad role the or terror also |
|
| 21:12 | in even powers of mhm. So this is kind of a scheme, |
|
| 21:21 | will talk about construction scheme levels actually for this um which is an extrapolation |
|
| 21:27 | the chances differences. You see the kind of triangle schema, you can |
|
| 21:34 | this in your first parliament for this of when it comes to the tractor |
|
| 21:39 | rule is for eight squared and then go to the next column here that |
|
| 21:44 | seen stage four and 86 etcetera that stopped. And I think that silence |
|
| 21:52 | for its moral is postulating the same as reconstructed. Well, it's kind |
|
| 21:59 | so it's the same format to get to go back and look at re |
|
| 22:04 | extrapolations conduct for derivatives. Now you from his Romberg integration for the one |
|
| 22:13 | , but you can see it's the structure and the same weight and it |
|
| 22:18 | from again that this is kind of formula for the track the songs. |
|
| 22:29 | so that gives rise to exactly the um schema as well, come for |
|
| 22:36 | and getting better approximations for integration for given age by doing India combinations of |
|
| 22:47 | that united in the first column that the source of the form. So |
|
| 22:58 | best is the same structures who did . So I think that's going to |
|
| 23:04 | or less the same exercise. But there's no difference except just using there |
|
| 23:11 | simple instead of the, the symbol the think about democratic expression because the |
|
| 23:18 | , the term looks exactly the same it comes to the trap is |
|
| 23:23 | And the whole thing is just to successive borders for powers or age they |
|
| 23:30 | the more active formula and the coefficients on the power. So um, |
|
| 23:38 | not on the coefficients in front so that's why we have the same |
|
| 23:46 | of the algorithm eliminate it where term this area ends up being so |
|
| 24:01 | yeah, continue as the in general looks just like the terror of being |
|
| 24:21 | powers so yes, right. That's and that's why it's best if we |
|
| 24:40 | say sure and I'm not going through , but Somebody did want to get |
|
| 24:47 | of the Air Formula four, the are low impact just has even hours |
|
| 24:54 | age. Stop it. So the thing you should remember now this particular |
|
| 25:05 | but that it's even for ourselves ancient system that would suck anyone interested. |
|
| 25:17 | it was pretty much and so I yes. So this is just one |
|
| 25:21 | thing and kind of think they have example of work. So they also |
|
| 25:28 | out, so how do you know sometimes the internet, Okay. Um |
|
| 25:39 | the table so one can look at particular expression and that production wait um |
|
| 25:50 | this formula here um and stick it um let's form it up so no |
|
| 26:02 | can show that this ratio should they behavior, it's this power is the |
|
| 26:12 | behaves in a way subject to the , that depends on the derivatives, |
|
| 26:21 | correct order depending upon um before you to go and jump on improving the |
|
| 26:27 | . Um Nothing welcome to interval. in quotes. The absolute value of |
|
| 26:40 | derivatives in the term that you are that. So arena or digital comics |
|
| 26:48 | too. The chapters on room. , so this is kind of the |
|
| 26:55 | . So as long as this comes more or less close to forward to |
|
| 27:01 | pharaoh was the number of columns. how many times supply the Richardson extrapolation |
|
| 27:08 | the Oh no ! So one can through and look at the expression, |
|
| 27:16 | fine. And then you can actually that uh can for m equals |
|
| 27:24 | The first thing you notice the first of the circulation and plug it into |
|
| 27:31 | ratio. But I said it's a test for whether that's circulation gives a |
|
| 27:38 | outcome. So on the budget in and then it was to manipulate this |
|
| 27:44 | like that. That's something that That's the four plus in the |
|
| 27:50 | Mhm. So it kind of Ah This is proved by example which |
|
| 27:58 | not been approved at least at And then this is just a reminder |
|
| 28:13 | what they've already said, remind you for and supporting the slide and then |
|
| 28:21 | the boat it shows the code for the Romberg type algorithm or extensions and |
|
| 28:30 | for them, not this one and only comment. They have you in |
|
| 28:37 | texas. Yes, you can go here and I think you can call |
|
| 28:41 | by columns. But the point of bromberg algorithm or it's an extrapolation and |
|
| 28:51 | our orders and things to get good before. So obviously not that small |
|
