| 00:24 | we're buying the shuttle conform to race . Yeah. Yeah wow. I |
|
| 00:46 | . And then you talk about the integration and next thing up there an |
|
| 00:54 | version Simpson's rule and I talked about time and then a few other aspects |
|
| 01:00 | integration. So just to remind you of what the who are they doctors |
|
| 01:11 | ruled and the Simpson's rule and point really have this status, the focus |
|
| 01:18 | the director. So I always like point out that the so they're higher |
|
| 01:27 | bounds or values I think of the that goes into the error term. |
|
| 01:34 | model that needs to be in order whatever errors from what so the |
|
| 01:45 | So if I have something like look this where the obviously the derivatives some |
|
| 01:54 | is in the beginning that means that need to have a very small age |
|
| 02:02 | order to get the activity you But that's I will not be um |
|
| 02:07 | court that's here. Where so if everything's are likely to be very |
|
| 02:13 | So so far in the persons we've about before last time that one would |
|
| 02:19 | to that's for you adapt or decide a change based on so that the |
|
| 02:26 | of derivatives kind of the worst section the of the function for the intervals |
|
| 02:32 | interested in. So that's not computational efficient. So that's what the ad |
|
| 02:39 | comes into play where everyone wants to to look at different parts of the |
|
| 02:44 | in different ways and then it gets decide on using different suitable length and |
|
| 02:53 | upon where you are on the So that's a simple idea. These |
|
| 02:59 | versions of integration benefits. So we'll it just show how you can develop |
|
| 03:09 | adapted version for Simpson's method and um coming next I think so basically yes |
|
| 03:19 | need to um underlying it at the office and then they used the error |
|
| 03:26 | for submit doubles and decided whether it good enough or not or what the |
|
| 03:32 | maybe and the relevant period. The that's the simple but the underlying all |
|
| 03:41 | them and then specifically how one can with that for them. The Simpson's |
|
| 03:47 | there is on the next few slides . So let's just do this in |
|
| 03:55 | basics. Same social they have uh is the symptom and then the other |
|
| 04:03 | and then the basic version and there's one into all this. Make it |
|
| 04:10 | . So then we basically have the area somewhere in the integral of the |
|
| 04:16 | , let's say in return. And again here it was the return, |
|
| 04:20 | just had one seat putting. So then um I kind of like simple |
|
| 04:28 | terms of principle to show how little where businesses okay for a look at |
|
| 04:38 | how the error of scales with the size in the algorithm and then just |
|
| 04:48 | the derivative plus all the other terms goes into the area terms. But |
|
| 04:56 | maximize whatever that value might be and it C. So I just taking |
|
| 05:04 | cap of what follows after the age divide itself. So now there there |
|
| 05:16 | stuff too get the activity idea is we're finding it should make his father |
|
| 05:25 | , for simplicity just divide things. as usual, so that means now |
|
| 05:31 | another saying some version. Where are now have operated? Yes, they |
|
| 05:38 | there. Sorry state. Give out evolving to have these four letters. |
|
| 05:44 | then they get another value for Simpson another based off now HP or the |
|
| 05:51 | they have the sites. So then with your heart, an error for |
|
| 05:58 | first half from A to C. another error from C to B. |
|
| 06:02 | that's why we get this. But few terms now in terms of there |
|
| 06:08 | it seems that operates but the size and then you manipulate these expressions and |
|
| 06:15 | the same idea. That's somebody's very returns here almost after Thanks to the |
|
| 06:26 | factor and make sure it's kind of the same maximum less. Yeah. |
|
| 06:33 | one single step. So that's what do. And then why can I |
|
| 06:42 | that? But you're wrong with this a little bit to figure out how |
|
| 06:46 | decide to verifying that is good So that's for these are the two |
|
| 06:54 | from the previous life for this on interval. I'm going to split the |
|
| 06:59 | into bats. And so that's the that relationship between the two errors for |
|
| 07:07 | in the morning. So now I'm minutes to figure out when they get |
|
| 07:12 | estimate of what the error is. saying that the true integral. |
|
| 07:20 | The one so that the coal seams you want plus it's earlier or the |
|
| 07:25 | sites two intervals Simpson his error and equal to the integral so that it's |
|
| 07:32 | this and get an estimate for the term. That is, there's a |
|
| 07:38 | between the two symptoms intervention values and proper factor 15 comes from, it's |
|
| 07:45 | comparing to do the subtraction sort of the uh this is 15 times of |
|
| 07:51 | times as large as despite from this . So In the 15 times error |
|
| 07:59 | oh my God, have the usable . That's enough thomas. And I |
|
| 08:07 | an estimate of error. And that's you can do 17 control. So |
|
| 08:14 | estimated error based on Um two versions the Simpsons are on the Say one |
|
| 08:21 | Winter Wonderland on behalf in Tomorrowland So if that's within the tolerance that |
|
| 08:30 | accept and we're done. And if not then you guys believe are going |
|
| 08:37 | repeat the subdivisions until you got something the area that's kind of a simple |
|
| 08:47 | creating something that is adaptive and kind self adaptive because you have underestimated, |
|
| 08:54 | compute as you go and then we decide when to stop science here. |
|
| 09:07 | and of course every time you refine and so you start with one and |
|
| 09:14 | you get two intervals but the error accumulate. So you get one area |
|
