| 00:00 | mm hmm. There's been a lot examples on everything. Every new concept |
|
| 00:20 | Yeah. So what are the types watching out there actually go to |
|
| 00:27 | Sure. Okay. So um presentable before the break, I started to |
|
| 00:36 | about ordinary differential equations. So that's I want to continue to talk |
|
| 00:44 | right? Yeah, it does. this is kind of a international last |
|
| 00:54 | too best. And so it's that's problem formulations where the problem is kind |
|
| 01:01 | defined by change or differential. So one knows the rate of change of |
|
| 01:09 | and then Bill starts the thing that thought you might be familiar with. |
|
| 01:15 | other thing is also Newton's second bomb this process. Change in velocity is |
|
| 01:21 | function of acceleration. So sometimes problems formulated in this differential form and for |
|
| 01:30 | of the action Simple Thing, there just one the question. But this |
|
| 01:37 | where ah things are over time for , they can always than the other |
|
| 01:45 | being in space and solution or a in some sense that one was to |
|
| 01:55 | terrible. Sometimes they use explicit notation sometimes just use it. You |
|
| 02:02 | everything big. And then I'll also this one the simple one where we |
|
| 02:10 | to solve this one can retest that kind of a problem with integration, |
|
| 02:14 | for nothing before. So it's just with buying up the or on the |
|
| 02:19 | hand side and through the integration. then of course that gets um it |
|
| 02:25 | often the but I never want that's function on the body that have at |
|
| 02:31 | see where we start and then there to think the grill um the right |
|
| 02:38 | side and so the most money break basically I believe that this value is |
|
| 02:47 | in the intervals of tangents and I to the end and that is what's |
|
| 02:54 | as the oilers method that there's some here. Um I'm gonna assume again |
|
| 02:59 | this is a constant On the previous . The other one that also went |
|
| 03:07 | Greece last time is then instead of this simple assumption, look at the |
|
| 03:14 | and using it. Doctors order. put it on the disorder role. |
|
| 03:20 | use the function values of the two and then there will be a range |
|
| 03:28 | . So that is this formula Oh my camera is off. |
|
| 03:35 | wow. Yeah. Okay. Sorry about that. So mm |
|
| 03:46 | But as we pointed out the problem this one we want fine Next the |
|
| 03:53 | . But we also have that one right hand side. So whereas in |
|
| 03:58 | one there is no beyond anything to where we are at time. T |
|
| 04:03 | this case both the left and the hand side has the same animal |
|
| 04:07 | So that this what it means to what? Not anymore. No. |
|
| 04:17 | . Um look in the middle I so it ends up being somewhere |
|
| 04:25 | No. So this was what's known an implicit formula where is this guy |
|
| 04:30 | here is explicit because it doesn't including future values. That is not |
|
| 04:37 | So we'll come back and most of things that we're talking about today are |
|
| 04:42 | methods or all the methods for its . Alright. So that's what wants |
|
| 04:50 | know something that they did not talked last time. Is this kind of |
|
| 04:56 | picture? So sometimes it is So this is simply look at some |
|
| 05:06 | of X values and some range of news and whatever you choose to plug |
|
| 05:14 | here. You can evaluate the function which is basically and tell them what |
|
| 05:20 | slope of a curve going through that would be. Now these vector fields |
|
| 05:27 | be used for a number of the I've talked about today but it's also |
|
| 05:32 | used in computer graphics where they generate these values and there's good trace depending |
|
| 05:40 | where you start show these different cards this spot is for different values of |
|
| 05:46 | . That the story. So depending you start whatever the solution follow some |
|
| 05:53 | and process. So then for So here is one of the |
|
| 06:01 | Have you can follow us generated values different points and sometimes you can inject |
|
| 06:07 | particles. A lot of the fluid people like to do. Yes, |
|
| 06:15 | particles and then follow the force Figure out something. Or I think |
|
| 06:21 | for these things from someone if you this part. Mm hmm I followed |
|
| 06:29 | brief mm mm mm mm mm mm mm mm. Well for us it |
|
| 06:36 | work. So you can follow basically of electricity language and that smoke and |
|
| 06:41 | how things evolve something that is being . There's another one I guess difficult |
|
| 06:48 | this one too. Yes sir. to give you some notion of what |
|
| 06:54 | very difficult sometimes. Mm hmm. and here's another one. Again, |
|
| 07:03 | back to this kind of a few . And the point of this one |
|
| 07:07 | just to show that someone gave some equations for all these um, are |
|
| 07:15 | of tricky in the sense that from Fargo these are basically a whole |
|
| 07:21 | of trajectory is the difference in the and the stolen values and this kind |
|
| 07:30 | vibrant shades in very different directions. for very small changes from the initial |
|
| 07:36 | don't need to be conscious is a bit about the properties of these |
|
| 07:40 | how they're going to behave. Not . But thank you. Stop. |
|
| 07:49 | another thing discovered in France is kind uniqueness of solution. The science of |
|
| 07:59 | process of things over our behavior In areas around value horses and supporting your |
|
| 08:04 | zero and 0. And it's some domain around both have really city range |
|
| 08:13 | time and in the solution. The if things are going overhead in terms |
|
| 08:18 | the function and its derivative, then can also say that solution is, |
|
| 08:26 | we're using their So it's unique. as long as these conditions, the |
|
| 08:36 | , wow. So now I talked some real methods for solving the differential |
|
| 08:46 | and What kind of unit three basic . It's always so serious is an |
|
| 08:57 | . Not necessarily good option, but is an option. Other ones are |