| 28:58 | . So basically, definitely this is point of care about. So you |
|
| 29:03 | want to do more functions because when go across the columns here, if |
|
| 29:08 | don't do any more function evaluations, the same number of function evaluations. |
|
| 29:13 | only thing that happens yes, you uh it doesn't show explicitly but it's |
|
| 29:20 | hola what's happening. And then the thing that happens that well, my |
|
| 29:25 | from one column to the next ah weights of the different function values you |
|
| 29:31 | . It's changed and I'll come back think so that's why it's here. |
|
| 29:40 | £7 kind of that can do trial error, you know, take an |
|
| 29:48 | and see what happens and that increased age. But yes, you said |
|
| 29:53 | , you know that they're so It's , 2, 4, 6, |
|
| 29:59 | here in this column. Then they estimate what they should need and they |
|
| 30:03 | need to go further down. And more functions about addiction and work in |
|
| 30:07 | column, they do function evaluations for person um to double the number of |
|
| 30:18 | more or less every time they moved increase. So so that's part of |
|
| 30:25 | show. Here's where the function of are and this is just a combination |
|
| 30:30 | what's your priorities completely. Yeah. , so this structure is called |
|
| 30:40 | It goes over That's all right. the most common something people ever |
|
| 30:57 | Yes. So the concrete for um for the book the integration of |
|
| 31:10 | , which one 15 points the next of the special again by age or |
|
| 31:21 | smaller. Aah and then what the result is straightforward application of the chapters |
|
| 31:34 | rule. And then in this case doing Romberg ah algorithm all the way |
|
| 31:43 | the right so that speaks to refuse the highest order accuracy you can get |
|
| 31:48 | the number of functions that are your you'll make. So that's the shows |
|
| 31:53 | that you get very good approximation in case for mr Uh huh is to |
|
| 32:04 | for 16 right function evaluations seven. I cannot versus to get the same |
|
| 32:17 | accuracy. You need to get some somewhere that's playing around many more functional |
|
| 32:26 | for straightforward. The first college aged and 6 and this causes the |
|
| 32:56 | Um So breaking the flow I would . But we'll talk about the general |
|
| 33:08 | on extrapolation in the general context of internship. Talking about the integration and |
|
| 33:14 | up with something. So this is shows like definitely it's just an idea |
|
| 33:26 | just fine and just a little bit messy. Is it's such a nice |
|
| 33:36 | of two comes of age. Uh . Some orbiter for south delta and |
|
| 33:44 | of age. But the same idea try to where There are three some |
|
| 33:50 | in that sense. So I was smaller than their smaller and governments that |
|
| 33:56 | want to grow the accuracy. They to get rid of the first things |
|
| 34:00 | and then they start to work on second time after you might eliminated the |
|
| 34:05 | . So just on the concept of you can still do the exercise on |
|
| 34:11 | state cutting H half. It's just now things are a bit last |
|
| 34:16 | And then so much thanks to you this format to space to the alpha |
|
| 34:22 | instead of the coefficient A They divided two or 4 but By 2 |
|
| 34:29 | And so that means you have to things to to the answer. Um |
|
| 34:34 | out the same proficient for the first of H and L. This abuses |
|
| 34:43 | the village. But if you've done now they can do that. Take |
|
| 34:48 | difference between these two guys and they a sulfa turned his fierce and then |
|
| 34:57 | Mhm. The first area term that the next 100 power over age and |
|
| 35:04 | you can keep doing that. The level of that, that's the |
|
| 35:10 | leaving term that it's ignored. Um smallest. So this is thanks |
|
| 35:23 | then. Um you this was the step start to work on. There |
|
| 35:33 | go. Okay, so so that's best of the general, the idea |
|
| 35:43 | the general extrapolation nothing named to the but even powers are in this case |
|
| 35:52 | will interval that pitch but it's a bit last year. Nice and simple |
|
| 36:01 | of But conceptually it's not in fact I'm not so sure. Yes. |
|
| 36:11 | could also be another conference for this form. Now that's the age. |
|
| 36:18 | it's one over some variable X. of course in the Lord to access |
|
| 36:24 | small market is one over age. I went over it so it's very |
|
| 36:30 | . So but now they can work the one over X. Term. |
|