| 09:22 | of the interval and you get another of the other interval. So that's |
|
| 09:27 | for simplicity you kind of say, fine. It's something to walk and |
|
| 09:31 | half the area of the food in types of the summer. See the |
|
| 09:40 | but this is not this sense. then there's just a little bit of |
|
| 09:46 | and respectful. This is true because their problem interval over the which the |
|
| 09:57 | happens on the supply seems into each of their hats and they basically get |
|
| 10:03 | return for each one of the So for the sum to be less |
|
| 10:08 | the absolute that you wanted in the want to ever take half of |
|
| 10:13 | A lot of energy economy. So also need to produce the error tolerance |
|
| 10:20 | the number of intervals four of victims the message about us. Yes. |
|
| 10:31 | also in the book there's the movie the national reversal according four billion. |
|
| 10:37 | almost and lastly picks up, has talked about way back in terms of |
|
| 10:44 | of these, if the algorithms they to have need to stopping criteria one |
|
| 10:50 | the error tolerance and one mr I to go on forever. So it's |
|
| 10:57 | true. Yeah. One something for number of some decisions if it doesn't |
|
| 11:05 | what you expect, Man, that . Mhm mm hmm. So the |
|
| 11:16 | questions on that. But the best . Use successive refinements. They've got |
|
| 11:24 | estimate for the error and then they it's good enough for now with |
|
| 11:31 | The very same one. You can the same innovation for integration for most |
|
| 11:36 | that. Okay. So this just more concrete examples from the book. |
|
| 11:46 | . So it's best to say Yes. You can obviously used the |
|
| 11:51 | age but that was not necessarily in of that of course. Mm |
|
| 12:03 | Well summarized movement and this works for same time. The next another set |
|
| 12:14 | rules known as the comments schools. some comments before for the back |
|
| 12:27 | So again started to talk about the the integration. He started with |
|
| 12:34 | Okay, okay fine. Doctors aren't is to think of it doing this |
|
| 12:40 | size. There's actually one be approximated function with a bunch of straight line |
|
| 12:47 | between the nose, useful integration. then use this uh concept instead of |
|
| 12:57 | you know and say, okay, at that. Finding an algorithm for |
|
| 13:08 | these ways vineyard kurds or functions He was in that idea and |
|
| 13:17 | okay fine. That's the quadratic Instead of straight line approximation and get |
|
| 13:23 | then it's exactly when they're after the . Some schools. So with this |
|
| 13:29 | concept that working with Interpol, waiting monogamy Als come down from the integration |
|
| 13:37 | . That's part of the icons. was there idea or principle views and |
|
| 13:42 | men from those rules and this just parents what I just said. So |
|
| 13:53 | one thing that's so or the doctors invested an arbitrary set of correspondent function |
|
| 14:05 | straight behind approximation for Simpson they took for each segment of all this kind |
|
| 14:14 | one continue evaluation and that taste for big point because we have a second |
|
| 14:20 | polynomial with three points in the second polynomial and use the midpoint between who |
|
| 14:29 | to use for the filling of the I've been following so at least you |
|
| 14:39 | coach rules but you can use the idea picked a bunch of points, |
|
| 14:44 | spaced points. A polynomial that is to the order of the points that |
|
| 14:52 | him and then develop the weights for function values in the way we did |
|
| 15:02 | we talked or derived the Simpson's rule use the lagrange formulation of interpolation polynomial |
|
| 15:09 | then integrated the cardinal polynomial property proficient be in waiting the function. So |
|
| 15:18 | the principle also for them And calls the one example and first became a |
|
| 15:27 | again. So and they said they to the grand formulation of Interpol integrating |
|
| 15:35 | . There was contingent on the integration because then uh the ways of the |
|
| 15:44 | yeah function values after nodes are simply implication of the cardinal for more than |
|
| 15:53 | get the proper weights All function values . Mhm Yes. So we sneaked |
|
| 16:05 | code schools It comes in two One that is known as cold someone |
|
| 16:12 | known as open. So the difference that the open ones do not contain |
|
| 16:21 | use the endpoints of the integration for contractors. Sword and Simpson where we |
|
| 16:30 | about them. They used the endpoint develop integration interval between two successful. |
|
| 16:38 | in the open formulas that those endpoints excluded so I can come back for |
|
| 16:48 | . So so here is just a of lines that Give two examples of |
|
| 16:57 | sneaked in some enclosed rolls and this not distribution. So and so that |
|
| 17:07 | cold suspect producing the general rules for they divide. Oh you're right. |
|
| 17:14 | will also depending upon It's affected two and the first of the polynomial. |
|
| 17:22 | they get the trappers or rules they their cultural. Another one is to |
|
| 17:28 | two end points in the mid points they did the symptoms and let's get |
|
| 17:33 | ah ruling India. They know the sets of rules for ah coast. |
|
| 17:41 | integration of divorce. There is also one that symptoms came up with this |
|
| 17:48 | I actually uses four points Including the phones and two and between and I |
|
| 17:57 | uh have more points and also have points in general you've got higher accuracies |
|
| 18:05 | . But when you get the composite to have many smaller intervals than from |
|
| 18:13 | , It became a page to the for this formulates four for this |
|
| 18:18 | The basic one in the morning. in this case between the sometimes both |
|
| 18:27 | third room um of Simpson then it's it is an age before one of |
