| 00:07 | right. Okay. So okay. will continue to talk about there are |
|
| 00:15 | examination um mm hmm. So and yes. Some of the area properties |
|
| 00:30 | said that we talked about which didn't time. And then and a special |
|
| 00:36 | on the matrices. All right. bearing in the application is quite often |
|
| 00:42 | it's discussing them separately or in detail degree. So, I wanted to |
|
| 00:49 | if I got to start to talk the next chapter in the book. |
|
| 00:53 | hmm. So I think I said last time that effectively what dozen elimination |
|
| 01:04 | transform the matrix. Hey, the that describes what's going on Into as |
|
| 01:15 | product of two matrices. It's popularly as hell. And you because of |
|
| 01:21 | shape of the matrices lower triangular and is that's right triangular matrix. |
|
| 01:35 | Everything like this. So it's just of obvious one left. Lower. |
|
| 01:49 | , couple of comments on that is in practice everyone does this thing. |
|
| 01:56 | you don't need the matrix a then you let l let you |
|
| 02:05 | They use the same memory space for new as it used for Now. |
|
| 02:10 | doesn't kind of work because both Ellen as diagonal. But basically two values |
|
| 02:23 | one doesn't really work. So you store the ones so they have So |
|
| 02:30 | know that Darren in the basic 1000 , what does it such that El |
|
| 02:39 | once on the diagonal. There are ways of doing it. But that's |
|
| 02:44 | of the most common way of Um A little bit of discussion about |
|
| 02:56 | don't want to work that it is terms of during the elimination. |
|
| 03:03 | So it's the next slide is just little bit messy or dance in terms |
|
| 03:09 | stuff on it that is just trying work out. So what the operations |
|
| 03:17 | the nature and say, how much is it in terms of some of |
|
| 03:20 | advice and actually device as well, ones really terrible because and finding the |
|
| 03:29 | that you use in order to multiplied stay there, the proper factor and |
|
| 03:39 | 20 multiplied by the factor and then . So established the fact that they |
|
| 03:46 | the device. So um doing characterization of Gaussian elimination is basically an |
|
| 04:00 | operation. So it's quite, computational expensive. Yes, So this |
|
| 04:09 | a it's quite images for simplicity and and my K is in this |
|
| 04:16 | this that's the two. To show progress you start. Okay, the |
|
| 04:24 | one is J was one. So that case you have N -1 |
|
| 04:29 | you need to multiply a force, you need figure out and one is |
|
| 04:33 | multiply and one for each of the ropes. And then you update The |
|
| 04:39 | -1 by N -1. Um sub . So mr left consumer business, |
|
| 04:48 | and by and and then there's a sub matrix That is one problem and |
|
| 04:53 | roll smaller because you know that with proper factor, everything is going to |
|
| 04:59 | into zero in the first column. you don't need to compute it because |
|
| 05:04 | , so then this Pittsburgh and mine on roles and then minus what problems |
|
| 05:08 | need to operate on. Sorry, , they need to multiply and whatever |
|
| 05:15 | send up bro. And then this results from what is in the current |
|
| 05:27 | example from that structure, there was role and then trying to multiply it |
|
| 05:36 | each one of the other roles And so it needs problems subject to |
|
| 05:45 | precise to count each fund in terms what it's like a corporation that |
|
| 05:54 | So once you're done with the first , then you move on to the |
|
| 05:59 | column and then there is the you don't trust with the first role |
|
| 06:07 | and this is the first column, turns into zero. So then you |
|
| 06:12 | down. So not only do you it And -1 by N -1 matrix |
|
| 06:16 | worry about. So that means that are N -2 elements below the diagonal |
|
| 06:25 | you move to the second column and you have to update the sub |
|
| 06:31 | So the sub matrix gets one column one roll smaller every time. So |
|
| 06:38 | until you're down in the lower right corner. So that's basically the summer |
|
| 06:46 | of ginger terms and the symbols square that are getting smaller and smaller. |
|
| 06:53 | it on out eventually will come with not too messy. So usually one |
|
| 06:59 | , you know, some of these the small in terms of this Since |
|
| 07:06 | . So the important part is not , the most important part is to |
|
| 07:13 | the fact that it is computational somewhat . It's cubed in this site is |
|
| 07:20 | in he's for the dimensions of the . Mhm. So on the other |
|
| 07:34 | , um and also does the same on the right hand side. So |
|
| 07:42 | have to process the right hand Also they get the right result. |
|
| 07:47 | when you do the multiplication and you also feel it on the right |
|
| 07:53 | . But if you just have one inside there is no square terms. |
|
| 07:57 | it becomes basically um never and And then number two is that multiply |
|
| 08:08 | subject comply. And then yes, you just start one element of the |
|
| 08:14 | . So the best concert it's N and that's the same. So we |
|
| 08:21 | this triangular forum and then on the and so then you start from the |
|
| 08:26 | one unknown and you use that unknown plug it into the equation above. |
|
| 08:33 | so that means to also get basically squared operations and do the back substitution |
|
| 08:40 | the right hand side. Well that's of the way the girls never mentioned |
|
| 08:51 | . And in terms of the computational . So here is scandal summarize about |
|
| 08:59 | . So that sources thank you the from the matrix And those numbers. |
|
| 09:09 | number of operations is hopefully independent on right hand side. So if you |
|
| 09:17 | more than one right hand side you reuse the factory ization for any |
|
| 09:23 | It doesn't depend um there's some stuff this is a ah yeah packages soccer |
|
| 09:36 | out there. They always tend to out the fact ization of the matrix |
|