| 36:34 | that all H. And then instead getting smaller it should get larger because |
|
| 36:42 | want this can be diminished, they converted to their ideas, it's in |
|
| 36:47 | case it's exposed towards infinity that this here. That's too this So now |
|
| 36:58 | of looking for convergence for some getting smaller than this convergence question, |
|
| 37:04 | mechanical knowledge and then the and stuff extend half your double X. And |
|
| 37:12 | that's the same idea. Just expressions then you got something new and eventually |
|
| 37:18 | we've got this is kind of the term now and get rid of |
|
| 37:24 | And then you can start to work the next reading from. Yeah, |
|
| 37:29 | approximation. Same idea about doing linear or whatever. So, and this |
|
| 37:41 | a complete example the numbers or that double X for each case here and |
|
| 37:49 | what the function like this and it 1 28. It was about something |
|
| 37:56 | So one step forward. Yes. you. Extrapolation idea. Then it |
|
| 38:03 | something ah well you're welcome. That's not some story that Okay, |
|
| 38:25 | think so. I don't know how Well, I don't have it on |
|
| 38:28 | spins. It's not what I project the chair. All right. |
|
| 38:37 | Um, we'll see. Try but when you try to Okay. |
|
| 38:50 | , it stays think but I know can manually can like right on the |
|
| 39:00 | for the Was one of the Okay, alright. Um, so |
|
| 39:12 | was about the general circulation idea. don't want to come back to integration |
|
| 39:27 | . Mm hmm, conceptually what they're in the comics is slightly different the |
|
| 39:34 | and attractive. So now like taking different you on the integration to try |
|
| 39:49 | ah mr driven by kind of a model or accuracy about the incident. |
|
| 40:02 | and this is who said that that's happened when they got or the approximation |
|
| 40:12 | the Stromberg, the idea of this and extrapolation. But the rule is |
|
| 40:18 | didn't even more function about the ever for different functions evaluations, they just |
|
| 40:24 | the white aggressively by doing the combination in the previous columns. So, |
|
| 40:40 | going after going to the role then if you just have one chapters on |
|
| 40:49 | there was just there for the function of the U. N. And |
|
| 40:54 | true for every we can manage our through for each one of these resources |
|
| 41:00 | them to take their average height. that's actually what it was. That |
|
| 41:08 | kind of a linear combination of the endpoints values. And for the shepherds |
|
| 41:13 | drool, A and B was the now. And you also know that |
|
| 41:22 | state took place the curve of the F for the straight line, the |
|
| 41:28 | of. So that was the tractor also what it says if to integrate |
|
| 41:35 | straight function, that is a straight between the two endpoints, A and |
|
| 41:41 | . R. A and B and . Then there is no error or |
|
| 41:50 | . So it's another way of looking it. So if in fact I |
|
| 41:55 | the function was this great times, is no approximation error and maybe the |
|
| 42:05 | error during the um occupies and all that but there's no approximation or translation |
|
| 42:11 | . They are starting to send annual finally and it takes years especially. |
|
| 42:18 | this is kind of trying to look integration formulas using this idea of what |
|
| 42:25 | of polynomial? Yes integrated, So for the trappers or rule the |
|
| 42:34 | is what's been your function from A B. It just takes the tractor |
|
| 42:38 | already calling for the function then you correct with no approximation. So so |
|
| 42:49 | look at that. So that means that's the first order polynomial. So |
|
| 42:55 | the purpose of road integrates any first and over correctly. Um First of |
|
| 43:05 | polynomial is a concept that 0° So that should be integrated clean. |
|
| 43:12 | now reduced instead of the arbitrary into 8:30 for simplicity of notation. And |
|
| 43:18 | we'll figure out how to make a In a bit but there's the tsunami |
|
| 43:23 | stay in speed from 0 to 1 sustained they integrate the constant bond and |
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| 43:29 | value is one. So that means the sum of it because function value |
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| 43:35 | influences one was a function. So this place they have this notion of |
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| 43:40 | combination of two. 10.5. This a plastic then this is a sure |
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| 43:48 | thing going straight down the street. are the one this X. So |