|
| 18:35 | less than ideal and you didn't gain order you game the system but the |
|
| 18:43 | factor of about 2 2.5 the cancer not too much can but if you |
|
| 18:48 | to Including the more points you get higher order polynomial five 4th audible and |
|
| 18:55 | approximation than you do And more after one. So now in the next |
|
| 19:06 | copy some of it's open speaking culture you're navy. So now examples but |
|
| 19:19 | means that this is kind of a point in an interval between A and |
|
| 19:24 | . It's not just so that means basically the council value actually point value |
|
| 19:32 | the function so it becomes a rectangle campus. And what is this on |
|
| 19:42 | method using now there are stop the . Fabulous. We have three |
|
| 19:52 | one from that standpoint. The first and the function value has fun one |
|
| 20:01 | the first internal for the second. so there's still a few Kind of |
|
| 20:07 | . So that's what you see is number three here 702 and two. |
|
| 20:14 | you have 33 intervals. So that's for them but there are equal |
|
| 20:21 | So and they another one that is fairly well known in the literature and |
|
| 20:30 | around with this movement that essentially gives the same or similar order of accuracy |
|
| 20:37 | the those same. Some truth. I think on the next slide I |
|
| 20:43 | to be talked about the trap his at the same time the rule. |
|
| 20:49 | Quantum and compare it to the area terms same order about scientific concerts |
|
| 20:57 | Mhm And the way and the hope that most of the time they don't |
|
| 21:03 | this for the same number of function . So you don't gain much in |
|
| 21:09 | stuff Using one or the other and on. Just three function evaluations. |
|
| 21:15 | is the error term. Just against function evaluations and the end of the |
|
| 21:22 | . But the point is that they about signing over excess integrating that and |
|
| 21:31 | can't do that with the post formula X equals zero is an important And |
|
| 21:37 | Sine X over X is on the . 00. And most software will |
|
| 21:45 | to do that for that. There serious expansion but if you use an |
|
| 21:52 | formula then you never evaluated for That something like so there's merits to |
|
| 22:00 | broken formulas but not necessarily for efforts be more terrible, particularly. Uh |
|
| 22:09 | . No or x values for the . So they're still worth. Uh |
|
| 22:18 | familiar with the open rules. Thank for holding. Mm hmm. That |
|
| 22:25 | to evaluate for being too All Thank you. Have an example. |
|
| 22:34 | . So but this is then using So this is the end of |
|
| 22:41 | Of -2-2. And they did this an open rule and it's the best |
|
| 22:46 | about In this case three notes are for the information. So the polynomial |
|
| 22:53 | these three points and but not until end. So that means we have |
|
| 23:02 | points. Um so we can get second order polynomial and so we have |
|
| 23:12 | distinct points. And then we define cardinal polynomial supported too. There's a |
|
| 23:18 | idea which is based on the noble and has written them down here for |
|
| 23:24 | constant 1st and 2nd. Ah our what they are and symptoms then just |
|
| 23:35 | this corner of the normals on the to the integral integration for the entire |
|
| 23:43 | intervals not just for the uh interval between the noble savages and interpolations service |
|
| 23:51 | case. The distinction in terms of , to get the endpoints of the |
|
| 23:58 | and not necessarily the extreme points of note. Uh huh. Park it |
|
| 24:07 | and then let's come on up the and then to get inside of |
|
| 24:14 | Not with that thing integration formula. Constance that follows from the information. |
|
| 24:23 | cardinal polynomial. So the information that wanted to choose and No, we're |
|
| 24:32 | already petitions for the function values and this particular case The function interpolation or |
|
| 24:39 | -10 and one. So that is that's the report. Yeah. But |
|
| 24:48 | is that's the big point of That was it proficient. So if |
|
| 24:54 | look at the table and have before some of the difference, the point |
|
| 25:00 | this place is for this. So have the mind of a for which |
|
| 25:06 | here from my industry, use of internet in the wall over which you |
|
| 25:11 | this platform so that investors can see this formula oh five ah for example |
|
| 25:23 | help. Right? So the general that is implement which is where someone |
|
| 25:32 | someone zone Mohammad. Mm hmm. . The question on this one. |
|
| 26:00 | . Um a lot. Oh this comes back from the nominal |
|
| 26:10 | There's nothing really different for novel and respondents shows playing around will have to |
|
| 26:16 | . You need to go to work standard in the border maybe 01 minus |
|
| 26:21 | to 11 is 2 to 13. example today. So yeah, just |
|
| 26:31 | through the things that variable changed on case. In this case 50. |
|
| 26:37 | is of this form on the left . It's 15 C&D. failing and |
|
| 26:47 | of the central point this way. is the stretching of the sending of |
|
| 26:52 | length of the interval. Yeah, see you not. Right. He |
|
| 26:57 | something So integration from C to They think there's something that it's the |
|
| 27:04 | comedy to be, which I think following woman inside working through the details |
|
| 27:09 | struggling it into dangerous. Mm All the values. Uh, expression |
|
| 27:19 | looks a bit messy but it's just confusing. Um, regarding girls. |
|
| 27:26 | you. So one more thing in of operation. So, so, |
|
| 27:39 | this is the, so after this building the grown folks. Um, |
|
| 27:51 | molds used for the polynomial in population was then used to, yeah. |
|
| 28:02 | awaits of function values. They all that for equal specs. Now I |
|
| 28:12 | after that of the normal interpretation, this another function that was started in |
