| 00:02 | Mhm. Uh huh. I like walk around. Ah I like |
|
| 00:26 | All right. Yeah. So um group finding next and uh yeah, |
|
| 00:39 | like the last time. Don't talk matrix method. Um That and um |
|
| 00:50 | these aspects of Newton's method. So of it is currently a new |
|
| 00:56 | And then I would say very anti method is known as Sissi can't method |
|
| 01:02 | clear what the relation is, It's very different. And then that's just |
|
| 01:09 | out something on a six point that set apart actually also very frequently used |
|
| 01:16 | sort of difference in nature. Hang since the supreme being covered some of |
|
| 01:23 | simple route farming methods that are very to implement, but they're not on |
|
| 01:29 | efficient constitutionally. So that's why in mouth used except for simple examples where |
|
| 01:37 | may be quite well even though they're converging as fast as these other |
|
| 01:48 | So this is pretty much what my said. I think so. |
|
| 01:52 | This system to this method so stressed time that it's very important methods is |
|
| 02:05 | frequently used because of its uh rapid to finding roads when it does, |
|
| 02:15 | doesn't always find them and get Um The way the basic method requires |
|
| 02:23 | , you know, the function that called it on the sides. So |
|
| 02:28 | can manipulate the function and that's what used in order to buy cemented. |
|
| 02:38 | because especially it depends on you being to find the derivative of a |
|
| 02:46 | So that's what this is very conceptually simple method. Um So it's simply |
|
| 02:58 | to happen in this case. Function coming on this five, you can |
|
| 03:06 | the tangent to the curve at the where you have to be the diversity |
|
| 03:11 | following the tangent to the point that tangent process the X axis. And |
|
| 03:16 | is Pandora. Next guest for where rule is. And then you keep |
|
| 03:22 | the process 10 to Swan And get function value of X one, find |
|
| 03:29 | intelligent etc and there's some reasonable selection zero. It's a good chance that |
|
| 03:39 | will converge but it depends how you X0 and the functions uh nothing what |
|
| 03:48 | is foolproof. Um and then we'll later about the show that if things |
|
| 03:54 | convert then it converts history. So question is simply um back in the |
|
| 04:03 | figure, how far you need to in the X direction from where you |
|
| 04:07 | zero in order to make the let's the height, go to zero. |
|
| 04:16 | all of the straight line and here's distance to go. So you'll find |
|
| 04:21 | new X in this equation. Such while accent these cases, That's |
|
| 04:31 | Is basically the straight line equation for tangent. So, so that's pretty |
|
| 04:38 | it in terms of new transmitter. any questions on this concept and will |
|
| 04:44 | examples, Talk about pitfalls and all that. Um So conceptually is very |
|
| 05:00 | . So the officer said, if made everything this method, we're going |
|
| 05:07 | use it throughout. That's another way looking at the transplanted, look at |
|
| 05:12 | taylor series expansion of the function at at the point where you happen to |
|
| 05:18 | your first guest of the where the is. And if you do the |
|
| 05:22 | serious expansion they have the function and distance to move from the point of |
|
| 05:29 | age um following the first derivative. you do the table selected passion then |
|
| 05:37 | haven't secondary. But it um the of the distance of the mortgage divided |
|
| 05:45 | two factorial and then higher order So yes, the first couple of |
|
| 05:54 | in the data serious expansion and forget everything from its square. They're not |
|
| 06:00 | then it was to have this equation this exception, what the new things |
|
| 06:05 | it does. So it's equivalent to doing it, take a serious expansion |
|
| 06:12 | keeping the first two terms and then with that. So that's and we're |
|
| 06:21 | to use this formulation in order to to figure out how good this mr |
|
| 06:25 | is. And that's again when I potatoes here function will be used in |
|
| 06:31 | to try to understand the quality of methods. Mm hmm. So so |
|
| 06:45 | is simply what the method is and have an interest next event and defined |
|
| 06:50 | function value that to find the And then you move according to the |
|
| 06:55 | at of the function at X seven keep doing that. And if you |
|
| 07:01 | it well enough, there's a good . Thanks for the comfort. Just |
|
| 07:10 | stuff. There is no convergence. if the heads well enough that it |
|
| 07:18 | the dominating method, it was very to do is obviously it wouldn't be |
|
| 07:22 | popular as it is. So I'm trying to scare you off. I'm |
|
| 07:27 | trying to make where the things So and we try to stress a |
|
| 07:37 | bit in this book and as part its benefits that tried to related to |
|
| 07:42 | able to do things and like a of coal. So this is very |
|
| 07:48 | but it doesn't, we can see what is the simple ftp or for |
|
| 07:55 | of X. So you have to that you have that expression for it |
|
| 07:57 | evaluate it, then you get an value and have an american value of |
|
| 08:02 | function itself. So this is kind let's open the distance, you need |
|
| 08:08 | go and then falling from this since want to get to diplomats zero. |
|
| 08:15 | then it's a very simple um this new two methods for us. |
|
| 08:22 | not complicated and it just stresses the whenever you need to write some |
|
| 08:29 | the maps that holding this course is going to be extensive history sally field |
|
| 08:34 | support in general. So there's a of things I guess to pay attention |
|
| 08:40 | this code putting the slides. So here's for the one condition that's |
|
| 08:49 | long if that's this condition is satisfied then quickly. So an idea why |
|
| 09:01 | statement is there. But which? right, asking Winston, but which |
|
| 09:15 | ? Oh, that's fun. Never . Why does one want to if |
|
| 09:27 | derivative is kind of small? Is it because the following statement is |
|
| 09:35 | of X minus are divided by that the day. So it is very |
|
| 09:39 | . It's gonna Yes, sorry, . And shoot quote unquote, almost |