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| 43:54 | integrates X from 0 to 1 is squared over two. Now the function |
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| 43:59 | of Um X0 is zero. And it's only the function value 10 |
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| 44:05 | So it's quite that one as So that's now. So now that |
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| 44:11 | that the effect the function value in that formula um for ex if zero |
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| 44:18 | zero so eight disappears. So we get mm hmm. So now we |
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| 44:24 | two equations ah two for the two , A&B. And obviously you will |
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| 44:31 | see that that day is also a . So what we got is |
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| 44:38 | to do the centerville correctly. Uh error than in the half of what |
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| 44:43 | got this formula which is exactly the is over. Uh but now basically |
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| 44:50 | by not just taking off as a is on the street plan and they |
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| 44:57 | say, well fallen order gets integrated . And we designed the format, |
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| 45:04 | the first degree polynomial, correct. then I can just change it to |
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| 45:12 | an arbitrary interval basically. So it like this formula. So for X0 |
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| 45:18 | why? It's a. And for the by the cuts being. So |
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| 45:23 | doing this variable kind of substitution Yet this formula from 0 to 1 |
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| 45:29 | supplied to norbit orient to both A tv. And then they get |
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| 45:35 | scaling factor because of the length of interval. Thank goodness for this |
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| 45:42 | So now the rest of the general or rule. But now I'm starting |
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| 45:50 | the correct integration of a polynomial, this case? Of course. |
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| 45:58 | So survival. So now. so the next thing is well okay |
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| 46:12 | slowly. So it's now I have by the way is not the |
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| 46:21 | the simplest for the integrate constant. this for me um this they started |
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| 46:32 | very simplistic. That means integrating step , constant function, staircase them |
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| 46:46 | Straight lines. And the next thing wow, maybe I want something that |
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| 46:52 | on a mobile first. So then same way the that time they pursued |
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| 47:01 | and now basically I want to attend bad idea. Sizing take now three |
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| 47:08 | um the two endpoints and then That's . We need three points to define |
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| 47:17 | quadratic formula. That's very simple. Kind of the simplest thing is just |
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| 47:23 | take a deep .15. There are ways of picking up boxes now that's |
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| 47:34 | . Um mm hmm. So now idea is that fish for the infrequent |
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| 47:41 | this function. But if the approximate in our academic then they want this |
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| 47:49 | of the normal to be Exactly. this is kind of the professional woman |
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| 47:58 | . And that means we have assume because when this We needed the three |
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| 48:14 | values or three notes and correspondent function to define the polynomial and they just |
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| 48:22 | to have um the function values at end points and then the reports and |