|
| 28:22 | more. Um, so I didn't much, but I showed them before |
|
| 28:28 | happens, people have equally spaced but that's not necessarily a good |
|
| 28:34 | So then there were the championship points resulted in a much smaller than. |
|
| 28:41 | this is just to say that sort one thing is to okay at |
|
| 28:46 | The entire wonderful aromas such as the . Another point is how do you |
|
| 28:52 | the board is not necessarily. And sort of thing in terms of integration |
|
| 29:01 | do something known as Gaussian selected for selected points ah and not television in |
|
| 29:10 | case. And I'm not talking about girls came up on the way of |
|
| 29:17 | selecting the notes, the gaps and I act person and they of |
|
| 29:29 | So I think this is put on line but also not. So. |
|
| 29:39 | yes. What's known as God's hand . So you stick it out. |
|
| 29:50 | if you choose the no being used of them is going to be used |
|
| 30:00 | knowing and interpretation of the function as roads of polynomial has particular properties. |
|
| 30:12 | when we talk about championship there was formula what the rules are the ship |
|
| 30:20 | ship polynomial in this case. Different . That's the that's the property that |
|
| 30:30 | this polynomial because Degree and Plus Then the integral of the interval. |
|
| 30:41 | with all the difference polynomial that are an order less than the order of |
|
| 30:50 | polynomial. Many of the cultural Mm hmm. Up to end where |
|
| 30:58 | . Is probably and personal. So you use a particular property that the |
|
| 31:07 | with respect to august normals is identically . Then that's a good choice of |
|
| 31:14 | . And then you find that polynomial those are then they know to use |
|
| 31:21 | population. So it's kind of two process wants to find you then too |
|
| 31:30 | the interpolation after he had found the . So that's to remember that it's |
|
| 31:38 | of step forward. So you know you can have. It's just saying |
|
| 31:46 | . So it is spicy basically increases order for accuracy by the power of |
|
| 31:59 | instead of being so the same center older. It's more than we get |
|
| 32:06 | that's over. Mm hmm considerably higher accuracy for the same number of hope |
|
| 32:20 | would like to do that. Um that's enough to be generic example kind |
|
| 32:27 | here. So it's symmetric interval of -1 last month. Um And |
|
| 32:38 | um we I guess decided to have a normal here of Degree three so |
|
| 32:48 | it's um three points to get the order from now. Two points gets |
|
| 32:58 | first order. Mm hmm. So for the normal so in a way |
|
| 33:06 | be try to find out well the uh yeah, 30 weeks X |
|
| 33:15 | So that should have three zeros. that I'm used for the population |
|
| 33:27 | So practice. Um We have them figure out in this case what the |
|
| 33:33 | are in this polynomial in order to a few that is suitable. So |
|
| 33:39 | you to be good then there should The Sisters 3 3 than the |
|
| 33:46 | And this guy up here where K 01 and two should be zero. |
|
| 33:52 | this is Integration of the constant zero 1 and K two. And you |
|
| 34:00 | have these properties to qualify to be him. And probably they want the |
|
| 34:06 | . We need to produce the same . No. So, so |
|
| 34:16 | we have four conditions. So we paramount degrees of freedom. And this |
|
| 34:24 | to choose uh These two coefficients are we have not simple for an overall |
|
| 34:30 | because it's ah since um this is the interval here so soon it's symmetric |
|
| 34:47 | X equals zero. That means these the meeting integration and other power of |
|
| 35:04 | . Um The green line is one 1 is automatically zero so that's kind |
|
| 35:11 | being used. Uh huh. Ah develop the coefficients here. So I'm |
|
| 35:25 | take first to the constant so if guys are zero then we'll have this |
|
| 35:30 | um and then again for the yes then we'll see. So where can |
|
| 35:43 | get this expression right? To take integral? We have two of this |
|
| 35:49 | . So let's say we're not consuming The integral should be 04. That |
|
| 35:58 | the texture first degree. So there's now what the gaps that seems um |
|
| 36:09 | quickly integrate this thing we get. So the integration our experience becomes next |
|
| 36:21 | so this part and this becomes extra . Just the old powers. Um |
|
| 36:29 | that's going in ah we Sure Um So we have C1 times basically |
|
| 36:40 | to the 3rd power. And that's of an answer obviously and then subject |
|
| 36:47 | for the lower power twice X. or three similar reports. Uh This |
|
| 36:57 | make it and mhm. You that's invested and if one chooses so one |
|
| 37:07 | to have one relationship here they say they have two degrees of freedom. |
|
| 37:11 | the C. One and c degrees one of them had decided basically ratio |
|
| 37:15 | been fixed but then you can fix the ratio And the number from this |
|
| 37:23 | was the curious what's best cheers um 5 -3 And then again zero is |
|
| 37:34 | problem. And the other two rules basically five X squared 93 equals |
|
| 37:43 | So yeah these two are the Um and there's and there's the three |
|
| 37:50 | groups so even actually And the group the best produce these two coefficients to |
|
| 37:58 | And then there's the choose the ratio c. one and c. |
|
| 38:01 | But and I seem to hear then still have three valid roads of the |
|
| 38:10 | . So now I missed it at point we're about to the point where |
|
| 38:17 | ah good choices for interpolation or department to interpolate the function. So please |
|
| 38:32 | coming now and integration formula that will was the root of the particular for |
|
| 38:39 | normal jews and then Q. Is unique. That was one choice and |
|
| 38:44 | is accused or these are by no the only choices of notes and the |