|
| 09:47 | by zero. That you should not um in a more conceptual by what |
|
| 09:52 | means. Again, this is So if the derivative is small, |
|
| 10:02 | what it means is this is the is the derivative. So it's small |
|
| 10:06 | . For instance, derivative is pretty parallel to the X axis. So |
|
| 10:10 | means you've got to move giant distance , technically it's just close to divide |
|
| 10:15 | zero. But it also means that method may not actually converge, generate |
|
| 10:23 | difficulty. Mhm. So that was then there is this other condition too |
|
| 10:33 | has it was as the distance, move the update on the X |
|
| 10:41 | If that's a small, then you that it's time to quit. So |
|
| 10:50 | a little bit of that makes sense . But it has to also be |
|
| 11:01 | conscientious or aware of that just because the update. The small does not |
|
| 11:08 | mean that they're close to because it says that yes you can keep on |
|
| 11:14 | this forever. But your estimate was changing. So it doesn't matter the |
|
| 11:19 | to the roof because you should you do the roof, you know the |
|
| 11:23 | , you know? And some of simple examples with you in the |
|
| 11:27 | we know what the route is. we can figure out what it |
|
| 11:31 | But in practice you're trying to find route because you don't know. And |
|
| 11:35 | only practical thing is then to start the updates this morning. So. |
|
| 11:43 | , so I guess it's an This is so this will be |
|
| 11:50 | right? So so it's an equation to find the rules. Right? |
|
| 11:56 | . So they put everything on one of the equal sign and put it |
|
| 11:59 | zero. And then to use the methods, we needed to find the |
|
| 12:06 | and this impala normally simple protective derivative special material squared minus two X times |
|
| 12:16 | . Tax four. So four X one. So if you move two |
|
| 12:21 | cubed squared plus three to the left side becomes minded. Stupid square. |
|
| 12:25 | that's this thing. So it's that simple. So now we have both |
|
| 12:30 | and F. Prime. And I'm take a starting point With a slight |
|
| 12:35 | three. So the issue of the , neutrons method is in the operating |
|
| 12:42 | . So we started plugging in That means x equals three in this |
|
| 12:50 | . So and then evaluate the it comes up to deny That point |
|
| 12:56 | then derivative turns out to the 16 x equals three and that means the |
|
| 13:03 | to go. Therefore that's prime the of X. There's this point |
|
| 13:10 | So that means that will now be new estimate and what the route is |
|
| 13:18 | , evaluate these things and you keep it step by step and as we |
|
| 13:25 | see that ah the steps that you get smaller and smaller. So at |
|
| 13:33 | point to this for french and see thought following the steps work of X |
|
| 13:41 | what X event is and see that increments just things that could be |
|
| 13:49 | very small. And the point was to try to find an X value |
|
| 13:55 | Makes the effort that's equal zero. they in some sense the error in |
|
| 14:02 | function value, we're close to the . So we can see here that |
|
| 14:08 | very rapid approximation of the estimate of and correspondent of the rules. But |
|
| 14:17 | will start the X values that the error in terms of the F value |
|
| 14:29 | down very quickly for this particular And so as I said, the |
|
| 14:36 | newton methods when it does converge, usually during the rapid and the one |
|
| 14:43 | to pay attention to in this case I did it in terms of the |
|
| 14:51 | as fear. So if it's not apparent for the first few days, |
|
| 14:58 | once we get serious for the third , you can see that basically that |
|
| 15:08 | diminishes very quickly. So basically if look at the experiment of the they |
|
| 15:15 | it basically ah doubles in America Let's get smaller about it. The |
|
| 15:24 | is 10 minus three squared 10 minus squared 10 minus 12. So this |
|
| 15:30 | what's known as the quadratic convergence of new respect. So the easiest |
|
| 15:37 | the by section and they've got a and it's sweet worse. And the |
|
| 15:44 | ones for essential have been here. by section was obvious. Right? |
|
| 15:50 | got the divided into solitude all the . So I think that's the |
|
| 15:56 | Yes, you have the area for iteration step in this case the area |
|
| 16:03 | done much more. Mhm. And over the home reason for the |
|
| 16:11 | subdued inspected finding votes to the Mhm. Okay. Okay. |
|
| 16:27 | so this was just a graphical administration this particular function. How is |
|
| 16:31 | seven difficult works. And it was pretty nice functions that wasn't too difficult |
|
| 16:38 | get good convergence and other examples coming . Mm hmm, correct. So |
|
| 16:49 | next thing. So yes. So was just one. And so how |
|
| 16:55 | you find the other rules? that's a bit tricky. So if |
|
| 17:00 | know, if you need this for to try to pick other starting points |
|
| 17:03 | for that starting point, the procedures . So it doesn't matter how many |
|
| 17:09 | you drive it, you need to a different guess as to what? |
|
| 17:17 | . What are the other rules? , the other routes happens to be |
|
| 17:24 | notes. There's nothing wrong with But how does Newton's method finally |
|
| 17:36 | So look here, Okay, this left hand expression and we have functions |
|
| 17:49 | there derivatives. So anyone sees the and finally comfort too. The problem |
|
| 18:03 | that if you, so again the value, both F and F prime |
|
| 18:08 | always evaluates to real values. So never going to move into the complex |
|
| 18:16 | . They have a real and historic . So in order to get values |
|
| 18:21 | routes in the complex plane, you to start the method also in the |
|
| 18:28 | . That's one thing pitfall and sometimes a deficiency or whichever restrict constant or |
|
| 18:37 | method that it never moves out of if it's a real equation, if |
|
| 18:40 | have a complex equation, but if have a real question then values will |
|
| 18:47 | continue to grow. So that's But so if you use software then |
|
| 18:56 | you should design. So they were all the roads. Yeah. If |
|