| 09:43 | routine and then there's um different routines view before running back elimination on the |
|
| 09:51 | hand side. The reason is again the characterization of the matrix is much |
|
| 09:57 | expensive and dealing with each one on right side. You want to advertise |
|
| 10:02 | work. Oh do you think we're matrix? This is to have more |
|
| 10:07 | one. Yeah. So that's sort structured all Uh huh libraries out there |
|
| 10:19 | questions on that? Okay. So a little bit from errors. |
|
| 10:29 | so this is a good. So was taken from before in terms of |
|
| 10:37 | this procedures local surgical, this is gun there work on the matrix. |
|
| 10:45 | ? The scaling. Mhm. Okay us is known to produce the first |
|
| 10:52 | . Outgrowing the sub matrix. I on value and subtracted from whatever it |
|
| 10:58 | and they're all you're working on. and then we do the same thing |
|
| 11:05 | case on the right side of the substitution and then response the background. |
|
| 11:12 | the thing when it comes to That and be of concern is this |
|
| 11:21 | market is multiplying. So um when can go bad? Yes, well |
|
| 11:32 | factor is particularly north or if the element that was using here, it's |
|
| 11:38 | than you divide by a small Um Yes, this is also |
|
| 11:43 | It may not be so bad, if it's not small things been kind |
|
| 11:47 | the worst and that's um and part the reason why I want to be |
|
| 11:55 | in how one does it kills an the boy that that's happened. Some |
|
| 12:00 | the other methods that I mentioned this householder method is also commonly |
|
| 12:06 | doesn't have that problem. It's automatically itself. So now I will guess |
|
| 12:13 | example on its illustrating what life But that's the wrong when a |
|
| 12:19 | K. Is for so epsom as all know that we follow the most |
|
| 12:24 | we use episode imitations. So it basically a small Um factors in front |
|
| 12:32 | the X one in the first All the other ones that one. |
|
| 12:37 | applying God's indignation, occupy this draw down with one of our |
|
| 12:43 | get the result one, so you see and then subtract it. So |
|
| 12:46 | get rid of the first ah variable the equation and then You multiply. |
|
| 12:54 | wanna over epsilon and subtract this is -1 over ε comes from here. |
|
| 12:59 | then we got similar on the right side. So now this episode only |
|
| 13:08 | , that means wonder where? So lord. And then what can happen |
|
| 13:13 | that they become so large. That of one and 2 is I think |
|
| 13:17 | more than we can actually lose track the monster that you So this is |
|
| 13:23 | of what? Here. Thank So once you do that and go |
|
| 13:30 | All right, so you end up next to mr more or less |
|
| 13:38 | The next one because of um you even Ex two is 1. So |
|
| 13:48 | is almost zero. And then you may multiply with a lot of |
|
| 13:52 | . So it's question what happens? it's going right along the way and |
|
| 13:56 | and that stuff and get into the . Which is not the correct |
|
| 14:01 | Yes. So, so um so , correct result is both X one |
|
| 14:11 | X two is about fun. So think yeah, that's the hook americans |
|
| 14:18 | property. And so in that case , It's on the sense of the |
|
| 14:24 | and one of the restaurants 2 to . And then In this case 16 |
|
| 14:31 | representation, that's what you get. then yeah, to have done this |
|
| 14:39 | Quotations with 20 times then yeah, when you have two days Take the |
|
| 14:47 | between two, a lot, you to align the guys and then we'll |
|
| 14:54 | the whole thing. And then at end mm hmm when thank you than |
|
| 15:04 | to the position that they have, you end up basically everything. Um |
|
| 15:13 | . The second one that I ever and become, wow. So that's |
|
| 15:22 | if you do that then. So good now. So this is |
|
| 15:34 | that's a solution is fantastic. So not what you get when that |
|
| 15:40 | So Harrison and what is the deal trying to avoid dividing by small numbers |
|
| 15:52 | the things that blow up? wow, mm hmm. Okay. |
|
| 16:00 | All right. I think that's what said. So, so what does |
|
| 16:09 | this example? So what was the not what I did on this side |
|
| 16:16 | . Change the order between these two . So now if I use the |
|
| 16:23 | row here, then you multiply that epson and then you get something here |
|
| 16:29 | is small. So if you lose of rounding, then you get that |
|
| 16:35 | . You have to be one uh this equation because again this year also |
|
| 16:42 | compared to the one. So now guess that takes full text index funds |
|
| 16:48 | equal. Wonderful. So, the here was to the boys um dividing |
|
| 16:58 | a small number to get the So this manipulation of the equations to |
|
| 17:07 | ah the role to use as what's as the prima draw the one that |
|
| 17:12 | used two to determine what the factors and then subtract from the other |
|
| 17:18 | It's almost everything. Sorry, that's most of this library scenes after that |
|
| 17:29 | you. They use one form of thing we make sure that dig out |
|
| 17:38 | results and don't end up dividing by numbers and the most common one is |
|
| 17:43 | as partial pivoting. That means your bet in the remaining sub metrics. |
|
| 17:50 | if you're going to this point so is that point. So you have |
|
| 17:53 | work on this one they're fine. largest value. The absolute value of |
|
| 18:02 | one of the coefficients in that So if you then use that specific |
|
| 18:10 | the events for you Yes instead of top row you pick this one and |
|
| 18:16 | means all the scaling factors will be than one because this was the absolute |
|
| 18:21 | values that anyone on the other Uh huh You know exactly one that |
|
| 18:26 | have more than one choice does same otherwise the rest of them are always |
|
| 18:31 | than so the partial Children in your the problem and they've shown that and |
|
| 18:39 | usually good enough one channel. So permutations of both those and columns and |