|
| 38:51 | longer longer depends on. But with accused of this is what we ended |
|
| 38:57 | with in terms of uh notes for population from And then the best to |
|
| 39:06 | what we've done before it induces your that we have no points. So |
|
| 39:14 | is integration. Yeah the branch for polynomial to get the very spot christian |
|
| 39:22 | already wanted and since it was supposed be integration now a certain degree Polynomial |
|
| 39:37 | have three points. So it should correctly. Like the concept of first |
|
| 39:44 | in the second degree so that what did is fascinating ah make sure that |
|
| 39:52 | get the questions so when we integrate these streets and to be someone else |
|
| 39:58 | , That's correct. So so now approved Same thing integrated constantly into its |
|
| 40:07 | power in section four of X. the door and then the gonna shoot |
|
| 40:17 | is the form that right? So plug in the function values of mm |
|
| 40:23 | The three notable points. So if integrate the concepts from -1 to 1 |
|
| 40:30 | get it to and then invested in value on function that is not one |
|
| 40:41 | this point at this point that if is a constant. So we basically |
|
| 40:46 | that's zero times one. The function is independent of X for one. |
|
| 40:52 | that's why we have just the sum it three positions now to integrate X |
|
| 41:01 | and um then we get X squared Itunes and apple ah then put in |
|
| 41:11 | upper and lower integration bugs and the because it there's a lot of functions |
|
| 41:18 | around 0 - zero. And now the function value that is the value |
|
| 41:27 | X. Ah less interpolation notices. value themselves chronometer. This is basically |
|
| 41:39 | that we have 4 -4, The function value 00 and zero stuff |
|
| 41:47 | like to get to the airport and corresponding provisions for the informed and similar |
|
| 41:54 | them integration of exporters fit in. now we have three equations. I |
|
| 42:00 | figure out 15 years Calendars from the and so we need to get something |
|
| 42:06 | looks like this. So now the to have also an integration formula and |
|
| 42:15 | we have an old points. We the weapons. So now ha that's |
|
| 42:23 | . The gas production formula. It's on this particular people excuse in |
|
| 42:33 | So yes obviously and then we can it and see if it works on |
|
| 42:43 | can do the integration here. So this is the gun and three points |
|
| 42:50 | things for interpolation. So there was quadratic polynomial. Um that was just |
|
| 42:57 | fifth but not at least three particular points. Not for once. Um |
|
| 43:02 | the Simpsons role that was 59 points eight point. Ah But now so |
|
| 43:12 | this case is too. So the The order of the polynomial Q. |
|
| 43:21 | three. So um so the N one was thinking still N plus one |
|
| 43:29 | five. So we can test So it integrates the fourth order problem |
|
| 43:35 | correctly. So get in here and best of the functionalities 64 that this |
|
| 43:47 | points to do that. The question the same result. And aggressively during |
|
| 43:55 | integration in this case the endpoint bad . And in principle it should hold |
|
| 44:04 | for The 5th order. So they to find out that it is |
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| 44:11 | So that's just part of the proof example that the plane problem is quite |
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| 44:20 | . That is correct. At least oh that's sweet. And then there's |
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| 44:30 | translation Frankenstein questions. Um so it seemed to go quite around without the |
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| 44:48 | times and previous lectures have come from was on your intervention to the minus |
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| 44:56 | squared to use. Ah no So the president, the former just |
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| 45:04 | imploding function values and change the interval final 1 to 1 to 0 to |
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| 45:10 | instead. That is what you want this protection. That's funny through all |
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| 45:17 | proper substitution what was on this side then what there are concerns uh some |
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| 45:30 | but then the end result them. the God Squad editor. Again there |
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| 45:38 | three points and this is um so f of -43 on the interval and |
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| 45:51 | midpoint and direct 10 points. So got these values and plug it in |
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| 45:56 | you get something like this. So pretty close to three. Well we |
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| 46:02 | two environment that he's using that script all this in the function. So |
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| 46:11 | is compared to what they had before rule there was 72. So this |
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| 46:19 | considerably better for the same and it be for the absolute last furniture by |
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| 46:29 | inc oh and breakfast also you can that first much smaller. Care about |
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| 46:40 | 12th Oh No 1's following separation on ground. So you know um for |
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| 46:58 | next what? It's on finding the any questions because some of the other |
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| 47:13 | the perfect. So All right. so the genre was some fellow that |