| 19:03 | would say you have a functioning like Hebrews were like the next word. |
|
| 19:14 | , can you put in measures that used? Yes, you can work |
|
| 19:21 | complex values. So that's getting an . But it's a find real story |
|
| 19:32 | that you can again, complex tools the nuclear battery but you need to |
|
| 19:40 | a way to ensure that to get the conflict. Okay, a little |
|
| 19:51 | discussion and I will not dwell too of it. But just it's not |
|
| 19:58 | complicated to convince you that in fact is quadratic convergence for these new through |
|
| 20:08 | exercise. I'm showing you how I convince myself it is a dynamic |
|
| 20:16 | It's a little bit. So the thing is basically to say, what |
|
| 20:19 | that actually mean? As a practical ? So the data convergence means that |
|
| 20:25 | is the route that this is the . So but as a convergence means |
|
| 20:32 | the new it's proportional to the square the previous. That's the notion of |
|
| 20:43 | . Yeah. They post on the of the square. So just a |
|
| 20:48 | example, Kind of talk to the and the big city and so on |
|
| 20:52 | assume that they're at some point just to the -K. The number is |
|
| 20:57 | kicking what you saw in this example after a few interactions. The function |
|
| 21:05 | that case that used the function value was 10 to the minus C. |
|
| 21:10 | then 10 to the minus six. so in that sense it was can |
|
| 21:16 | for example, in that case it all the rule that the area was |
|
| 21:24 | to the square of the previous Now the next thing is to do |
|
| 21:32 | reasoning of our Newton's method actually does potential convergence um by the structure of |
|
| 21:40 | method by example. So in this we have nothing error at the end |
|
| 21:47 | the first error versus how it's related that previous um it was square with |
|
| 21:56 | constant of professional Itty. And in case you can try to figure out |
|
| 22:01 | this constant is or put an upper on the constant for this. Doing |
|
| 22:08 | exercise will come back to using dictator . And that's so then I'll show |
|
| 22:16 | next where this thing comes from It comes from looking at this table |
|
| 22:22 | expansion. And in that case the time they didn't include and it took |
|
| 22:26 | change of serious expansion, suspected was ignored the term that was secondary. |
|
| 22:33 | was the function itself from the first . That that was it. So |
|
| 22:38 | is related to what was not included the the first term in the se |
|
| 22:45 | Sears expression everything. So now I'm to manipulate this expression attractive convention that |
|
| 22:55 | is true for this particular value on car. Alright. So here's test |
|
| 23:02 | the new next whatever in the next . And here is the new transfer |
|
| 23:09 | for you where we are at the situation of the aspect of the |
|
| 23:15 | Uh huh. And this now it's minute. Rewrite this a little |
|
| 23:21 | This Armanis extent is the current And then we have the double |
|
| 23:27 | What's this becomes the first. So the other terms in this expression for |
|
| 23:34 | error in them next iteration. So, um, now that's, |
|
| 23:45 | that. Then we're going to use take a serious expansion and the road |
|
| 23:51 | the current estimate plus the error in estimate. Um, so now we're |
|
| 24:00 | the s the taylor series expansion and he comes up. But that was |
|
| 24:05 | age in the previous slide. Some back and then the first time that |
|
| 24:11 | included in the Newton's formula South post . So now, then I'm always |
|
| 24:25 | , this came from the busiest, ? This is the spot. |
|
| 24:29 | the current in the new era is to the previous era and the functional |
|
| 24:35 | and yes, manipulated this investor make comment. Denominators have multiplied the country |
|
| 24:45 | you get this impression and the whole . It's the bicycle. And that's |
|
| 24:51 | order to get things to look a bit. So you never This expression |
|
| 24:57 | a sentence there. I think we're to use. So that's what I |
|
| 25:01 | saying. Then it turns up and can rewrite things Now. So vegetarian |
|
| 25:08 | one and the first piece here. if we look at the topic, |
|
| 25:15 | , we know that this expression is fact equal to minus this expression with |
|
| 25:20 | . You're supposed to get it to . So that means that this was |
|
| 25:25 | of moved in here. So now basically have an expression for how many |
|
| 25:34 | is related to the older. So is related to the square of the |
|
| 25:44 | error with the must apply being dysfunction here. That in itself does not |
|
| 25:55 | . So in principle, if they down this Anthony, did you know |
|
| 26:00 | it's proportional to this forever from the interruption? Not too hard to basically |
|
| 26:08 | myself that businessman as cordelia convergence and little bit more trying to figure out |
|
| 26:18 | bad can this get. Uh That is uh this this as |
|
| 26:28 | Can this spread? Scientists said they come previously said quadratic convergence. There |
|
| 26:36 | a new era is proportional to This constant depending on delta here and |
|
| 26:43 | in previous square later. So the to let this constant. And this |
|
| 26:51 | a question which will put the opposite him to remember. Mm hmm. |
|
| 26:58 | . Mhm. Well, basically here you are right. So we tried |
|
| 27:01 | get an estimate for that. And by and here we have the |
|
| 27:08 | So that means the error in its no worse than the maximum of this |
|
| 27:16 | the minimum of the denominator. So is simply just um better than you |
|
| 27:25 | basically abound for how the convergences. because of the assumed that the function |
|
| 27:32 | is well behaved, they know that derivative is not going to go through |
|
| 27:38 | roof and go crazy. But it means that depending upon her dramatics. |
|
| 27:45 | function based the value of this thing on it, wow ! Various sort |
|
| 27:52 | high temperature. Mm hmm. Second disturb batory first derivative is a slope |
|
| 27:58 | other one so quickly. So the depends on that. But it is |
|
| 28:06 | of the reviews. And I want to stay on this phone. So |
|