|
| 18:45 | civil works fine in terms of the day correct reasons. Yeah, formally |
|
| 18:54 | but the american dream doesn't so Yeah just put him in the car |
|
| 19:00 | what is that? So these are And usually I think that one assignment |
|
| 19:10 | is supposed to write you know and factories asian but it's only like has |
|
| 19:16 | loops and start. So the some the major chosen. So you need |
|
| 19:24 | think that I have to do with parts of the so ah no, |
|
| 19:34 | just like your mark for since this science class is the by having the |
|
| 19:46 | pivoting things are becoming data dependent because next step depends on which is supposed |
|
| 19:55 | yes and you don't know this value you are effectively progress and and updated |
|
| 20:04 | cause that's not known from the That's the value. You keep computing |
|
| 20:10 | updating us progress with the previous So that means that the dependency in |
|
| 20:17 | cold and that makes us money. difficult to stream and paralyze things because |
|
| 20:25 | comes in the defendants. That's why of the other methods like that. |
|
| 20:31 | are not as stable but they don't the problem with things growing up. |
|
| 20:37 | why mentioning currently on the first is method and givens method that are the |
|
| 20:48 | flow is data independence. So you need to worry about the physics and |
|
| 20:53 | obvious that's deciding factor against the Okay. The National Financial Services. |
|
| 21:03 | that's why it's good to know about other methods for that recently. No |
|
| 21:11 | . Mhm. Okay. So generally computer sciences aspect of nelson elimination and |
|
| 21:23 | should mention this. Okay. So terms of linear and your brother is |
|
| 21:32 | commonly used stores everything is known as in your algebra subroutines of loss for |
|
| 21:43 | and they come with the number after plus blame and it just says the |
|
| 21:49 | of followers. It's just the number nested loops that is involved during the |
|
| 21:55 | time in your So anything of actors lost everyone and the things that are |
|
| 22:03 | major issues when special made some 11 to to those, those and |
|
| 22:11 | um last number three that is making one of the things. So all |
|
| 22:17 | the vendors out there for the processors d in philadelphia and the United |
|
| 22:25 | including the few members like doing Um optimized implementations on these basic linear |
|
| 22:34 | functions for the but also open source that are quite good. Check |
|
| 22:43 | B well optimist as well for different and that you can get from and |
|
| 22:49 | great. So this is good to . So now the different operations here |
|
| 23:02 | over a lot more 23 have different in terms of how well they can |
|
| 23:10 | for the computers. So when you about he was an elimination in that |
|
| 23:20 | there was this includes operations to do collectivization But in 86 basically n squared |
|
| 23:31 | . So that means on average do n operations for data. So that's |
|
| 23:39 | important thing to realize because that means . Ah both are going to china |
|
| 23:50 | folks then you can potentially make good of the memory authority in your |
|
| 23:59 | The principal will review something because they again thanks to the operations and elsewhere |
|
| 24:07 | that means direct is all well you benefit from caches, caches. That's |
|
| 24:15 | good some guests managed to reduce. For our safety lookouts, level one |
|
| 24:26 | do basically add three vectors is that's one operation for data items. So |
|
| 24:33 | doesn't matter how hard you try. can't really benefit from fashion. So |
|
| 24:40 | why for things like matrix smoke petition tell you characterization goes in elimination one |
|
| 24:50 | to structure, they told so somewhere there is embedded, they may be |
|
| 25:02 | . That's what you think would use fascists and that's what you can |
|
| 25:11 | So it was a Yeah. So you know, everyone knew |
|
| 25:24 | So now I'm just casting and turns maybe it's meant to Spotify first. |
|
| 25:32 | in this case it's one of the on lots of the matrices. A |
|
| 25:40 | roles and a few problems. Um , so then you try to select |
|
| 25:47 | blue boxes here. So they to the load the squares and then basically |
|
| 25:56 | proportional to the number of elements. then you have they small legacies that |
|
| 26:03 | based on cute operation of IT data so this is the way the structure |
|
| 26:12 | matrix matrix multiplication. So it can do things so to do this for |
|
| 26:20 | , so that the blue box, it's a level one cache and then |
|
| 26:24 | I think you move off that fits to cash and so they get set |
|
| 26:29 | so simple three. That's the Matrix Matrix multiply it turns into this |
|
| 26:36 | are the size of the matrix is in a few plants corresponds to the |
|
| 26:44 | of the Mhm. And so So this is just working out the |
|
| 26:57 | in the blue box and the your basically reused, possibility or benefit of |
|
| 27:04 | is proportional to. So the number roles columns and the um this is |
|
| 27:17 | so when um I love that's how then work on things. So you |
|
| 27:25 | instead of doing things single roll by column. Mm hmm treat this with |
|
| 27:32 | lot here and you can work through guys one x 1 for instance and |
|
| 27:37 | on these columns but they don't update green until they have dealt with all |
|
| 27:43 | problems and all these roles and then this is sense of being now is |
|
| 27:48 | matrix matrix multiply where the kind of the dimensions in the is the number |
|
| 27:57 | columns and also refused together. This kind of the trick that people use |
|
| 28:05 | get fast everyday conversational. Yes, right. We'll make questions on |
|
| 28:20 | So that's something to know. I'm yet for fun. I guess it's |
|
| 28:25 | perspective. That's all it is. died for have fun um student and |
|