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| 47:22 | it up. A procedure for finding would seem common audience and there is |
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| 47:33 | a list in the amount of polynomial This was around me previous an example |
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| 47:42 | it for the the best that it after the first formula for defining Oh |
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| 47:51 | which you can use the rules probably through this version formula has this property |
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| 48:01 | when we integrate them in perspective Morneau up to par les sunday the order |
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| 48:09 | a few functions. Yeah. The property is zero and then use |
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| 48:18 | roots but not now. Okay, that is stable spectacle for ah the |
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| 48:34 | what are the rules? That's why sometimes called the because it's based on |
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| 48:44 | for enormous Judd's because don't start see it does. The open I think |
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| 48:55 | wanted to remember except that terrorists are and their tables worth that and finding |
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| 49:01 | but you know you are and what correspondent the weights are for those functions |
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| 49:08 | ways to get our integrated Cardinal No one else for the nose of |
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| 49:19 | . There is a geometry problem. . Thanks to it's not that simple |
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| 49:27 | values so don't guess principal. So , sponsors so far. Yes it |
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| 49:42 | three equal space points and that was . Not necessarily for the tractors on |
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| 49:50 | road but the simple the reform that use for the space point, the |
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| 49:57 | rule. Um It was that's been the midpoint for each interval but then |
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| 50:04 | intervals of different lengths and the composite . But for this again each some |
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| 50:11 | . The midpoint was the 164 defining . And then for the new England |
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| 50:21 | wall for generalized institution actually different and notes for each little something terrible. |
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| 50:30 | then there are still in a war equal respect And he saw the generalization |
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| 50:37 | must move beyond that achieved 30 points that was the house ah majority rules |
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| 50:43 | fund interpolation points during this for enormous the special property integration was enormously and |
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| 50:57 | the reversal form and and these nights is a okay fine market generalize it's |
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| 51:08 | fun. Sometimes very useful that you the integration is back to some form |
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| 51:14 | awaiting functions um trying to integrate the of functions and generalized what they've done |
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| 51:26 | terms of getting this economy integration. this is that's the same way but |
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| 51:36 | through the integration respect what the to the enormous too. So that now |
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| 51:44 | integral as the mystery trump represents. was earlier. I just assumed that |
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| 51:49 | boy compunction was and then to the of what watch it you have some |
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| 51:57 | we wanted to do. Yeah from software. Um Yes. 2016 for |
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| 52:21 | their expression of someone to find Yeah. No with respect to higher |
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| 52:28 | better than this are given. And then went to the doctor |
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| 52:37 | Okay. I don't know. okay. Got any questions on the |
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| 52:53 | . Okay. Two starts again demand trickery to being watched. Wow |
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| 53:06 | So you don't have one. So we talked about saint sonics or |
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| 53:14 | one of the tricks was to lose say the serious expansion and then The |
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| 53:21 | of the sign it's over except for . Also next was small and it's |
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| 53:26 | serious punch. Another thing was used approach rules avoid using that picture um |
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| 53:35 | value. And he was yet another to use valuable substitution to the this |
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| 53:47 | right here. And that sort of for ah T. and between zero |
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| 53:56 | one that was zero about to Mm Hmm. Um quarterbacks from |
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| 54:06 | Then you get to the T squared of the so find the derivative of |