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| 48:30 | of a narrative with the traveler's sword with again the evaluation function you can |
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| 48:41 | and I think that's one of them the next year is nice but you |
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| 48:44 | also I think it's the polynomial is using the lagrange functions. Then there |
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| 48:55 | this cardinal polynomial and they used function the nodes used them for collections. |
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| 49:04 | in the spirit of the ground formulation wow different in the polynomial because the |
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| 49:13 | polynomial has depended on X. The it only depends on the chosen noble |
|
| 49:20 | but it's not that. So that the functions are just used in the |
|
| 49:28 | they worked to plug in Lagrange formulation have to get the constants then those |
|
| 49:36 | have the function values and A. B. And C. Will he |
|
| 49:43 | to the integration of the cardinal Oh Mhm. So no I |
|
| 49:53 | For simplicity. Now we have three . So it's kind of convenience and |
|
| 49:57 | a good point Correspondent. Except for and the Q. and points. |
|
| 50:03 | person nine is 1 for the It kind of is symmetric standard in |
|
| 50:08 | morning. So now and this idea it's location. Now we should have |
|
| 50:16 | The three function values there are such if f. Of X happens to |
|
| 50:21 | a quadratic polynomial then the integral All right. Same thing again. |
|
| 50:31 | we integrate the constant X. And squared. So these are the Up |
|
| 50:36 | the 2nd 3 Polynomial. And that from -1-1. Then The concept the |
|
| 50:49 | is intermodal length is two and the is one of the two. And |
|
| 50:54 | we plug in the function value. is one of all the different arguments |
|
| 50:59 | extra asbestos function values of three points one. Such especially some of the |
|
| 51:07 | . Um text and integrate text that's negative side on the positive side of |
|
| 51:14 | equals zero. Equal areas such as builders, zero. Um The Function |
|
| 51:22 | of X -1 is -1. So . So again Xterra at the midpoint |
|
| 51:29 | from and similarly from values from the value. So now we have three |
|
| 51:37 | for three and also we can figure out what these professionals are and then |
|
| 51:42 | the A. N. C. this sign and they have all of |
|
| 51:49 | . So so this is not what but this centers off it is of |
|
| 51:57 | form and it was designed such that quadratic pulling almost identical except approximation. |
|
| 52:05 | the approximation ever happens between from the in the integration of the pulling |
|
| 52:14 | So then we're just gonna stand into arbitrary point. So the manager gets |
|
| 52:22 | is from A to B. That's expression that looks like this. So |
|
| 52:28 | not two different um the weights are of difference compared to taking other chapters |
|
| 52:40 | there was half of it to the points and then it was just a |
|
| 52:46 | by the live. So the whites is yes ma'am. So the interval |
|
| 52:54 | of length that could have this food being monitored. They provide about |
|
| 52:59 | So this is the right, And then the formula has three. |
|
| 53:03 | that's fun. The sixth. and this formula is known as same |
|
| 53:13 | alone and that. So it's still simple, but it turns off go |
|
| 53:23 | inside somewhere, but there's an expense me order to report, but it's |
|
| 53:31 | ah more accurate than the chapters and not very much more complex. It's |
|
| 53:39 | , but my phone. So then the very simple mitigation format. But |
|
| 53:46 | advice you seem because it's a lot than, and it's not for |
|
| 53:53 | I know. So that's an example where things. That's right. It's |
|
| 54:03 | very subtle symptoms, picks up a bit more about courage and about and |
|
| 54:13 | we have against the americans, some too. So, So we have |
|
| 54:19 | investigation only six square which arm with and and see what happens. |
|
| 54:30 | ah, so I just standard into 0 to 1, do the very |
|
| 54:37 | one per episode for the chapters or . And that's just basically two endpoints |
|
| 54:47 | provided by harsh and that's what you now on the steps of the soups |
|
| 54:52 | . We also have the net point the proper waiting years and I will |
|
| 54:56 | the value that is quite different from use this pair of function that turns |