| 28:33 | well together implement this. Same somebody and the point here. So there's |
|
| 28:40 | of things here in terms of attraction the big performance with that from this |
|
| 28:45 | . S. B. and another is sort of the power consumption. |
|
| 28:50 | if you do this thing tonight in of our consumption you can see dealing |
|
| 28:54 | lights and sound versus the actor factories this is time on the for |
|
| 29:01 | you can pretty much follow what goes in terms of how the power dissipation |
|
| 29:08 | over the execution, wow. So have been from the interested, there's |
|
| 29:16 | bunch of slides, see how um algorithms uh big um I think it's |
|
| 29:35 | the other things turn to you might across is so called left and right |
|
| 29:44 | algorithms. And what I've talked about is something the right to open |
|
| 29:50 | I don't know what it does is for this life. So on the |
|
| 29:57 | side of things, I mean is you go first progress on left to |
|
| 30:02 | . So as long as we update to the right left hook and the |
|
| 30:11 | for different things because you don't really to update this being part of the |
|
| 30:20 | all the time. The only thing need to know is when they're going |
|
| 30:25 | work with these separate columns than they updated. So I left looking algorithm |
|
| 30:33 | first working on the green park and goodbye. Whatever works is related to |
|
| 30:43 | part. When you start to work this bluish section, then it's best |
|
| 30:50 | start to produce all this information on the elimination factors that you have. |
|
| 30:58 | the progress to this point. Then justified this part. So that's what |
|
| 31:03 | of the left looking someone in here look to the left and see what |
|
| 31:06 | don't need to do to get this out there. Mm hmm. And |
|
| 31:12 | upon the member system I would and factors of the computer architecture turns out |
|
| 31:22 | , sometimes it's credited to the left and sometimes that right looking and sometimes |
|
| 31:27 | fact we both for different levels of just something that you want. |
|
| 31:38 | No questions. Um, that's there go. You didn't ask for |
|
| 31:57 | Yes. Yeah. And this is playing it on talk about. So |
|
| 32:08 | get method that is oh very common selected agencies. And it's just simply |
|
| 32:19 | simple majority simply got some elimination. advantage of symmetry. So because symmetric |
|
| 32:28 | That's a lot of things in So you don't necessarily work need to |
|
| 32:32 | on both. So that's what the it take to safety factor too confirms |
|
| 32:40 | memory of work the first telling you supposed to related. So in In |
|
| 32:48 | case so in that case you don't the one on the diagonal of Alaska's |
|
| 32:53 | for the health. But uh, sure that in that case Ellen do |
|
| 33:01 | have the same values of the So that means they want to store |
|
| 33:05 | of it because it's symmetric. That a transports are the same. And |
|
| 33:12 | also means that the two things new left. No one less triangular. |
|
| 33:19 | then um elements are all the Meet the monster. And then there |
|
| 33:30 | other variations as well of the Never so and then questions on. |
|
| 33:49 | the next thing is the personal maid friday. I'm going to make this |
|
| 33:56 | special version of that has known as and and and more general constructed major |
|
| 34:04 | . For me. It's about the . This is now. No. |
|
| 34:13 | um So right. So the point that they have an interest is super |
|
| 34:22 | different dimensions. If you don't really the whole matrix store the diagonal |
|
| 34:28 | You know that there's no point restoring for computing on it. So the |
|
| 34:34 | specialized algorithm. Them too. The that you know a lot about the |
|
| 34:39 | of the Oh yeah. So So yeah. Many solvers for parts |
|
| 34:56 | differential equations that somewhere in the steps china or uh system. So many |
|
| 35:05 | . So anyone that uh some other may work on partial differential equations always |
|
| 35:12 | across the special understand systems and we use it later on when we talk |
|
| 35:21 | compliance. Um, it was the systems and something used to figure out |
|
| 35:29 | proficient inspiring approximation. So welcome back on. Yeah. Okay. So |
|
| 35:40 | what it looks like. So obviously of winter cereals. So as I |
|
| 35:45 | , those people do this for basically feedback cruise. Um the southern super |
|
| 35:51 | and the diagnosis. That also means it's not a whole lot of work |
|
| 35:57 | do in this case. Because they need to to turn everything into the |
|
| 36:02 | parliament to zero. It's just they and that's it. And when you |
|
| 36:07 | that, the only thing you need update the the second one because the |
|
| 36:12 | of the zeros and I think it's and then of course you need to |
|
| 36:16 | on the left cancer. But it's , that was not much work. |
|
| 36:25 | , um, um so this was much the same. So this because |
|
| 36:33 | , you still have to worry about . Again, it's all about this |
|
| 36:37 | moment. If they're nice to then one doesn't everything either. So |
|
| 36:43 | was that? What? And that out to be the case. Sometimes |
|
| 36:49 | of these traditional systems. Mm So the next one also shows because |
|
| 37:01 | so little work. So in this now there's not into direct victims is |
|
| 37:06 | like two operations on the natures in elimination steps. So in this |
|
| 37:14 | But everything is of order and So what? Mm hmm. |
|
| 37:28 | this is just consuming the gold. guess that's not spectacular about that. |
|
| 37:37 | that's why china angle system. So , it's important. Again, because |
|
| 37:44 | the significance of them in all You know, this particular special |
|
| 37:53 | I tend to ask questions on Either on assignments or make terms. |
|
| 38:02 | This is um china an important as department they're gonna dominance against means that |
|