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| 54:16 | . Then as derivative of squarely And so let's go ahead and get |
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| 54:26 | of the phenomenal. So That means x suspects the two attacks T. |
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| 54:35 | pretty. So you have digital T top of them get on the |
|
| 54:44 | Can we get the simple expression works some learning can be this type |
|
| 54:49 | So this is kind of the variable to you can use for instance the |
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| 54:57 | integration formula no whatever. Um once get this so the for many of |
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| 55:15 | functions that kind of victory that's the for the warning um transformation or also |
|
| 55:28 | . Oh this is a summary of I said. This is about in |
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| 55:35 | case for all this integration formulas. think this way of rewriting them and |
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| 55:40 | about them is to think about it a systematic way has fallen on the |
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| 55:47 | . And then the second branch formulations get the dependence of coefficients of function |
|
| 55:59 | by having separate don't find somebody and forget it better actors than sequence space |
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| 56:12 | . The university was english and spells performance the service for enormous next example |
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| 56:32 | and yes. Yeah. Mhm And then we'll send them information tomorrow |
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| 56:44 | then Yes. No and that's a not you one polynomial fits for that |
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| 57:12 | rule was essential to self control and as linear approximation of a function instead |
|
| 57:17 | just the states and promoting the walls that was and for Yeah it's also |
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| 57:31 | um for its internal segments some proper about formations of that whatever they choose |
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| 57:44 | the last time. Yeah, that's yes. So this idea after uses |
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| 57:55 | prize approximation comes back to all my was going on instead of the polynomial |
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| 58:07 | for a large number of points of example and the more the number of |
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| 58:16 | that means contributes to errors. Not well they hate. So it's better |
|
| 58:25 | have Many approximations. Piecewise No order than one fire woman. So 100 |
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| 58:39 | to be paranormal. 600 points and reverse results than they deserved. Simple |
|
| 58:45 | normal. It's so that's just thankful . So yeah. Thank you. |
|
| 58:56 | . So as I said, So that's well the next step for |
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| 59:04 | is about spying and finally face there's find the collection of no order at |
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| 59:17 | in your meals on the water for different polynomial at different intervals and then |
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| 59:28 | these enormous distance up in the most Then you have a much nicer than |
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| 59:37 | 71 for normal with all the Yes, that was the question. |
|
| 59:47 | , I remember hearing that term when, when and when it comes |
|
| 59:51 | like making curves. They basically use in drawing applications to make precise |
|
| 59:59 | Right? So that's what it So let's see what don't have very |
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| 60:07 | like probably Find one common somewhere back . All right. Yeah. In |
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| 60:19 | twenties. Yes. Yes. so here is. So I flew |
|
| 60:29 | the next subject of all that. have a bunch here to get this |
|
| 60:33 | wise um straight line segments and in case depending upon what whatever the next |
|
| 60:42 | value here and the straight from is you get something where it's not very |
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| 60:49 | when you go from one segment to other. Yes they are connected. |
|
| 60:52 | there is no job. No this . But as you can see it |
|
| 60:56 | another straight time. You know there's continuity saying the derivative. Um That |
|
| 61:04 | necessarily also teach you here is another but it could potentially. So what |
|
| 61:09 | with finances in addition to your serving interpolation between points. You also put |
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| 61:18 | on this kind of the smoothness when move from one segment to the |
|
| 61:23 | So the spines that specifies there behavior you transition from one polynomial approximation to |
|
| 61:31 | next. So you can in addition specify that it's continuous. Like it |
|
| 61:37 | be, you know, underlying the one's linear but it's continues. So |
|
| 61:45 | next thing is to specify, I the first delivery thing also to be |
|
| 61:49 | same on the on the left side , that's on the right side. |
|
| 61:53 | that the tangent is kind of a move around. And then you can |
|
| 61:58 | , well I also want the second to be the same to the left |
|
| 62:02 | right. That gets you different order restrictions on it polynomial stuff and get |
|
| 62:11 | smooth 1st. And that's what photographic . And you don't want necessarily to |
|
| 62:19 | it like a bunch of flat surfaces you want to have a conservatory but |