|
| 55:04 | to be the function that this integral . And then you can see that |
|
| 55:10 | is actually okay, they're much better and the practice now that wasn't quite |
|
| 55:21 | comparison I would say because and the rule will use three function evaluations. |
|
| 55:30 | we did one more function evaluations so would say. So this to be |
|
| 55:36 | fair one should have used to track . They especially have the same number |
|
| 55:43 | function evaluations because that is more apparent me. So I think that you |
|
| 55:50 | do that in the book but so you actually use um the trap is |
|
| 55:56 | instead of one same number of function but trappers or rules just to get |
|
| 56:01 | better result from this, you get but this is still much so that |
|
| 56:08 | that Yeah bye. Using a different of the three function values. Ah |
|
| 56:19 | is in the Simpson waking of the that is compared to if you use |
|
| 56:25 | chapters always here then you still have of these two values and then to |
|
| 56:30 | not before but the function value of meatball there, they still don't get |
|
| 56:38 | good an approximation. So that would Yes for 1/2 in the mid |
|
| 56:46 | And whereas these days So the 16 one point and to thirds of 46 |
|
| 56:54 | the leg weights the hit points and differently then twice chapter one or to |
|
| 57:02 | results. So it's the waiting that makes a difference. Yeah, so |
|
| 57:12 | guess it's a than looking at so it took this formulation and trying to |
|
| 57:22 | interpolation polynomial Now if you have three and I choose basically 40 and age |
|
| 57:32 | age 2 3 points. So it's quite plus minus H No one has |
|
| 57:38 | But from 0 to 2 age then got this cardinal Obando meals that has |
|
| 57:46 | length of the For the distance between three points age for a pair of |
|
| 57:53 | and there are two main towards so are the cardinal polynomial. So then |
|
| 57:58 | we integrate the function values as you , they don't change the X. |
|
| 58:04 | are given for the global points. it's only the cardinal polynomial surfers. |
|
| 58:11 | variable X and y integrate this place this for the gap. Okay. |
|
| 58:19 | okay. So yes, yes, the function values. And then to |
|
| 58:24 | integration to work it out and that's this is going to work it out |
|
| 58:30 | somewhere along the lines. Eventually you this form of your simplified so so |
|
| 58:37 | the third one can start in this . It was convenient to use from |
|
| 58:42 | ground formulation because yeah, um the polynomial has nothing to do with the |
|
| 58:50 | that so that this was just a separation and you can find the weights |
|
| 58:56 | this formula for the integration that we or a B and C. S |
|
| 59:05 | waste for the function graduates. They out directly from the integration or |
|
| 59:11 | So we got these tractors often. this also shows that the symptoms exactly |
|
| 59:25 | greats. The director. Mm S So this is a good let's |
|
| 59:36 | about any questions so far. So also different integration formulas out there. |
|
| 59:46 | , talk about it. Also expect and sometimes there's no less quality moves |
|
| 59:52 | it's all place around with the white , different function values in the |
|
| 59:58 | The intensity. Thanks. Mm So now I would like to show |
|
| 60:04 | the accuracy is of the same sense all. And I'm kind of on |
|
| 60:10 | stage four and we'll see if I'm . Um, so again, go |
|
| 60:14 | to this simple taylor series expansion for same role. We have The 300 |
|
| 60:23 | from there. They point and they for us At the midpoint is four |
|
| 60:28 | the endpoint is one and divided. you take the steel suspension or functional |
|
| 60:38 | passage to age. Then they tried walk down the expression or the right |
|
| 60:46 | side of the topic. Creation. think. So now they want the |
|
| 60:51 | hand side that is up there, fire. But it's just having these |
|
| 61:00 | equations Plus this is more than proper . So understanding, not multiply this |
|
| 61:08 | and this 54 and the system So straightforward competition of They were serious |
|
| 61:23 | this expression and an expression of this . And what it looks like. |
|
| 61:30 | not done yet more. Now we've something like this. So now we |
|
| 61:40 | the right hand side. Now we look at what the left hand side |
|
| 61:45 | like for that. Ah, they this form on the color. I |
|
| 61:54 | the answer derivative, the capital And then they're going to make it |
|
| 62:00 | okay, two Plus H version. , so we still have a small |