| 28:20 | just manipulating it a little bit Here's what the question you had The |
|
| 28:26 | one and for convenience also try to there is this pressure was Down to |
|
| 28:39 | times the sea of that. This order to get something that take this |
|
| 28:47 | . So if we have, put together the city of was one |
|
| 28:57 | the at the end then they got delta, delta. And then you |
|
| 29:05 | just see that it eventually converges because this quantity is less than once or |
|
| 29:14 | number of power. Is that It does converge. It is |
|
| 29:24 | Okay, I think that was. then some concrete examples. So, |
|
| 29:35 | Newton's my opinion. Um, is way of doing reciprocal zor division if |
|
| 29:44 | like. Um, division is kind hard things to do compared to add |
|
| 29:55 | multipliers. So yeah, man, think too much about it red |
|
| 30:04 | But if there are some of them then they have to take them out |
|
| 30:08 | be commissioned. Um and as well square root and some other functions. |
|
| 30:16 | , uh huh. And if you up working in particular groups and companies |
|
| 30:22 | Bill Barber and I had the darn to new reciprocal square roots and trig |
|
| 30:30 | . Some of these things that comes most programming. So one thing is |
|
| 30:36 | the formatted something such a solution is to their reciprocal that they want. |
|
| 30:43 | in order to evaluates the reciprocal of or one of the T. You |
|
| 30:49 | form the function that is one of minus D. Because yes, if |
|
| 30:55 | find a route to that equation, means that X ends up being the |
|
| 31:03 | apartment please. So that's and so just do the usual thing and find |
|
| 31:10 | derivative neurotransmitter. So here's the function . This is just the question we |
|
| 31:18 | . So this is basically what you to do and you can simplify this |
|
| 31:23 | and that's what I based on the hand side. And then so it |
|
| 31:31 | so most harvard out there have making multiple areas are relatively simple compared to |
|
| 31:39 | , the division unit. So this two shows that you can Deal with |
|
| 31:44 | occupy and as subtract unit, one supply. Um And what subtracting another |
|
| 31:52 | soldiers one way that you can use existing things and then you basically keep |
|
| 31:59 | it and you got. Um And one manipulation in order to these are |
|
| 32:09 | it's also significant. So they kind try to get numbers not too |
|
| 32:16 | So you don't lose them. And did this in your questions but this |
|
| 32:20 | one way in which you can do and as a practical matter. Since |
|
| 32:27 | , um I'm talking about implementation issues it's the computer science class. The |
|
| 32:35 | what one does is we'll have a look up to get the first to |
|
| 32:41 | or several bits and then return it to get the rest of it and |
|
| 32:46 | of the pandemic convergence, can I take one or two restorations after they |
|
| 32:52 | a look up to very high Mhm. Any questions? Yes. |
|
| 33:08 | . Square root as it was another . Yes. So there is substance |
|
| 33:12 | of the square roots. Find any such that the solution to the question |
|
| 33:16 | the square root of the variable. . In this case that they're trying |
|
| 33:21 | find and we have the expression again terms of the function itself and its |
|
| 33:27 | and get a new expression and they keep going and so I have convergence |
|
| 33:33 | we basically have left hand side equals the right hand side. This is |
|
| 33:38 | you get the convergence. So and you can manipulate these expressions of what |
|
| 33:43 | question means. This perfect start the that you started with divided by the |
|
| 33:51 | value two X. The convergence value the square root of the best. |
|
| 34:05 | is a way also which you can . Thank you. Okay, square |
|
| 34:15 | , it's over there. I mean definitely has some references again in part |
|
| 34:23 | these examples up when this is inverse square roots. It's another |
|
| 34:28 | It's similar thing. Yeah. But reason I know this, I worked |
|
| 34:34 | the computer industry and have to do effectively to do things sufficient. Any |
|
| 34:45 | on those things. But it's Well, of course one has to |
|
| 34:49 | inventors are suitable function tools, good . It is that you know exactly |
|
| 34:55 | you want about it to be that looking for. Mhm, mm |
|
| 35:02 | Okay, practical things. And if ever have to do it. AH |
|
| 35:10 | more example apparently. Okay. So have a new question and we'll keep |
|
| 35:16 | up the evidence and the value of function uh starting at externally correspondent it |
|
| 35:32 | . You have to do estimates. follow up nothing magic. Uh Exactly |
|
| 35:42 | the first basically To rebuild this The function value is one and the |
|
| 35:48 | visits. It's fury against sort of is too. So If the 1 |
|
| 35:57 | is a half half and then we the next one, plug it in |
|
| 36:04 | get the new sets of values and just keep going. So nothing magic |
|
| 36:15 | this function is is not quite as as only observes and it's nice crossing |
|
| 36:21 | X axis. So if they don't the hatred somewhere here, It started |
|
| 36:28 | one that was the first guest X is one. The X one ended |
|
| 36:34 | feeling it has and then next to there have been three. So it |
|
| 36:44 | from one path 23 And then I back to the 1.4 so one can |
|
| 36:52 | that difficult work started 1/2 the number , 2, 3. And then |
|
| 36:59 | went back to something at one point it really was and a little bit |
|
| 37:04 | to where we started. But there's quite a ways from the actual zero |
|
| 37:14 | . Mm hmm. So okay, all. Ah the test. So |
|
| 37:25 | did you convert? Well then run to the right. I'm not kidding |
|
| 37:32 | to find Yeah, blessing and it right. No, no. All |
|
| 37:48 | . Let's see. So here is point and I will see and then |
|
| 37:56 | something I guess at 1.4 um kip's of not looking tips for isolating compared |
|
| 38:08 | where we started. But it turns big time and that's fine. But |
|
| 38:16 | just an illustration of the convergence, necessarily mon atomic, that eric gets |
|
| 38:22 | every step of the way but eventually it depends on the function practically |