| 38:13 | absolute value of the elements or their on the diagonal is at least the |
|
| 38:20 | in some of the off diagonal. . It's kind of nice to know |
|
| 38:29 | discussing the donation procedure preserves the diagonal and if it's a diagonal dominant, |
|
| 38:35 | don't have to worry about David. again some of the facts, the |
|
| 38:40 | preserves a nice property is um I'm just trying to everyone this |
|
| 38:51 | Mhm. Well, yeah it is Oh and a special case of the |
|
| 39:01 | uh it is known as triplets And what's special about them is that |
|
| 39:10 | the values um The animals are the , corresponding the and all the force |
|
| 39:19 | elements are the same when it's symmetric well. So when this is the |
|
| 39:27 | , you can take advantage of this and you can save yourself the fair |
|
| 39:33 | that works because the sun goes Yes, there was an elimination procedure |
|
| 39:39 | lettuces of this type ah because if gonna dominance again then it turns off |
|
| 39:50 | after a bunch of steps regardless of size and the matrix. Ah the |
|
| 39:56 | values that they want to operate on become sufficiently small. So they are |
|
| 40:02 | America the same. So you don't to proceed to go through with |
|
| 40:06 | Metrics, studies in a 1000 roles columns. Try lying on matrix. |
|
| 40:11 | they took its form uh after a position depending upon how diagonally dominant it |
|
| 40:19 | but Graduate four is not uncommon in solvers Then it basically takes six steps |
|
| 40:29 | the china angle solver regardless of the of the system. So um and |
|
| 40:35 | you have any double position, yes it takes For the same only like |
|
| 40:39 | steps. So that means you can the amount of work considerably. |
|
| 40:45 | An agent to have these properties and also something that's commonly used. Um |
|
| 40:54 | okay. So that's just the text I guess that's it now. The |
|
| 41:07 | things that happens also the excellent There is a little bit more generalized |
|
| 41:15 | system. So not just the elements as column five name that animal and |
|
| 41:27 | to off. Yeah, I know and below. To the extent of |
|
| 41:34 | admin system solvers are also something that's that you start there. Mm |
|
| 41:41 | Particular to not much specialty is just you don't need to do pivoting then |
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| 41:52 | can just go through the process and just more work on the friday |
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| 41:56 | But there's still a couple million number operations. Para one comma. So |
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| 42:05 | it's a couple of different places. looking at the spectrum in our |
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| 42:11 | I don't want to. This will happen but this is possible. So |
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| 42:24 | one way one can do it is turn it into something that looks like |
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| 42:30 | triangle instead of having single elements to some major seats of blocks, all |
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| 42:40 | political system. So many times he in they are turned into the block |
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| 42:46 | out of the system and then dominantly formula and you can just leave the |
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| 42:52 | procedure as he did. Didn't try arguments over. The only difference is |
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| 42:58 | instead of the elemental operations, some it ends up the multiply operations development |
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| 43:07 | are supplying our correspondent. So so is the way I also love to |
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| 43:15 | fun. Um Mhm. And against packages that that's throwing the attempted the |
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| 43:23 | I guess it's just um what are doing in this case? Uh for |
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| 43:35 | can't happen. They don't necessarily again the universe. But this these steps |
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| 43:43 | so this is a divide operation that be multiplied by the inverse. But |
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| 43:48 | practice what to do you this will the D on the left hand side |
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| 43:52 | to solve the system of regulation that so you don't normally the the universe |
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| 43:58 | be inspirational song people. And this something I think the next picture of |
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| 44:07 | years just showing um of these Gangel for systems, it's here. So |
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| 44:23 | you have a measure of some form space into the end small doctors and |
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| 44:33 | you have something that some form of equation, some of the best things |
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| 44:40 | uh also take miss processing convolution kernels that looks at the values of neighboring |
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| 44:49 | . So that means they only looked simplification us neighbors. So then you |
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| 44:56 | higher if your label things that role role. Yeah, I don't |
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| 45:04 | Yeah. So the points to the and the right time successive numbers or |
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| 45:13 | but see points above and below the . This point the they are as |
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| 45:23 | away as the length of rope. , so that means from this point |