|
| 62:06 | to roll then they can just stay same suspension of the capital and |
|
| 62:15 | this form of expression and then we that um, integrate from 8 to |
|
| 62:19 | 0. So this is double zero derivative or capital F is lower case |
|
| 62:27 | . So, And then it's There's lower case for one. Go |
|
| 62:33 | . The insurance. Mm hmm. of this was coming. So |
|
| 62:45 | we have the So this is the A. two H. So this |
|
| 62:51 | kind of this expression And um, what putting in the fact that this |
|
| 62:59 | here and this is that's the workplace . So we best support to |
|
| 63:05 | So that's just plugging in the So that's how So this is the |
|
| 63:10 | thing of the left hand side. now what is it going to do |
|
| 63:16 | take the difference what they have this inside right here we have expression for |
|
| 63:26 | using table series expansion and then now have it on the left hand side |
|
| 63:33 | both left and right hand side. friends are this thing and the left |
|
| 63:42 | side. And so this is the . And put that here. But |
|
| 63:53 | warrants. Yes. So they have expression that um so anyway, you |
|
| 64:03 | compare terms here hopefully. So these the same and the right left hand |
|
| 64:09 | and that was the right hand This term is also the same and |
|
| 64:13 | is the same. Um so the term that is different is in fact |
|
| 64:23 | . So now I said that in of the integral this morning, Israel |
|
| 64:29 | exactly material Simpson role for the interval an error term is mm hmm. |
|
| 64:42 | , so that was from a small but in general they want to do |
|
| 64:45 | larger interval. And then we apply send some jewel to a bunch of |
|
| 64:50 | access. Oh, it's called. that's why you and am I will |
|
| 64:57 | him the next time. That's all to have a bunch of these. |
|
| 65:01 | one order it disappears because of the of things that are proportional to |
|
| 65:13 | So, so in other words, just passed it. No, something |
|
| 65:24 | is determines the that's a long Um, that doesn't have a |
|
| 65:33 | Sorry. Yes, that's one another five which is, it's one symptom |
|
| 65:39 | whole thing. But you want the , all the things you divide the |
|
| 65:47 | between A and B into some intervals then you apply Simpson's rule for each |
|
| 65:52 | of these sounds into all the same as So let's start from the beginning |
|
| 66:00 | year. Any questions so far on , on this today, I'm showing |
|
| 66:05 | electron for me it's it was the trick that was used plus before and |
|
| 66:14 | ah left hand side. And use anti derivative and the privacy of expansion |
|
| 66:21 | on the left hand side and also a series expansion on the right hand |
|
| 66:26 | , look at the difference and then downwards the first. How about |
|
| 66:39 | wow ! All right now you don't just want to stay in some form |
|
| 66:46 | the interval all the intervals. And the results of the symptoms smaller. |
|
| 66:59 | that divided the interval between D. . A. And to end |
|
| 67:06 | And for each one of these segments going so far simple something. So |
|
| 67:13 | million years with the kind of symmetric I'm now the midpoint for each of |
|
| 67:19 | Simpson's rule. So that's an And then some of you're insulted of |
|
| 67:26 | Simpsons. We're going supply for these in cigarettes. So this is exactly |
|
| 67:35 | the sense of to do a bunch intervals over small segments to age and |
|
| 67:41 | . And we just divide the integral into something else. For informative, |
|
| 67:46 | memory. Very soft for all the , smaller intervals, no approximation stuff |
|
| 67:54 | approximation happens in the interval for the interval. Where are we going to |
|
| 68:00 | this? So so now we're looking during the integration and replacing this uh |
|
| 68:13 | the interval of length to each for Simpson formula which is left standpoint report |
|
| 68:24 | right. So this is basically to -2 and this is why I'm on |
|
| 68:32 | 1610 and some of the age the thing is right one of the two |
|
| 68:43 | points of this million to one and in the middle The Vitamin D three |
|
| 68:48 | after transforming it will some interval of function. So now that's why the |
|
| 68:57 | comes in it's in formula but it's now the truant developments the Simpsons for |
|
| 69:03 | smaller so what kind of player all this expression? Ah Now get the |
|
| 69:17 | form. So if you take one these sub intervals and he takes the |
|
| 69:27 | summer vegetables ah The two endpoints of subject of all is in both the |