|
| 38:31 | It does converge. Mr Amberson was second derivative divided by the first derivative |
|
| 38:38 | your c pathetic conversions. So depending the function, it's not always as |
|
| 38:45 | and easy. So what is And also again, so these were |
|
| 38:56 | , there's a real but the question it's real starting point does not touch |
|
| 39:03 | other. Okay, yeah, you want to say about convergences too. |
|
| 39:10 | gave this example here. So, , yeah, I think the text |
|
| 39:19 | the slide is pretty obvious what I . So here's just a couple of |
|
| 39:23 | worth what things can happen. So this case it's the case that the |
|
| 39:29 | would not converge because it never gets the left side of the big because |
|
| 39:36 | river, they were almost pointed from thing. They will not find they |
|
| 39:47 | because the starting about relative to where root is turned out not to be |
|
| 39:53 | good. There are other kinds of are this that correspondent this little code |
|
| 40:01 | ? Is that the tangents are If exactly zero then well, you can't |
|
| 40:07 | America do it. But it also that uh, okay. And this |
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| 40:16 | kind of pathological case but that could happen. Uh, you got the |
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| 40:22 | and one point where you are. maybe you started here and you've got |
|
| 40:26 | derivative, we'll go follow the sierra , you get function value and then |
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| 40:31 | happens to be a derivative actually brings back to where you started. So |
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| 40:36 | get kind of the cycle that principles last forever and practice that not necessarily |
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| 40:44 | likely to happen because in America So defendants are not necessarily to bring |
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| 40:50 | back to exactly what it is could . But not that So it depends |
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| 40:59 | on their function. I think we end up here. Mm hmm. |
|
| 41:07 | . So then the other case in of uh things to be aware of |
|
| 41:16 | terms of Newton's method as that may the case that not only is say |
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| 41:25 | first two limit zero at the but it could also be that the |
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| 41:29 | derivative is also zero for more than . In that case, things may |
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| 41:36 | behave as well, but when it to know that there is several of |
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| 41:43 | derivatives of the function, including the itself 10 at the same X value |
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| 41:51 | and modified. And Newton's method to they better. So we'll know the |
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| 41:56 | or how many of the function F its own derivatives are zero at the |
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| 42:02 | of F. There. Put the . M for the number of uh |
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| 42:10 | are just a number of function derivatives the function Chancellor. So here's a |
|
| 42:18 | of examples of functions that have a of more than one. That means |
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| 42:27 | for this one, both the function . Uh it's uh the river do |
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| 42:36 | there. But the roads are I guess this one. So that's |
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| 42:44 | that that one is the rule to the derivative and get to expand its |
|
| 42:48 | . So the derivative is also um . Some of the, what we |
|
| 42:57 | multiplicity too because the root chemistry negative zero. I have this in the |
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| 43:02 | one. And for this other function has to be not only that function |
|
| 43:08 | also 1st and 2nd derivative that constantly . Ah They're also and then they |
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| 43:16 | them some sort of guards and So to your area around the actual |
|
| 43:23 | that tells a little bit tough It is to Randall for uncertainty |
|
| 43:34 | So next is best generalization or when transfer to systems of equations and questions |
|
| 43:46 | far the transfer consecutively simple are my and in most cases decent case and |
|
| 43:59 | . It's converted very rapidly. It on the D circle yes on the |
|
| 44:05 | but it's also true. So the methods we talked about last time perception |
|
| 44:14 | there first guess is don't bracket Yeah have the same function value. They |
|
| 44:22 | be drafted the route but then the fails because um So then you have |
|
| 44:26 | make new guest system. Try to out If there is a zero |
|
| 44:32 | But if the function values are the at both endpoints, even if the |
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| 44:40 | itself have zero crossing between them, have to say before and even number |
|
| 44:46 | the message will have. So That's a certain amount of blocking starting |
|
| 44:55 | . It simply wants as well as right. So if they have we |
|
| 45:05 | a system of equations. But their because we're trying to find routes on |
|
| 45:12 | linear equations are supposed to linear equations reduced the calcium elimination. I think |
|
| 45:18 | in this case we have non linear If we have for the previous example |
|
| 45:25 | just one function that was pulling all in all the cases that I showed |
|
| 45:30 | . Um But so now we go dinner. So we have a bunch |
|
| 45:35 | equations that describe different aspects of a and they depend on a number of |
|
| 45:43 | variables. Yeah. Show your formula very simple to generalize and then a |
|
| 45:52 | bit means. So now we have of a vector valued function in the |
|
| 46:03 | that you know, they respond to family and what I mean, factor |
|