|
| 09:02 | I would say using the intuition about white fields because you want to construct |
|
| 09:11 | . But basically looking at the vector and you're sampling in the vector |
|
| 09:17 | And the third is things going back from the early space today is how |
|
| 09:24 | do integration Over the small interval from plus H. And looking at an |
|
| 09:33 | formulation of in their own ladies. now there is a serious methods. |
|
| 09:42 | There was serious expansion, you straightforward as 15 many times by |
|
| 09:47 | And depending on how many terrorists an in gold, you've got the method |
|
| 09:55 | some order and the order is determined the highest derivative. That could be |
|
| 10:00 | in the expansion. And then there some truncation follows. So I'll give |
|
| 10:07 | a couple of examples of using the series expansion. And then if the |
|
| 10:15 | is not used. But it's useful as always trying to understand how accurate |
|
| 10:22 | only have two lives. They're going go back use and trying to figure |
|
| 10:28 | the other methods farthest back up in of the taylor series expansion. So |
|
| 10:37 | hmm. Okay. The first thing I am equals man, just include |
|
| 10:41 | first derivative. So basically these three , which is what we did |
|
| 10:47 | Which is this obvious method. So what having a function that ex prime |
|
| 10:53 | given. That's the function itself. that's the first two terms is X |
|
| 10:58 | H of X crime. But X is given a F. So this |
|
| 11:02 | just following the slope of the green . That's what the prediction prediction would |
|
| 11:08 | then or what the function value X a small incremental same time from |
|
| 11:20 | So that's the only matter. And the facilities, gold and everything. |
|
| 11:24 | not I don't think anything to comment just there. And here's a little |
|
| 11:29 | of an example um uh the shortest of the Higgs. And as you |
|
| 11:37 | see, so you know that either access the function was delivered in |
|
| 11:42 | the function itself. So we know this is exponential functions. Um I |
|
| 11:50 | on this block here we have the is the actual be today. See |
|
| 11:59 | the other one is step by step the owner smith So why are the |
|
| 12:08 | always, you know, why So one thing we can see here |
|
| 12:29 | almost at the gap in the blue . The true solution and the productive |
|
| 12:38 | girls in the gap becomes quite real the volunteers as for why this |
|
| 12:48 | Yes. Is it bad? Ah one of the initial points it was |
|
| 12:55 | little bit under where the actual value . And because of the nature of |
|
| 13:00 | exponential equation, if it's a little lower than what the ultraviolet they're |
|
| 13:05 | That's correct. The tangent at any point, right? All of the |
|
| 13:11 | , the current trends up. But tangent is a straight line. |
|
| 13:14 | So if you progress from one point the next along the tangent, it |
|
| 13:18 | always come to be under the actual . And that then accumulates that you |
|
| 13:23 | going and then for the next one already start below and it gets |
|
| 13:29 | So but one other inheritance. Not great property or all the methods that |
|
| 13:39 | trying to tend to accumulate as you , overall. Mm hmm. China's |
|
| 13:44 | taxes practice the independent variable team of . Mm hmm. Yeah. So |
|
| 13:55 | in order so that was kind of . So that we talked about |
|
| 13:59 | So them just including the say first couple of terms usually doesn't give good |
|
| 14:06 | in savannah. The option is to better accuracy to use more terms than |
|
| 14:13 | mhm. And the thing that that that truncation error is much following |
|
| 14:20 | So yes, I'm doing it five example. Yes. But in this |
|
| 14:28 | it's almost like that feeling differential There's someone here. Yeah. What |
|
| 14:35 | use more terms in the taylor series in heaven. In this case we |
|
| 14:41 | through Potatoes years method have ordered four including 4th definitive as well. So |
|
| 14:49 | just starting out and taking the derivative X prime users, the second derivative |
|
| 14:55 | then from changing roulette sector so we all these expressions for the higher order |
|
| 15:03 | and then we plug it in and it is the taylor series expansion and |
|
| 15:08 | from there in this expression In this and then stopped here in the 4th |
|
| 15:15 | techniques areas and but I mean stepped of the things we need to have |
|
| 15:24 | expressions and if you're lucky like cheater it's an analytic expressions that we can |
|
| 15:31 | and the olympic television and this is enough to use my hands but if |
|
| 15:38 | a complicated function or we don't have nice and addictive expression. One that |
|
| 15:45 | to do it from America differentiation Before it's not kind of sensitive like |
|
| 15:51 | do something they want to avoid because that they have put on errors and |
|
| 15:58 | self but he was kind of the result here. It was just police |
|
| 16:03 | to charms and this is amusing. the approximation of the previous line. |
|
| 16:09 | then we get the value that is and and that's what they just to |
|
| 16:18 | that its approximation errors in the formula not the american position. That makes |
|
| 16:24 | difference is that higher order position They use cauliflower position. People are very |
|
| 16:32 | evaluation of this year's national workforce. fact that these two were called just |
|
| 16:38 | just trying to show that it The difference between these two are not |
|
| 16:44 | around though. In terms of the position it is the approximation area, |
|
| 16:50 | difference between Mr So it was just it's critical that we use for this |
|
| 17:00 | that. So they kind of wrote has this kind of order or nested |
|
| 17:07 | being and I think that's a minor . So the one thing, It's |
|
| 17:12 | of worth $20. What is the thing is this one? So what |
|
| 17:19 | did here to find all the instances of doing an instrument of both. |
|