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| 45:29 | this point now, that's basically the of rules and columns. So they |
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| 45:35 | spread out. But in this little example than the best responses, but |
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| 45:40 | best way to find them just then one of the roads, it's |
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| 45:46 | but it was just trivial example for things can come from. And then |
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| 45:53 | think that's what happened about there was elimination on dancing systems. Yeah, |
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| 46:08 | for the simplest things are not mentioned the book as far as I |
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| 46:12 | Um, otherwise most of the and optimizations have tried. Yes, sort |
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| 46:23 | matrix matrix multiply some operations when it to the something. Something for him |
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| 46:33 | of computer science. Oh, mm hmm. We'll have more on |
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| 46:43 | the next thing. I agree with . No, I think I never |
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| 46:47 | all of that. So. And next thing is not to talk |
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| 46:54 | We'll find some any questions on the otherwise. Mhm. So it was |
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| 47:09 | sent by methods. Some simple and not so simple. Um Oh, |
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| 47:19 | heard of anyone on these methods. ? All right. Um, I |
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| 47:28 | say in practice probably the most commonly method is symptoms. Probably one |
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| 47:35 | Um, I'll start them with a section. It's fairly straightforward. Mm |
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| 47:43 | . Um, so we have tried find the rules or the function always |
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| 47:51 | the arguments for a function for The value of X for which the |
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| 47:57 | is zero the move so thereby section for as well and the different it's |
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| 48:08 | and I have so in this case I have two points A&B. |
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| 48:15 | The function value of the nbr of sign. And again, for me |
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| 48:21 | is a continuous function at least in well behaved. That means somewhere between |
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| 48:28 | and B. It is across the , the X axis already function function |
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| 48:38 | 10 years old. Somehow it goes a to B. And obviously then |
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| 48:42 | passed across somewhere it can cross multiple , but it just across at |
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| 48:50 | So that's what to do. So this section method I guess here and |
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| 48:56 | some examples, you know, just can be multiple crossings operational guarantee. |
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| 49:04 | , yeah, they want to guarantee that there's a police report so they |
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| 49:13 | . Ah The other thing I guess should say is yep, mm |
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| 49:19 | The fact that if the function values both on the same side positive. |
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| 49:23 | this right hand graph doesn't mean that is not the zero crossing, But |
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| 49:30 | cannot conclude a certain that there will one because they don't really know how |
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| 49:35 | function behave. They only think about two positive values so you can't use |
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| 49:39 | . I think that maybe there is simple process. No, so that |
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| 49:44 | only kind of works this by that's the final rules. If the |
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| 49:49 | values of A and B are office , then it was someone transaction is |
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| 50:02 | the bike comes from this perspective the between A and B in half and |
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| 50:09 | the function, the midpoint between A B. And once you know this |
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| 50:17 | of the function at this midpoint, can determine whether the rule is to |
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| 50:21 | left on the midpoint of to the of it and then you just repeat |
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| 50:27 | process once you say okay fine, not the new rules here and then |
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| 50:34 | will try to so reduce the interval brackets. Mhm. So that's what |
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| 50:45 | , nope. So figure it You know whether it's to the left |
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| 50:53 | the right and then just proceed and come along continuing close enough to the |
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| 51:02 | that we're happy with close enough to position and it has to find the |
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| 51:09 | structure. So you know that the is between in this case the midpoint |