| 46:07 | their function. Yeah. Our dependence the collection of independent variables X. |
|
| 46:17 | evaluating its straightforward but then they also to find the derivatives because they had |
|
| 46:24 | . Prime and reconstructed and that becomes little bit more involved. So 1st |
|
| 46:33 | show you yes, formally what it like. And then I'll get example |
|
| 46:40 | concrete. The formula has has said not have effective value function F. |
|
| 46:46 | houses effect equations for the different We have independent variables that are also |
|
| 46:56 | given the vector notation include all the components. So this once again, |
|
| 47:02 | we needed the derivative and um we by the derivative. So and gets |
|
| 47:10 | . And formally then we just write down the derivative of inspector valued function |
|
| 47:17 | respect to all its arguments and its . And this formally looks like this |
|
| 47:22 | the functions. So now this is matrix and this matrix is known as |
|
| 47:29 | utopian. Ah Yes. Anyone ever the expression Jacoby on the board is |
|
| 47:36 | context. Yeah. Isn't it typically for training change of variables to change |
|
| 47:44 | into different the sensitivities. That's So what do you think of it |
|
| 47:53 | two D. Um So you have function describing the surface and then the |
|
| 48:02 | in that case tells you then again point the tangents and in the |
|
| 48:07 | Direction and the tangent of our direction two independent variables X and Y are |
|
| 48:13 | function describes the servants. And I'll complete examples to make it tangible. |
|
| 48:23 | normally the kind of Yes, it's my mother. You actually do |
|
| 48:26 | Well let me show you and some how you find the Jacoby in or |
|
| 48:31 | matrix that is something from taking the of each one other components of the |
|
| 48:41 | valued function F. Um But then can again do that based on the |
|
| 48:46 | of the taylor series expansion for the two terms thomas. The way we |
|
| 48:52 | Newton's method in a different way. For just a single function. And |
|
| 49:00 | what is it? What are they ? 2, 0. Like we |
|
| 49:04 | before. And then they found out step that is not effective valued |
|
| 49:09 | There's simply failure equation kids. Um one. said that I'll show you |
|
| 49:22 | out to show you an example if called me but I said when I |
|
| 49:29 | about Gaussian elimination, Yes. You to solve it in your system |
|
| 49:34 | A. A C equals B. . Normally you can write that's equals |
|
| 49:39 | inverse of that you can speak. he also said rarely would you like |
|
| 49:43 | be explicitly form the inverse? So reduced cost determination to solve and find |
|
| 49:52 | . So it's the same here. don't try to form the inverse of |
|
| 49:56 | A cobia and major sports applying to the entity that tells you how much |
|
| 50:02 | , update uh the vector X. that this is a matrix. And |
|
| 50:09 | it's a vector valued function, so is the vector sum. The vector |
|
| 50:13 | you a vector so that tells you much each of the components of X |
|
| 50:17 | supposed to be updated to get to next district. So unless you solve |
|
| 50:24 | then and that gives you your which is how much you want to |
|
| 50:30 | . Acts to get them next Uh huh. The rules to the |
|
| 50:40 | man. Mhm. So, I the next test to the concrete examples |
|
| 50:46 | believe. So um it's just for equations and three unknowns. Let's see |
|
| 50:58 | Okay, that's much more concrete. there should be a strike to |
|
| 51:02 | So yes, this nominally and then so again, slow here. This |
|
| 51:08 | about its function F depending on the variables. So now perform the |
|
| 51:16 | One needs to take the derivative of one of these functions with respect to |
|
| 51:19 | one of its independence variables somewhere based . Ah When you take the derivative |
|
| 51:26 | F. One with respect to each of these variables, you're going to |
|
| 51:30 | three entities and you get three entries each one of These roles in the |
|
| 51:35 | . So you get the three x matrix in case. But in this |
|
| 51:40 | that is mm hmm. So against is formula taking the partial derivative with |
|
| 51:47 | to each one of the arguments. then look at your three x 3 |
|
| 51:51 | . That is your Kobe. And know, I guess the complete example |
|
| 51:59 | is a concrete example. Then stop questions. So here's an example then |
|
| 52:06 | this is also non immigration in this right squares against financial functions and products |
|
| 52:13 | variable friendly questions. So in order follow this procedure, not using mutant |
|
| 52:21 | , we need to take the derivatives each one of these equations respectful All |
|
| 52:26 | three different independent variables. Hopefully on next slide was it's done. So |
|
| 52:37 | such as a. Mhm. we have here the function F then |
|
| 52:42 | as vector valued function goals. And that the derivative with respect to X |
|
| 52:50 | , X 2 and X three Derivative X one is 1 X two is |
|
| 52:55 | and X three is 1. This I take the derivative with respect to |
|
| 52:59 | one that's two X one derivative with to its due to extreme heat to |
|
| 53:04 | and such And then this one derivative into the ecstasy to the X |
|
| 53:10 | So that doesn't change Derivatives are expected.x For the 2nd 1 is X two |
|
| 53:20 | with respect to X one again is three. So this is now The |
|
| 53:26 | with respect to their worry about X They have to respect to X two |
|
| 53:31 | just a single mobile X one same with extra. So that's for the |
|
| 53:36 | derivatives with respect to each one of independent variables. Four. I don't |
|
| 53:43 | so jay is the program which is the derivative of the function f Respect |
|
| 53:51 | the three independent. And then the was to find um we don't do |
|
| 53:59 | universe but that's the issues universe of this salute update but we do it |
|
| 54:07 | solving this system of oppression that gives an edge and then they find that's |
|
| 54:24 | . So any questions on that. yes gets kind of more complicated. |
|
| 54:34 | the principle is thinking let's go. really? So yeah, so I |
|
| 54:48 | I'm depending upon starting points and get solution. So and this next |