| 17:28 | is O. T plus H which an infinite position is the same. |
|
| 17:33 | every time you get the round off then the round off and that's why |
|
| 17:39 | computing the new T is larger than using democracy complications. That gives you |
|
| 17:49 | the accumulation of error and figuring out the next issue. But that's one |
|
| 17:56 | . Think about the one that told more events accumulation. There are some |
|
| 18:02 | kind of activity from this star. right. Mm hmm. Yeah. |
|
| 18:15 | one thing that I guess I've been before about the taylor series expansion. |
|
| 18:23 | the local transportation error in a sense it's completed T plus H. It's |
|
| 18:30 | from the starting point. So basically local truncation error means you have an |
|
| 18:36 | . Um, in each step of algorithm will complete the two plus H |
|
| 18:41 | plus two. This one in every , in that case you have a |
|
| 18:46 | plantation error And then the greatest example was 4th order. So that means |
|
| 18:52 | local translation is only and then potentially you don't care the older and the |
|
| 19:00 | . The fact that there's no Alright. And then suggests. And |
|
| 19:09 | those are the approximation error in terms the expansion and as I mentioned the |
|
| 19:17 | then the results? Yeah. Right. Right. So here's the |
|
| 19:33 | line is well that means you need evaluate the function itself and the need |
|
| 19:39 | evaluate derivatives and if you have an depression, that's okay. But if |
|
| 19:44 | don't then numerical differentiation and that tends amplify. So that's why taylor series |
|
| 19:56 | . I'm not leaving used in solving function. So instead on the lights |
|
| 20:05 | methods that only the values for function and not effective. So that's what's |
|
| 20:15 | up next. Um And to me understand these methods, it's helpful to |
|
| 20:23 | of it in terms of this we on board I guess I've got a |
|
| 20:30 | . Mexico side. So um but so now yes, my friends are |
|
| 20:44 | on based on sampling this place and trying to figure out how and |
|
| 20:54 | Yes, these are known as methods mr came up with the initial idea |
|
| 21:02 | how to construct accurate higher order Just based on the value additional the |
|
| 21:13 | or the right hand side of the the question. So it's so I |
|
| 21:21 | to talk about how to do that it's a little bit or something. |
|
| 21:26 | wonder that it is an honor that's on the then we have the intuition |
|
| 21:33 | please for me. So let's show . Let's get down to the |
|
| 21:43 | Um So this is you then use function evaluations and depending how many you |
|
| 21:52 | for, how you use them the um explicit methods. Again unknown future |
|
| 22:01 | but I can give you very good . And the most popular put the |
|
| 22:08 | is what's the most 44 that means this covers up to the 4th order |
|
| 22:17 | in the expansion has a local transportation . Good morning. five Lights example |
|
| 22:27 | report. It's very simple. So one does is kind of a little |
|
| 22:35 | similar idea. House one being the integration in that case. Use the |
|
| 22:41 | of function values and then they try figure out the weight of the function |
|
| 22:46 | in order to get a polynomial corrected a certain order will pull him |
|
| 22:53 | And then there was a construction waited wait for function value, so |
|
| 23:00 | speak. So to me it's a bit of an idea was about to |
|
| 23:06 | and how we spent about constructed. and then they see a little bit |
|
| 23:11 | within language but it's kind of both favorite series function for digital things that |
|
| 23:19 | used for the fitness, serious methods then um we have function values in |
|
| 23:27 | uh back to space at zero. basically their functionalities and best families will |
|
| 23:35 | you the main here where they go little bit but it's saying the extract |
|
| 23:41 | direction and maybe it's a symbol in K direction and that means what that |
|
| 23:49 | . The right hand side here becomes function of two variables. And in |
|
| 23:53 | to figure out how to interpret whatever combination of function values used in terms |
|
| 24:03 | the taylor series function director from this . Um One needs to look at |
|
| 24:10 | take the serious expansion of the function and this. Well the domain between |
|
| 24:19 | and T plus H in terms of , whatever they arranged, they won't |
|
| 24:27 | things. But I'm just warning little things somewhat. I believe the expression |
|
| 24:35 | functionality. So so the first thing done too. Look at, |
|
| 24:45 | this series of attention. Um and the notation has been switched around a |
|
| 24:52 | times and then decide. So I . And and expertise that would |
|
| 24:59 | And this thing is general two variables and Y. And this is You |
|
| 25:06 | to identify one what's what the best them on your heart. And now |
|
| 25:13 | function of two variables. When you to material uh say the serious expansion |
|
| 25:18 | you have to treat it with respect both of the arguments for the function |
|
| 25:24 | irrespective something your age and something else . And then the increments of the |
|
| 25:31 | look unnatural voters to arguments of the And formally one tends to write it |
|
| 25:37 | way. Um, this is from unraveling a little bit conventions. So |
|
| 25:44 | , because of two arguments and its derivatives because the beach is not the |
|
| 25:50 | office backwards. Mhm. So I know if this is from a general |
|
| 25:59 | . Another courses you have a deep for accident with respect to y and |
|
| 26:05 | take a serious expansion. Then there a distance from moves and each one |
|
| 26:09 | the variables on this from moving in X direction and saying. And that's |
|
| 26:18 | of just a normal our thing. we had to take a serious expansion |
|
| 26:25 | factorial and the denominator. So when exponents of the Parenthesis zero than the |
|
| 26:39 | 1. So that the first term especially just the function itself like mm |
|
| 26:49 | . For the single, A normal variable. Then the next one. |