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| 51:18 | the and if you know the route be closing in the B or the |
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| 51:26 | points. So you don't necessarily known they think better than the length of |
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| 51:32 | interval to oppose possible. That means best of the length of their final |
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| 51:42 | in this. Ah Having processed is position by which family. So that's |
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| 51:55 | . What is that song? Just them, What do they want? |
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| 52:02 | it will converge in a reasonable number iterations. But um you may |
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| 52:10 | you probably need to put the suffering because it's the curve is 35 then |
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| 52:20 | defense pretty much. That's of course done this very not too hard. |
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| 52:26 | takes that many iterations to. We a reasonable position about this firm pretty |
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| 52:33 | follows thing X. Axis. Thank , clicking games of all. And |
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| 52:41 | can be very slow convergence on access females speaking starting criterias or that |
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| 52:51 | So so hard to actually works, . So yeah. Alright, so |
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| 53:14 | guess now you step by step on case. So on this case mm |
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| 53:23 | . Try to find out if there a room In the interval from 0 |
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| 53:27 | 1. And we have the functions we can evaluate it both zero and |
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| 53:33 | . And it turns out the function plus one, X equals zero and |
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| 53:39 | minus one Extent was one. So know that there must be a |
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| 53:46 | it's been one. So then the step is then again Look at the |
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| 53:53 | between zero and 1 for X. it will also evaluate the function at |
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| 54:00 | X. value them. So and it turns out that the function value |
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| 54:05 | that is -2.37 - five. So which side of the lake Fund fund |
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| 54:18 | it's true. What? This is zero and .5 for between 45 and |
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| 54:33 | . Mm hmm. But zero and . Ah zero 2.5. Yes, |
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| 54:44 | That .5 is negative and at once negative. So we don't know that |
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| 54:48 | be rules, but there's no guarantee very smart. On the other |
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| 54:53 | between 0.5 is positive at zero And . So it must be on the |
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| 55:02 | hand side or to the left of midpoint. So then I just through |
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| 55:12 | thing. So the next time around midpoint of the 0.5 and now it's |
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| 55:20 | positive value. So that means um at zero and possibly Uh .45. |
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| 55:29 | that means it has a very negative . So now it's actually on the |
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| 55:35 | hand, all of them, there's need points. So let me take |
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| 55:39 | just keep doing this. I guess gonna start doing this probably so |
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| 55:48 | So that's basically said look at whatever length of the interval is. And |
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| 55:54 | they take the midpoint of where I born and then look left and |
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| 55:58 | where is my team? And in case I guess 20 iterations, they |
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| 56:07 | something that is more or less single . So any questions on that? |
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| 56:18 | . Right. So it looks like about the complexity of this um mm |
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| 56:26 | . Like theoretically in a bad thing you can have anything number best |
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| 56:29 | But where are the machines that it's limited by the epsilon? How do |
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| 56:34 | , what was that? The man steps you can do before you hit |
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| 56:37 | a human? Um Yes, um yes, so I guess |
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| 56:52 | wow, latino capsule is single position not to to the minus 24. |
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| 57:03 | just knowing that it's working 16. . 2 20. It's like, |
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| 57:21 | , not so The Truth of the , it's roughly 1000 wins for that |
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| 57:35 | and thousands and thousands of us. . Mhm. And otherwise user |
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| 57:53 | Okay, that's a long. That's term of what they're doing. |