|
| 69:36 | and right something. So so this is the and for the sake of |
|
| 69:46 | previous interval and this is also right point of the figures and this is |
|
| 69:52 | left hand from from the next thing . So that's why you live with |
|
| 69:57 | guests. Now the waist this four the mid points dr Alexis Simpson install |
|
| 70:09 | they each fill sub interval. The appear fights especially left hand point and |
|
| 70:19 | and right hand front in and So now instead of having the two |
|
| 70:25 | staying on the way The same but it's up to the same chapter Sawyer |
|
| 70:33 | divided by two The Total vol three instead of just something the internal function |
|
| 70:43 | I think it's weighted by 424242 instead 24. So to me it's a |
|
| 70:49 | simple performance also kind of so memorize Simpson worked as opposed to the tractor |
|
| 70:56 | its various simple parents speaking. So Right so because now yeah that's fire |
|
| 71:18 | of this book potentially the error accumulates all the different sub intervals and subject |
|
| 71:27 | and end times pages blank for intervals can produce one power here but they're |
|
| 71:38 | so that's what becomes age to the in there. So so that's solid |
|
| 71:48 | it's for a very simple scheme. on the very simple example of one |
|
| 71:55 | record for the for the virus expert . The difference in waiting for the |
|
| 72:02 | number of function evaluations Stadium that That's fine. Mhm Thanks. Evaluating |
|
| 72:13 | expression instead of just having them waiting or by to market time of 4 |
|
| 72:19 | instead of best not to find one us having them up very little. |
|
| 72:30 | are you doing? Multiplying by two forces shifts amongst by in terms of |
|
| 72:36 | binary systems. Very simple. You that cell free. Right. So |
|
| 72:46 | can see not only is oh jihadis four instead of those two possible proficiency |
|
| 72:55 | the 15 smaller and smaller. So works out. Okay. Stop |
|
| 73:08 | So um too many great informer's Yeah, that's something else. |
|
| 73:30 | So So we talked about the tractor rule, the basic 1 um two |
|
| 73:40 | we got this for this kind of then Now two steps from to grab |
|
| 73:48 | swords. So the interval 9 15 divided by two. And that they |
|
| 73:55 | for the endpoints and and then we the new points and this is the |
|
| 74:02 | . And if we then your insides users hamburg algorithm based on the |
|
| 74:12 | So now what happened? two points H&H of the two With a step |
|
| 74:19 | one and 2 trapezoid And then we do one step the Richardson extrapolation of |
|
| 74:28 | step of the Romberg. They use to to compute an approach estimate. |
|
| 74:36 | if you manipulate these two expressions you discover the actual against things. So |
|
| 74:42 | noted its exact same. Waiting other values best to forget if you use |
|
| 74:50 | a fun mix population on the so this also shows up and they have |
|
| 75:01 | but but these weights find out something adequate to they should before the same |
|
| 75:09 | up as Bloomberg God it's all the . But now we're kind of awesome |
|
| 75:21 | for starting for deriving the 60 based for normal skin. That's actually that's |
|
| 75:29 | into the second order, There's no instead of just looking at Stromberg algorithm |
|
| 75:41 | expressions and yes, you know the term. But driving the waiting positions |
|
| 75:50 | for the function values based on what normal to want to integrate correctly. |
|
| 75:57 | gives me a better intuition for what . It does a good job on |
|
| 76:07 | same thing on the market. So not really. So that was that's |
|
| 76:19 | think for today so we'll continue to a little bit more different and progression |
|
| 76:27 | us. It's time. Yes, had to come and so on. |
|
| 76:35 | back to that. If we talk compliance in other words, you're looking |
|
| 76:42 | this the chapters old rule. Integrated voice senior functions correctly. The Simpson |
|
| 76:51 | basically into it functions correctly which consists peaceful, he's still in the |
|
| 77:03 | If you think of it as the is successful the next interval and have |
|
| 77:09 | totally different political enormous success. The of political enormous because that happens to |
|
| 77:14 | your function then it's integrative. It rain. Ah It's obviously it works |
|
| 77:27 | , it doesn't mean that it's a flexible in terms of what options from |
|
| 77:31 | integrated function for the normal instead of single and it still doesn't Okay. |
|
| 77:50 | questions. Oh yes. |
|