|
| 26:55 | that one. And then that's kind probably pretty much direct correspondence to the |
|
| 27:01 | variable that age times deliberative respect for thanks to gravity with respect to Y |
|
| 27:09 | what other functions and the next term is um, it's just a |
|
| 27:19 | you know, um, evaluation invested branch of the science itself and then |
|
| 27:27 | for higher orders. Okay, But in this case you got one |
|
| 27:33 | sustained squared times The 2nd Derivative Even the experts Operating on the 2nd |
|
| 27:42 | and then the same thing pointer And then that first product in the |
|
| 27:49 | way for me, dysphoria or all parenthesis. So when I'm on the |
|
| 27:57 | guy and someone was going to use these things. All these previous |
|
| 28:04 | just taking the biggest serious expansion now we have the terms would worry about |
|
| 28:09 | then there is the local transportation or this case is the term we do |
|
| 28:15 | think road in the expansion and then to make it look like flat. |
|
| 28:21 | hmm. Ah function and value is a song range between X and X |
|
| 28:29 | H and Y. And one. . Like it wasn't before and the |
|
| 28:33 | taste, it was between the extreme they've already won on the subject to |
|
| 28:40 | . Now it's kind of the square it's not to mention not directly allow |
|
| 28:49 | to. That thing's a little bit combat and write them down and transportation |
|
| 28:58 | don't know if the derivative or has respect the facts from the substrate. |
|
| 29:02 | something counselor. But the derivation both the ones and they're straightforward and footage |
|
| 29:21 | dr right down instead of using some sign the best affected first derivative |
|
| 29:31 | the first time was with respect excellent can respect why and then. |
|
| 29:39 | The second or the third term roller taylor series expansion. Now that was |
|
| 29:45 | has been huh? Okay. That's here just replacing these things with the |
|
| 30:00 | this one and then we can continue you want to go and cancel |
|
| 30:07 | But then as soon as its especially these two Only going one |
|
| 30:20 | This takes 10 0. So those drops out and this is zero. |
|
| 30:24 | those stopped, stopped up. Nothing Alright. So now we get to |
|
| 30:33 | question on the best notation. And I was trying to do the method |
|
| 30:42 | method the only two. And this we're here to show the machinery used |
|
| 30:51 | order to use construct an explicit method arbitrary order. But once you get |
|
| 31:02 | the second order becomes cumbersome. So is not really used much. This |
|
| 31:10 | what's while you use and also desire but in order to show that construction |
|
| 31:18 | to work. So again, this of the vector field in this case |
|
| 31:29 | the laws will get the distance in respective independent variable fee. And that |
|
| 31:36 | some range for the dependent variable. . So in this case for this |
|
| 31:47 | I'm gonna clip from that for the to we're going to use the function |
|
| 31:52 | uh where we are basically down here the corner um D and access the |
|
| 31:59 | points and you know, t the that's straightforward. And and then we're |
|
| 32:07 | going to use some other function value the domain between T and T plus |
|
| 32:13 | and X X plus in this I am thanks. So whatever they |
|
| 32:24 | things we don't know is where we to sample the function value. So |
|
| 32:30 | have an alpha here that's somewhere The and three plus H perfect and ex |
|
| 32:39 | and the X value is being picked this particular way. So now the |
|
| 32:44 | is to figure out what is alpha what should all find Barras, B |
|
| 32:50 | what should the weights fear be of primitive values and restoring point and wherever |
|
| 33:01 | sampling is. So we have four . We need to figure out the |
|
| 33:08 | of these two and where is this of And then so then just using |
|
| 33:17 | two expression and problem 18 this for , that's something that looks like that's |
|
| 33:25 | to show up on the next time um Alright, so here is the |
|
| 33:33 | and the bottom as assembly showed up the points on Tuesdays, thousands and |
|
| 33:38 | and then we have standard taylor series and I said well they don't really |
|
| 33:45 | to use this one but we're going use have to select the Ws and |
|
| 33:51 | outputs and bearers in order to match number of terms and this famous series |
|
| 33:59 | , the accuracy um in the morning then we're going to the other cover |
|
| 34:11 | . So so um and it is that is for the crystal method of |
|
| 34:20 | . Yeah. They have already shown the preliminary stuff for hard evidence. |
|
| 34:30 | the function since we need to have expression here. Right? So we |
|
| 34:34 | the function respected to arguments. And they need to take a serious |
|
| 34:39 | Function with respect to both the first the second argument in order to be |
|
| 34:43 | then to compare it with the Yeah you can have some of those |
|
| 34:52 | here and there correspondence from what they what they want and how they interpret |
|
| 34:59 | this. That was former doctor this was coming from and now using what |
|
| 35:06 | have so without um we have the with respect to the first argument. |
|
| 35:12 | that's this one for each time first . And then you have the |
|
| 35:21 | Thanks the derivative with respect to the argument. But when we get to |
|
| 35:26 | renovation, both respect property, Thanks. This is the question |
|
| 35:37 | We have a transportation. So now conduct trying to compare the blood rather |
|
| 35:48 | this expression in. That corresponds to part here appearance. So we have |
|
| 35:55 | . h. 72 H. Kindness expression here and they can plug it |
|
| 36:02 | in and clean things up. We Wh and then we have another |
|
| 36:09 | Coming from this expression here. So was W a few times there. |
|
| 36:13 | that's the kind of system here. this from the W two times engines |
|
| 36:21 | then we'll get the rest of the . This gives us now ah using |