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| 58:00 | Thank you. Yeah. No Okay. Mm hmm, mm |
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| 58:15 | That's one. No problem. Can necessarily a simple section. This is |
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| 58:24 | different place. Examples of detail, they may want to look at it |
|
| 58:34 | case. So yeah. So the it's a very simple procedure and so |
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| 58:46 | easy to use, but often it to be somewhat so in convergent. |
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| 58:51 | that's why it's not necessarily the dominating news. It's good to know. |
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| 58:56 | it's simple, you know, it's just figuring out things are quite |
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| 59:03 | How do you figure provided? So well known, you can have a |
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| 59:11 | , you want single position, you the length of the interval they start |
|
| 59:17 | and then whatever ah sleeping in Then you know the length of the |
|
| 59:25 | so that the fact that the roots out how many steps to take to |
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| 59:30 | the position 1st. Straight Forward to Left. Mm hmm. Right. |
|
| 59:41 | this is just a concrete example. magic about that. What about the |
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| 59:48 | , the position that they have and difference in the length of the |
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| 59:53 | Then again check it out really what still needs to get and they get |
|
| 59:58 | great one little bit circumstance. and that's what that's for example was |
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| 60:19 | okay. Ah it's just they know convergence against that, the notion of |
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| 60:28 | process and that's the difference with because they have at the time most commonly |
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| 60:42 | methods and methods that's a highly converting the number of gigs and stuff and |
|
| 60:57 | talk about the next son bullying And then there are some tweets from |
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| 61:08 | by section efforts that one is not his regular falsity. So it's kind |
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| 61:16 | false move on. Yeah, I'll you my back. So it's kind |
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| 61:22 | a tweak to try to improve the somewhat. So, so what it |
|
| 61:37 | is he said. So instead of to explain this in the liberal than |
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| 61:45 | basically straight down as they come between two 0.7 obviously in this particular example |
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| 61:55 | imperative to summit point that so no that it wouldn't necessarily work for often |
|
| 62:04 | gives them a little bit of a . That's a good point. Um |
|
| 62:11 | Count. That's not as a regular . And his face was figured out |
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| 62:18 | the big question for actually I think we're using the congress triangles are coming |
|
| 62:26 | figure out what the C. Value where for where the streets are in |
|
| 62:34 | se and Bse where it crosses several for that. So it's also fairly |
|
| 62:41 | expression for the embassy um from So I um so yes that's just |
|
| 62:54 | excessively doing this and we get the for doing that. Uh plumbing is |
|
| 63:05 | mason police. Someone think it's straightforward. It's again just using um |
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| 63:13 | equation and then start to index the step. No. Okay so okay |
|
| 63:31 | um illustrate that now and there's a tweak on that one. Mhm. |
|
| 63:45 | That and tried to find crossings not by doing successive. So they have |
|
| 63:57 | this street team has won like three right then. The next step would |
|
| 64:02 | for the regular policy would be then find This dashed line and find that |
|
| 64:09 | crossing. Now in this modified version it uses the previous slope line to |
|
| 64:19 | . But um this year or find value for which you want to evaluate |
|
| 64:24 | function particularly Whether zero crossing is. in this particular case, if you |
|
| 64:30 | this modified version of the regular falsity this thing, this crossing point is |
|
| 64:35 | to the actual root then they would proceeded yes. Using the normal model |
|
| 64:44 | . People have all kinds of trickery they have tried to do. They |
|
| 64:48 | still fairly simple algorithms for the proceeding finding the rules. But then the |
|
| 64:55 | is delaying and using the previous soap of the new. Of course I |
|
| 65:03 | where they're if it overshoots it could done. So it's not a |
|
| 65:09 | Again, that is always his It worked often enough for people. |
|
| 65:19 | are you using 123? Yes. . I guess that once it was |
|
| 65:39 | today I went fast. Any Uh huh. Mhm. Mhm. |
|
| 65:58 | . So the next time that says was mostly talking about this name, |
|
| 66:02 | matter that has again for that known quadratic convergence. And that's why it's |
|
| 66:10 | very popular method because it Yes you . That was the number of |
|
| 66:18 | Mm hmm 74 About time about So you know, just mask it |
|
| 66:25 | often used for implementing things. That root function. So condition programming |
|
| 66:33 | So how do you get for groups out that used in that case because |
|
| 66:43 | wanted to the best. Mm So it was actually done with breakfast |
|
| 66:49 | that is combined the tables, vodka the victim is trying to get the |
|
| 66:57 | bits. No tables. Yeah. , mm hmm. Yeah, I |
|
| 67:19 | yes, actually. Uh huh. |
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