|
| 54:55 | I'm so and I guess this is much summary what I said is called |
|
| 55:03 | simple iteration formula and it proved that was basically because I had a convergence |
|
| 55:10 | the next time your roads they can the convergence by using the fact and |
|
| 55:20 | , baseballs um vector valued or system motion. So I was talking about |
|
| 55:32 | second method that is really simple variation . Okay, so the simple variation |
|
| 55:47 | simply that yeah, you don't have way of all the previous examples that |
|
| 56:00 | complete. They had a simple analytic for the function and we could symbolically |
|
| 56:08 | politically find an expression for F. . Now sometimes that's not necessarily |
|
| 56:17 | You can still have an expression for . But it may be difficult to |
|
| 56:22 | an analytic expression for the derivative that's . So then you need to find |
|
| 56:28 | way of America then try to approximate derivative. That's what they say it |
|
| 56:35 | . It's best they think of again series expression for that crime or for |
|
| 56:43 | of X. Take the first two between the nutrient iteration and basically that |
|
| 56:49 | approximation of the derivative is simply the of the line between please do exercise |
|
| 56:57 | sentence. Just shut up traffic on next. So that was. And |
|
| 57:04 | the second method instead of if you have the way of directly evaluating it |
|
| 57:10 | an olympic expression then you can numerically it as the derivative of the curve |
|
| 57:19 | at the point. Thanks So this what's being done is basically you take |
|
| 57:25 | successive illiterates and you look at the values of those two successive hatreds and |
|
| 57:33 | find the slope based on those to of the rules and the corresponding function |
|
| 57:39 | . And that's what you use as approximation of derivative here. So it's |
|
| 57:44 | of in this case kind of lagging bit in terms of how it changes |
|
| 57:49 | along. So this is basically the of X ray and then the derivative |
|
| 57:57 | to be a denominator. So now basically have the difference divided by the |
|
| 58:01 | in the exhibition. So this expression should take more of these guys |
|
| 58:08 | Then it's an estimate on the slope the line for second finest simply the |
|
| 58:17 | between Newton's method and the second person that instead of using an expression for |
|
| 58:24 | derivative that we can evaluate you form estimate of the standards from the kind |
|
| 58:34 | the previous system. But that means convergence is not quite as good but |
|
| 58:41 | close as well as they need. true. What definitely. So there's |
|
| 58:59 | just text inside and the same thing a piece of soup pickles figure out |
|
| 59:07 | how it works I think. Ah I think it's another example then to |
|
| 59:16 | the convergence and at the beginning it's little bit slow. Like it can |
|
| 59:22 | within Britain's method but then we start pick up and most of been quite |
|
| 59:29 | extremely has illustration I have in motivating expected right? And I'm not going |
|
| 59:42 | with him detail about. It turns that they have a correspondent things that |
|
| 59:49 | had in the business method, The is because now um they're using also |
|
| 60:00 | points and testimony in the roots. instead of the square of the current |
|
| 60:06 | , you have the product of a and current theater to get an estimate |
|
| 60:13 | what the next area. So in sense it's not quite the square Since |
|
| 60:18 | a product of two previous, it's quite as rapid but it's close. |
|
| 60:25 | it's also very good. Nothing than job. Have a way of directly |
|
| 60:32 | . But there were mistakes. Mm . Then I not through required in |
|
| 60:45 | but I have a couple of slides showing try to commence to say |
|
| 60:49 | In fact this is true. It's same depression. So this was the |
|
| 60:56 | pattern. This was the correction are on the same account of the slope |
|
| 61:00 | the line between two successful estimates and this expression of it. And then |
|
| 61:08 | up the aerobatics, the road minus estimate. And the Minister Carmen is |
|
| 61:16 | expression of here for X and plus plug it in and manipulated. That's |
|
| 61:22 | that looks like this. And then can rewrite that expression to make it |
|
| 61:28 | to what the market wants. um, so this will have a |
|
| 61:37 | up here. And then I used . So this is kind of the |
|
| 61:42 | second expression the inverse of the So the two last points. And then |
|
| 61:50 | of yes, I can pull this out and then when I get this |
|
| 61:58 | this was AM -1 deployed like to these things in And that means what |
|
| 62:05 | get left from party and It's So you got the N -1. |
|
| 62:09 | it's very funny. This is just these expressions for convenience. So if |
|
| 62:15 | can put some balance on these two we have once we wanted something proportional |
|
| 62:21 | the product of the current is released the next slide is just going through |
|
| 62:29 | exercise and playing with this expression and again favorite, serious expression. And |
|
| 62:38 | pretty much similar to what I need Newton's method. And in the end |
|
| 62:43 | put the band on this funny expression I generated that stuff just related to |
|
| 62:49 | second derivative affection. So. Mm . Absolutely. So that was pretty |
|
| 63:00 | I think that in terms of the method that is almost as good. |
|
| 63:05 | even if you don't our position to the derivative analytically the computation as long |
|
| 63:13 | we have while you're evaluating the Mm hmm. You know, some |
|
| 63:24 | of expressions here. I'm not mentally so you can find them. They're |
|
| 63:34 | high rapidly. It actually converges with constant confront. Um but the super |
|
| 63:47 | is not quite quite direct but but was proven on the previous side but |
|
| 63:50 | not Turns out to be this expression . Um, so one man is |
|
| 63:56 | with the five divided by. But turns out not, but it's certainly |
|
| 64:01 | than linear light. The dissection, experts. Any question a second method |
|