|
| 36:30 | idea of everyone and put the method sampling things in this box from the |
|
| 36:36 | you want in a form which basically have the function values of T and |
|
| 36:46 | . Which is the same thing else the Fearlessness expansion. So now we |
|
| 36:51 | compare what is from the method dollars . You can compare directly when the |
|
| 36:59 | is expansion. So here is again from the previous line and here is |
|
| 37:07 | famous series expansion which you're using the we now have and then the ex |
|
| 37:15 | minister that follows here. Next double is everything Inspector the little things and |
|
| 37:23 | we have the second um which is expression and pointing it in. So |
|
| 37:34 | with the system Using it for the Series six function using the same location |
|
| 37:42 | the country. So now we want and 8 to match as far as |
|
| 37:49 | can. And that is going to us figure enough. Right? So |
|
| 37:55 | have here looking identification of terms in H times F. In the taylor |
|
| 38:00 | expansion. So the one journal could it. W one plus W two |
|
| 38:05 | to be one. So we know this is a constraint on how we |
|
| 38:10 | these two guys and then we have here for them in front of a |
|
| 38:17 | comes in property that means our car two needs to be passed on similar |
|
| 38:25 | the next turn. But this gives , so when you're choosing um these |
|
| 38:32 | here such that did you have to it? And that's an equivalent to |
|
| 38:38 | us in this method 42. If notice here, there's three questions therefore |
|
| 38:46 | . So they have one thing going freedom that we use um once we |
|
| 38:51 | about that. So one word is to choose um cool, it's W |
|
| 39:02 | they are equal and a half and it follows the and so on. |
|
| 39:09 | in this perhaps a in order to again looking at a serious expansion of |
|
| 39:20 | involved in the Iran contra formula and trying to match it to the taylor |
|
| 39:25 | functional XD plus eight. And that us the second order, I don't |
|
| 39:34 | . So that's the procedure. And you can choose to use um more |
|
| 39:40 | or more points in the where are rectangles Oregon between and plus um some |
|
| 39:52 | depending on how what So in that . So I'm gonna start with |
|
| 40:01 | the senior to drive to see that just did the function rather in the |
|
| 40:08 | was K one. And for the one we got this point there was |
|
| 40:12 | older method and therefore then looks at alphas and betas and the Ws turned |
|
| 40:21 | then one yards. The points where function value, mm hmm. Is |
|
| 40:28 | in the domain continues the extra and that's the slope of the curve. |
|
| 40:36 | that corresponds to a different starting point exploring that's coming and things. So |
|
| 40:41 | is there value or came move At point. And then if I had |
|
| 40:48 | choose in K2, one would end here. Mr And at this point |
|
| 40:56 | this Indonesia point, the result probably and then they're gonna cut down method |
|
| 41:02 | order to that is just average Those predictions. And at this point here |
|
| 41:10 | our prediction in this case for But then obviously loved it so much |
|
| 41:16 | prediction. The Either or yourself or other one. It shows in this |
|
| 41:22 | for example, in this space This is your skills and one they're not |
|
| 41:29 | , I'm just trying to help So this is just pointing out that |
|
| 41:36 | his arbitrary choice of people, 71 72, the half that want to |
|
| 41:44 | all of it from this basic sounds . 2/3 since then this time with |
|
| 41:48 | fear stop all. That's important. just shows the 3rd degrees of freedom |
|
| 41:58 | use. But there still was. , you talk about the thing that |
|
| 42:05 | is the dominating version I would I'm going with the methods more informal |
|
| 42:16 | there but they also he's on the . So any questions on that. |
|
| 42:31 | ? Okay, so book and I it's just quite cumbersome weight should |
|
| 42:43 | Um So the method of warding ford this structure is on a simple structure |
|
| 42:52 | vaccinated but for the moment to infused value where they are not. Then |
|
| 42:59 | also uses the functional again halfway along the access to half of them. |
|
| 43:11 | so they so any curve that goes that takes it to the point, |
|
| 43:19 | follow desperately tangent of X TNX. the point as well as on the |
|
| 43:27 | phone. And then we evaluated the in the morning to then we use |
|
| 43:37 | scope again and for half um same . But now they evaluated us following |
|
| 43:48 | standard against different derivatives and to follow derivative to the very end point and |
|
| 43:58 | gives you another um their ability or . And then you do within your |
|
| 44:05 | of these sports sample the views. But this wait where the two endpoints |
|
| 44:17 | about halfway. You have 25. then it turns out for this particular |
|
| 44:27 | if that's something that has the local area of And that's why it's kind |
|
| 44:33 | the 4th order methods of matches Um terms of the data, serious |
|
| 44:43 | Oh, so yes, wow ! . Yeah, because this was the |
|
| 44:54 | methods and following that, that was other one choose kind of a new |
|
| 44:59 | in terms of the interval. See hope so. And then once occur |
|
| 45:06 | almost up and then we get the at that point and then use that |
|
| 45:12 | to go. So for you to the third Um They're everything that there |
|
| 45:19 | actually three. That is extraordinary. the slope given by K two and |
|
| 45:27 | was mm hmm. 12 hours from And then. four function values. |
|
| 45:40 | what a wonderful thing. Mhm. these guys. And then I went |
|
| 45:48 | Giving the combination of these four So that is what they're Of course |
|
| 45:56 | slide is doing. They would just the point suggested government approximation foursome analytically |
|
| 46:03 | now 4 20 method given their their . Mm hmm. Yeah, that's |
|
| 46:16 | that's charlie's brother. Whether sampling forms in terms of food. So halfway |
|