|
| 62:26 | small. But then the polynomial is one of them or the reporters. |
|
| 62:32 | they don't grab around the block smaller pieces to do the smoothness conditions so |
|
| 62:45 | we have many low order for no else? Government's fault domains. But |
|
| 62:50 | their schools. So that is what talked about, constructive. The different |
|
| 63:00 | . So they have this property but also seeing extra force derivatives. There's |
|
| 63:12 | limit on how many derivatives of course the same on the left and right |
|
| 63:18 | side. They're not ending up with single pull and all. So that's |
|
| 63:24 | phones so far. This is seven paranormal. Left and right. And |
|
| 63:30 | want the 2nd river. They've also this movie actually wonderful. But we'll |
|
| 63:38 | about that. No. Mm I'm going to look at one |
|
| 63:53 | Did they kill the guards predator? right now. I'm busy just like |
|
| 64:03 | for every like point that you're making approximation of. You've got to make |
|
| 64:08 | you want that approximation. You can anything you can do with that. |
|
| 64:17 | , I just picked it to us . The best buddy you can have |
|
| 64:20 | paranormal city. The only point is , the, the higher best for |
|
| 64:30 | . Yes. What are you? so that it still is over even |
|
| 64:36 | not change. I can look after . Wonderful. A normal I |
|
| 64:41 | Oh good question. So this is gun. There are so it's just |
|
| 64:54 | for every increase order that's being One more hope. Right so |
|
| 64:59 | This simply for the normal approximation We got three points. We have |
|
| 65:07 | points. So for every water you one more point. Yeah. So |
|
| 65:15 | where is the idea that the so like an at the end of term |
|
| 65:23 | points within the interpolated fall now then how what degrees of polynomial visits? |
|
| 65:33 | Is that I remember it has to like a specific or did he like |
|
| 65:41 | them? Thank you. Yeah. we'll see if I ask questions. |
|
| 65:56 | highest order upon the normal is unique a number of Um whatever number four |
|
| 66:05 | one order less than the number four needs. So if you have three |
|
| 66:13 | you can fit an inordinate number of order form enormous but only 1/2. |
|
| 66:22 | once you try to get the order paranormal of northern and central points then |
|
| 66:27 | degrees of freedom and then they're normal points such as okay. What? |
|
| 66:36 | it goes on from the incident. are three points. It doesn't mean |
|
| 66:41 | is necessarily but that is falling normal if these three points are on the |
|
| 66:47 | lines but you end up getting it's the first order from the north. |
|
| 66:52 | what There is no way of defining than a second order a few |
|
| 67:03 | Mhm. And the number of rules a polynomial equals the number of disagree |
|
| 67:13 | the following all of that stuff. looking for evidence. Yes. Yes |
|
| 67:19 | the crosses. Yes. Excuse our . And process it twice. There |
|
| 67:25 | two rules. Their post or you talk to you. First owner is |
|
| 67:33 | crossing. So the number of The rules from the number of rules |
|
| 67:38 | the melanoma. Is that most of degree of the north. So this |
|
| 67:45 | when we talked about They wanted a number of so they want three operational |
|
| 67:54 | . Then you need three routes because points. So if you want the |
|
| 68:01 | points we need the development. Mm . Get four points. We need |
|
| 68:08 | choose. So in that case the of the polynomial is tied to the |
|
| 68:15 | of collection points that they want. hmm. That's also that defines the |
|
| 68:23 | of polynomial consensus point. The operation Different 53rd Degree polynomial to those |
|
| 68:31 | And asked them to find the accurate . Again connected so. And then |
|
| 68:50 | . Mm hmm. I think sometimes song. But this is the expression |
|
| 69:00 | . So this is based on this So the rules are the interpolation and |
|
| 69:06 | number of rules on the number of points determines the degree of the uniquely |
|
| 69:16 | the degree of the polynomial defend defines personnel was you know, simple |
|
| 69:25 | That's the point. First degree. without a requirement Simpson normal export in |
|
| 69:35 | of home. The composite note. that's it accurately In spite of the |
|
| 69:41 | in the home and a music class that was in general the degree of |
|
| 69:50 | is connected to The number of inspiration . This is definitely true. |
|
| 70:00 | So this is where I have to about. It is I started collecting |
|
| 70:07 | dr Simpson. The higher order polynomial our police still segments better than the |
|
| 70:19 | . So what they want that are accurate. They want. That's also |
|
| 70:24 | integration or no other forms. I the question now it's adopted. You |
|
| 70:31 | choose some very cleverly downstairs. They need as many. I think that |
|
| 70:37 | can take formula in the effort. in this case this instead of caring |
|
| 70:46 | something proportional to cover and then there's that is more best and so |
|
| 70:54 | This place to get The higher order eight or higher power of eight. |
|
| 71:00 | certainly the temperature. The pictures are equally with the party basic choices. |
|
| 71:14 | order polynomial higher accuracy and the normal . If you want determined the number |
|
| 71:22 | interpolation points that you need. And if you want to be on the |
|
| 71:26 | instead of picking them equally spaced to them as roots for this performance. |
|
| 71:35 | again these polynomial is about the need the minerals constructing such polynomial. Uh |
|
| 71:45 | the finding falling almost that has this . But it's not. So the |
|
| 72:00 | behind the rules concept the complicated mechanics questions. Alright. So that someone |
|
| 72:30 | you. Oh |
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