| 64:13 | one more I'm going to quickly Oh, okay. So the fixed |
|
| 64:22 | method is and in a different way thinking about that. Um so it's |
|
| 64:33 | know, I want to live in and I talked about um you |
|
| 64:39 | your reciprocal zor square rules. So of finding a function root for F |
|
| 64:48 | equals zero, they tried to find solution to an equation of this |
|
| 64:56 | So it's known as a fixed point because it should plug in the right |
|
| 65:02 | of X. And then to get value of X factor. So that |
|
| 65:08 | it's a fixed point. You'll never out of it. So I have |
|
| 65:11 | solution um for this equation no matter many times you try to value cable |
|
| 65:17 | she always gets the X factor and an option 6.9. And then I |
|
| 65:24 | couple of examples of this just trying show them in this case. Productions |
|
| 65:33 | left hand side of the equation and store thing. It's the right hand |
|
| 65:39 | and they should go around to the of in a way subset converges implementing |
|
| 65:45 | points where you have the solution you it. So then it's a bit |
|
| 65:54 | creativeness and taking there's function there for that they got to find a solution |
|
| 66:02 | come after the another equation D. X. So I start when you |
|
| 66:08 | the fixed point to this equation, in fact have the extra the solution |
|
| 66:12 | definitely. So it's kind of a step. But sometimes these represent They |
|
| 66:20 | better than your strategy approach for x zero. Okay. And so here |
|
| 66:28 | is. Um huh? In fact the work starting with G. Of |
|
| 66:39 | . And then trying to figure out our correspondent function is. But normally |
|
| 66:44 | I start with this and I'm trying figure out what G. Two use |
|
| 66:49 | order to find a solution to but it's not too hard to see |
|
| 66:55 | The party of actually equals two Then what you get is in fact |
|
| 67:03 | ah you can sort of multiply That's so this has put this equal |
|
| 67:08 | X. Multiply the X. Then get the next square turn on the |
|
| 67:13 | what on the other side? On equal side? And they have to |
|
| 67:17 | you have Exxon in fact and she them. Um So that means that |
|
| 67:24 | X. That satisfies that this is to access. In fact also satisfying |
|
| 67:30 | -X -2. Again, that expert on the other side. Yes. |
|
| 67:36 | this is mm hmm the solution to equation equals X. Pfft for |
|
| 67:43 | It's the same value what we So now one can sort of indirect |
|
| 67:50 | this, which I think it's the . So here is all of this |
|
| 67:57 | down just warming up. Zero Is supposed to be prophetic. Central |
|
| 68:01 | X. So then you have the interest to find a new threat and |
|
| 68:07 | keep working on it. And eventually , you got to Something is close |
|
| 68:13 | two and we can see that actually you a solution to this equation two |
|
| 68:19 | is 4 -2 -2. So that's . So through it is a |
|
| 68:25 | This was a different way of finding . The fixed point iteration and in |
|
| 68:33 | case you see the firm but it the convergence. Mm hmm. And |
|
| 68:42 | need to find another as well. this case different starting points. So |
|
| 68:52 | This is pretty much what I So yes, it continues it. |
|
| 68:57 | method is this is a good It's called convergence. It's the second |
|
| 69:03 | is the second method, but six methods may also be quite good, |
|
| 69:10 | something you need to be creative and figuring out what functions G to |
|
| 69:16 | Two. Yeah, the same solution to the function. That's when I |
|
| 69:22 | here. Yeah. So, Okay, yes. So this starting |
|
| 69:37 | is always a tricky thing. And you don't have any in traditional knowledge |
|
| 69:46 | the function, I guess a good point. So there are still issues |
|
| 70:00 | it turns out I guess in practice know enough for function stuff. Thank |
|
| 70:08 | rely on this method. And normally , the second or the fixed point |
|
| 70:13 | to solve systems along in their Mm hmm. I don't think there's |
|
| 70:23 | much more that I haven't said Mhm. So what? Yeah. |
|
| 70:37 | any questions of myself. That's what am today. And then of course |
|
| 70:43 | Canada, I can't Yeah. Combined to you don't need here. |
|
| 70:50 | they might start this, I'm And by section idea and then switch |
|
| 70:55 | newtons to get traffic convergence towards the as a way of trying to find |
|
| 71:03 | starting point. So, but I have any advices. We'll have to |
|
| 71:13 | stomach points. Worry about that. once you have something this is |
|
| 71:22 | And also submit this method, we definitely find it most packages after software |
|
| 71:28 | . So, because it's such dominating and usually take that tab and the |
|
| 71:36 | successful packages there very well implemented. too dealing with loss of position in |
|
| 71:44 | and very robust. Mm hmm. is carefully implement. And because things |
|
| 72:01 | fail, it's a practical measure to to have, need to have both |
|
| 72:08 | different stopping criteria. one is define it usually is in terms of |
|
| 72:17 | . So some of the updates, you get small enough stop but you |
|
| 72:23 | tend to need to have a max trump. So it doesn't from |
|
| 72:31 | Thanks to Newton's method, it's mainly of their behavior. They may convert |
|
| 72:40 | but for the increments gets um not rapidly enough that the update criteria doesn't |
|
| 72:50 | in the recent times when you need have. So that's that was also |
|
| 72:57 | case of Arson the little gold. was when I talked about the dissection |
|
| 73:03 | . That's it. Both the maximum and that doesn't have you much. |
|
| 73:10 | prevent you from the cold open for long time. Um to be only |
|
| 73:18 | on the convergence criteria and sometimes they forgotten works. Some functions may have |
|
| 73:30 | slow convergence and they may run for . Yes, of course you should |
|
| 73:37 | when a terminal is based on iteration , step count. They should clean |
|
| 73:43 | up and figure out what the error . So you know it is and |
|
| 73:46 | retract it for some other situation. . Okay. That's it for |
|
| 74:03 | Yeah. That was My guest next four. Yes. Yes, mm |
|
| 74:17 | . Mhm. Okay. |
|