| 46:23 | 10 points. Yeah. So the point and comes from your point. |
|
| 46:33 | those are limited. The space has Donald Trump More than four. Mm |
|
| 46:43 | . Um Right. So this is so can anyone no location matters having |
|
| 46:57 | order message. It's advantageous comparative. in primary something will happen. I |
|
| 47:08 | to start again. It's not totally to compare them directly in terms of |
|
| 47:19 | in the sense of in this case gonna put them happens before. It's |
|
| 47:23 | function evaluations and they're all your Only one. So it's obviously it's |
|
| 47:29 | work. In addition to the linear of us, we should have felt |
|
| 47:34 | that much work but supposed to use normalizes it for the same number of |
|
| 47:40 | evaluations instead of using age in order would take ages before. So that |
|
| 47:47 | the same number of function evaluations as filter method before the accuracy, Eventually |
|
| 47:56 | Manufacturing 16 whereas ages recently small The is you know, 1 16 verses |
|
| 48:05 | skilled and unskilled, a lot less one of his experience. It gives |
|
| 48:11 | much better efforts for the same amount work and that's for them. The |
|
| 48:19 | I want was used before june when look at what I really wanted approximations |
|
| 48:25 | it? That means you get more on this sport. Congratulations for |
|
| 48:38 | Any questions on that. Mhm This is just a good example for |
|
| 48:57 | what kind of so I want to about information from the fact that for |
|
| 49:16 | functions very rapid changes in some things and very still changes in some things |
|
| 49:23 | all. And then they wouldn't, makes sense to use in the tickets |
|
| 49:29 | wants adjusting. Mhm The size and true for all of us as well |
|
| 49:35 | the rapid change not to get largely very small step size, nothing changes |
|
| 49:48 | nor did they economize or work. the kind of some form of transportation |
|
| 50:02 | . So that is the combination of former running back on the airport portal |
|
| 50:11 | another twist on it. Um That's supported for the method and he has |
|
| 50:21 | fish older method and combine those two that's from two years to come up |
|
| 50:26 | something that is good to help So appearance. Another version of something |
|
| 50:39 | is also 4th on the method. , that in this case it's to |
|
| 50:44 | five functional evaluations all by itself. not perfect. That's interesting because now |
|
| 50:50 | is fire function evaluations these guys here we don't get any older methods. |
|
| 51:00 | , but it turns out that if one use this five function about |
|
| 51:06 | one more Then one. In fact 100 organized. And as you can |
|
| 51:14 | from these proficient here and how to things together. It's not entirely trivial |
|
| 51:19 | figure about irregularities now that we have patients you can actually and you can |
|
| 51:28 | it also in the sampling points here very traditional Children. So, |
|
| 51:37 | so now it's absurd. They can one more function evaluation using you previous |
|
| 51:46 | violence. Bye. That disorder method means something that has a local transportation |
|
| 51:57 | . Orange to the sex. How does the prediction of the next |
|
| 52:05 | for this now instead of from the most popular. So mm hmm. |
|
| 52:20 | those things you just flanagan will point of the center for example for this |
|
| 52:32 | a very important and different to the . That's right. I just tried |
|
| 52:39 | did somebody say something. Mm I am born the um, |
|
| 52:48 | Also Jonah. Therefore the So, what that can be used is what |
|
| 53:03 | use is um, I'm gonna cut four or 5 which is moving the |
|
| 53:09 | of before and Also is um, know is the 5th order. Six |
|
| 53:21 | values. And by giving both the and the 5th order blocking the estimates |
|
| 53:29 | error. And then like local error the exhibition and then continue stuff for |
|
| 53:38 | choosing what step size best to use it's fun. Um Sure. Hello |
|
| 53:54 | arrangements here's a gun. The function using these coefficients on this and point |
|
| 54:03 | that's the way they did it this example as well as bye. Um |
|
| 54:10 | an age. Listen Multiplying not for one but it has to be scanned |
|
| 54:17 | the step size stage that has been descriptions. Aah outside here and then |
|
| 54:25 | don't need the native side but then have to take care of it from |
|
| 54:32 | following how they get things the rest it in here. This train one |
|
| 54:37 | that this came of age and the um how far removed And the |
|
| 54:47 | Direction depends on age and that's not them not in front of that is |
|
| 54:54 | here and two um As one of so open. But what this four |
|
| 55:07 | method does is and differences. Um nice fact that that Spielberg version is |
|
| 55:15 | it uses the same Approed 1 2 In the 4th and the 5th quarter |
|
| 55:25 | some of those um I'm not these five conditions, I'm not using differently |
|
| 55:32 | year at long more points. So are actually the very same function. |
|
| 55:41 | open that is You know, those orders including because there's no change in |
|
| 55:50 | . The five points when you want have one more order backwards. |
|
| 55:57 | But if you look at the difference these two estimates based on the fourth |
|
| 56:02 | , the 5th order that gets an of the ever the next part and |
|
| 56:12 | we can use that. So if error is larger than the then you |
|
| 56:18 | redo it with a smaller step sons if there is smaller than you accept |
|
| 56:27 | larger area than what it's being done can increase the steps up. So |
|
| 56:34 | is something this is hardly used then error estimate that we get from using |
|
| 56:48 | both the 4th and the 6th The only additional work is uh for |
|
| 56:54 | both. The fourth and the fifth is you do one more function of |
|
| 57:01 | than yesterday to find a simple expression then that's usually not much of the |
|
| 57:08 | . Most of the work is in function evaluations but we're doing a little |
|
| 57:13 | more extra work and getting the four tom wong. So The only hard |
|
| 57:23 | is usually arguing six. There were doing this linear combination here at that |
|
| 57:30 | . The extra one they do is one, one function of our innocent |
|
| 57:35 | one and the rest of it is trivial in terms of work. So |
|
| 57:39 | get up eric estimates the very little work. In addition to that what |
|
| 57:53 | on has been done with increased series otherwise. But as always should have |
|
| 58:00 | grounds on how large texas and as step sites as well as the number |
|
| 58:05 | steps gone forever. So this is good example. I'll have to do |
|
| 58:21 | . It's a remarkable thing. Mm . Using the area to control the |
|
| 58:30 | . Have some comments on the Wow. So this subject. So |
|
| 58:48 | one so west David says expunging these or in terms of very popular explicit |
|
| 59:02 | from other stuff cover next. Those used very expensive. There's more explicit |
|
| 59:15 | I talked about. Okay. Yeah not talking about that woman or orders |
|
| 59:26 | or five. That's something the racially methods are supported. Yeah, I |
|
| 59:37 | have any but this is hold down workforce methods to song. Thank you |
|
| 59:57 | usually separation would approximation order that the is from you know these remember, |
|
| 60:14 | thanks their analysis And then also we talked about so again This one is |
|
| 60:33 | today's um coming back to it again terms of their analysis, remember this |
|
| 60:40 | finding is showing solution effects and all methods talking about don't um yeah, |
|
| 60:54 | in this case it's kind of exponentially and as we know that investments ah |
|
| 61:01 | changes in restoring point because of external . The difference between the cars grows |
|
| 61:08 | a goal and depending upon signing our error as in the argument that they |
|
| 61:15 | always to go and make a struggle growth on the Memphis unable to shoot |
|
| 61:20 | understood something. But it's still a in that area between whatever you want |
|
| 61:28 | the numerical solution growth. The next . That's a little bit Analysis on |
|
| 61:37 | points and this show on the 5th another negative exponents. I'm sorry. |
|
| 61:45 | they are referenced tickets. That's Huh? So there's a couple of |
|
| 61:54 | interested to figure out. I want son um convinced myself what if things |
|
| 62:05 | exponentially growing and the era of the technologies suspect. Yeah. What the |
|
| 62:14 | . That could be a small change time. This small chance gets |
|
| 62:20 | making this resort and what has been . I think outside it. Just |
|
| 62:26 | to prove it. And then on outside. Mhm. First this was |
|
| 62:34 | differential questions in the next final Pete for that local sensitivity to inspect |
|
| 62:42 | distance initial conditions. So 100 take driven it is a derivative of the |
|
| 62:52 | on the right hand side. And my comfortable currents here in terms of |
|
| 62:58 | hmm. For basic person and actually and they can send the order of |
|
| 63:05 | not. So that's the one get this case, the derivative since um |
|
| 63:15 | and in fact the derivative with Yes. That one has to protect |
|
| 63:19 | derivative with respect to X. and derivative as perspective. That's the first |
|
| 63:30 | respect to the 2nd organ and And then they riveted on the second |
|
| 63:34 | in respect to its dependence. S then we have also respect to the |
|
| 63:41 | argument in terms of and I was first garden offers the starting conditions tests |
|
| 63:53 | he does not in any way with necessary independence of this. No, |
|
| 63:58 | only thing we have to worry about transportation to make a simple one call |
|
| 64:09 | . Yeah. And one of the this part. Yes. For you |
|
| 64:19 | yeah everybody The movement is this one for teeth among your best of best |
|
| 64:26 | fortune and others exponential the question. from playing with this one when you |
|
| 64:33 | something in solution that looks like this the exponents of this financial is injured |
|
| 64:41 | this function to all the time. nonsense if then derivative. Yeah. |
|
| 64:55 | up to so this derivative here is then unfortunately. Yeah. So it's |
|
| 65:07 | . So Then it is bounded away zero. Continue to choose so they |
|
| 65:17 | Yes, that the relative respect of second argument this here is positive, |
|
| 65:24 | possibility and complex one more question. just if you're shopping for financials for |
|
| 65:44 | can take you the government will send no. So talked about three different |
|
| 65:56 | straightforward use of Tennessee's expansion, It evaluation of derivatives and you can choose |
|
| 66:02 | a number of affirmative one but not you for the accuracy that it needs |
|
| 66:08 | and more during christmas and if you an electric expression and will be |
|
| 66:15 | The then who is an american approximation ? It's not usually a good |
|
| 66:23 | counseling can profiles errors and they use interview staff one the space between where |
|
| 66:35 | are and the next point and I accusing the example points and the weighting |
|
| 66:41 | the southern points then you can go again. The same methods, expensive |
|
| 66:48 | that are whatever order they wanted The third version is too quiet tonight |
|
| 66:59 | the best going back to one of smartest guys. Yeah, next nobody's |
|
| 67:09 | look the solution local solution for the sons in time as using the current |
|
| 67:18 | plus the integral of the right and of the question. So it's been |
|
| 67:26 | . The holy method was best to when this was a constant. |
|
| 67:32 | so that's one way of looking at And the other one what's on the |
|
| 67:39 | side say well is that news? formula mr the unknown is also used |
|
| 67:50 | the right hand side. Someone formula actually solve the problem. So now |
|
| 68:00 | the baths and we looked at development methods, one approach was against a |
|
| 68:11 | sample. Um The function in the and tractors don't actually end points the |
|
| 68:18 | son role. Do we have points version and it points but the whole |
|
| 68:25 | was that within your combination of function . Ah that had a property and |
|
| 68:32 | upon how many functions that we choose is correctly integrated the instagram or if |
|
| 68:41 | was important polynomial of the 1st, and 3rd order we could design depending |
|
| 68:47 | how many sampling points to Maria Again, there's a bunch of function |
|
| 68:55 | besides coefficients for linear combination. That some nice properties in terms of the |
|
| 69:02 | of the next. So now we're to do the same except it was |
|
| 69:11 | different. So I wanted to do Let's get started, 1.3 different in |
|
| 69:22 | of solving the ease compared to this Doing integration of 20. So so |
|
| 69:32 | can be, it doesn't show in formula. So it's a little bit |
|
| 69:42 | talk to me story and start thinking . There is a difference between straight |
|
| 69:50 | Between two points. Yes. It's it's it's I mean it's not the |
|
| 70:04 | , the thing that is different and going to be used in this production |
|
| 70:09 | for solving all we need. And the one, this is basically |
|
| 70:17 | progression to start. That's saying something equals zero and then you want to |
|
| 70:24 | it all the time. So I them a point when you want to |
|
| 70:31 | the local staff, they have a maybe start at zero and evolved and |
|
| 70:40 | I'm going to go from an orbiter whatever time be whatever. 10, |
|
| 70:48 | 11. Well, you know you already evaluated things history so you have |
|
| 70:56 | history. So I want to do common ominous feeling. You can use |
|
| 71:02 | you already know what they call a of things in history as well as |
|
| 71:08 | kind of prevention future. So that's scenario compared to integration where your best |
|
| 71:17 | have A. And B. And don't have any trying to do that |
|
| 71:26 | of between and and then you choose ages in order to get sampling |
|
| 71:34 | But now we want to use things this integration will require continue to be |
|
| 71:42 | onward. Oh it is then what coming to say? So the difference |
|
| 71:51 | that at some point obviously not from start you have no history. Ah |
|
| 72:01 | some time we have so this quadratic something different to start them and you |
|
| 72:09 | no history one once you have His dreams from the states, the |
|
| 72:13 | states time T. Then we already evaluate what makes some he might be |
|
| 72:24 | the previous canyon spends over two back back etcetera for something once that. |
|
| 72:31 | in that case from the medication here want to do it where we |
|
| 72:39 | But then we're going to use it your combination or all prior function |
|
| 72:46 | Um two Yeah. Then he has make the process. Mm hmm. |
|
| 72:56 | is best for saying that please honor . Mhm. There's some F. |
|
| 73:02 | the corresponding values of X. So these are known as defining your |
|
| 73:11 | time team. That response I thought the that for gays were unknown and |
|
| 73:19 | just started then figure out how to the weighting of the figure. Thanks |
|
| 73:26 | three plus coach. And that's known this evidence that supports thanks. So |
|
| 73:39 | do you find them? The coefficient then we'll go back to so Not |
|
| 73:47 | do standard interval 0 1. That's many times generic. Nothing when that |
|
| 73:54 | to choose that changes things. But this case we are Until zero in |
|
| 74:02 | case or at this point but then have a bunch of prior Bank one |
|
| 74:08 | there was so that is the a of things from the most recent and |
|
| 74:15 | the back in terms of function by are introduced to approximate this for this |
|
| 74:28 | Amanda probably the integration formula. And order to get this method of the |
|
| 74:36 | order. The point was that you choose um the words properly then if |
|
| 74:45 | integrated up to the starting order, on how many points you choose, |
|
| 74:53 | that should be integrated itself. So next form when you're on the bottom |
|
| 74:58 | just using the memorials. So the so far is the binomial here Our |
|
| 75:05 | formula and up two and -1 if have deadline is one um or endpoints |
|
| 75:12 | the storm. So this is the good one over or um so it |
|
| 75:20 | means thanks Are the arguments are our from zero 2 1 -10 which is |
|
| 75:31 | is here is that this is there's here is simply this one will be |
|
| 75:39 | argument and that shouldn't be true for power Opportunity and runs 1st quarter. |
|
| 75:53 | right. So that was fun just get this our system of the |
|
| 75:58 | And while I don't know what the hand side of the integral is that |
|
| 76:01 | integrated is one of the art. we know but the the left hand |
|
| 76:07 | here is and then system of equations A. Is that function values from |
|
| 76:19 | expression of angiogenic and then it's an salt for seasoning. So this mr |
|
| 76:27 | council reservation or whatever they want to is known. There is no it's |
|
| 76:32 | and we can solve for C. the 4th order. So this is |
|
| 76:40 | of what this smoke them formula does some explicit method. They did not |
|
| 76:50 | the income of one used just the set of values zero and where we |
|
| 77:00 | and we can design decide whichever and on how many points you seriously. |
|
| 77:06 | . And then I was transformed medicine a silent order as well and as |
|
| 77:13 | as performance. So for what's the values should be. And that's when |
|
| 77:22 | time is up. So we'll continue this. So these are one |
|
| 77:27 | It was known as the victor character that are another set of popular methods |
|
| 77:34 | someone. So but the predictor character have an implicit component from that. |
|
| 77:44 | you use this to be a prediction then then they pretend that the prediction |
|
| 77:50 | is what it is. And then find an implicit method to that correction |
|
| 77:57 | next But of that's known as the then it was the director of I'm |
|
| 78:11 | talking about that. Thank you. you. Thank you. Yeah. |
|
| 78:24 | . Mm |
|
