| 00:02 | Okay, thank you. Mhm. things works? So today continue a |
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| 00:11 | remarks about that's fine. And then , well, there were a couple |
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| 00:21 | things. I didn't dude structure and I will continue to talk about. |
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| 00:32 | the subject of the class. And in particular a little bit about |
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| 00:38 | known as backward error analysis, something as Horny as usual, when it's |
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| 00:44 | Chapter one of the books and I have a couple of more slides in |
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| 00:52 | slide deck that is uploaded. So not going to comment on all of |
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| 00:55 | slides. But I said the one here that was simplicity. I just |
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| 01:03 | out by last time, but there's number of yeah, in a way |
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| 01:11 | values in my plan. That's they avoid trying to use for variable |
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| 01:17 | That's pretty much what it's like. tend to say stop. And one |
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| 01:22 | the things that was on the side I as a constant is kind of |
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| 01:26 | coded and available with high precision in place. And I think that's pretty |
|
| 01:33 | it In terms of comments on this , that's like, you know, |
|
| 01:37 | guys talked about most of these um only thing that I think that did |
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| 01:42 | comment on was there's also a function the for a race which are kind |
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| 01:50 | convenient construction. So there's a bunch but building functions and we don't talk |
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| 01:55 | them all in the class here, as to the convenience of using maps |
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| 02:01 | been manipulating matrices then. I think one perhaps it's not totally intuitive physiologic |
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| 02:14 | , operators works on the race. so that has these two. |
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| 02:20 | I mean all the celebrities to compare with each other and there were call |
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| 02:26 | morning. So for this particular for example, with you Small, |
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| 02:31 | x 2 matrices, it operates column . And then just the condition for |
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| 02:38 | pair's elements going down col so for any this that means that the condition |
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| 02:46 | somewhere in the column there are some at least of a that is less |
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| 02:53 | the corresponding to be. So for particular cases that honestly this number is |
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| 03:00 | smaller than the corresponding number. But this number that 20 years. |
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| 03:06 | that's why the condition is true for first column, but it's not true |
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| 03:11 | second column. Because in that case becomes there is comparison results in a |
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| 03:17 | vigorously, on the other hand, all means that this condition needs to |
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| 03:24 | all corresponding pairs. The best points falls for the first column because |
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| 03:32 | In this case, the first government column of A and B. Uh |
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| 03:38 | element is the biggest of us for of not just something to be aware |
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| 03:45 | how that works. And then we another example here. That's um for |
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| 03:55 | , um it's not totally intuitive, for example. So there's a very |
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| 04:01 | logic and uh where they do that different things, depending on what the |
|
| 04:08 | for access compared to a couple of . So the question here is What |
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| 04:16 | in these four cases and in So that's and I want to ask |
|
| 04:22 | so for the first case A. . Here, what do you |
|
| 04:28 | Why would be 500? Okay. makes sense to me. But that's |
|
| 04:38 | what McMahon thinks what. So we try another one and we can try |
|
| 04:51 | find and then I'll tell you how thinks about this. So fine. |
|
| 04:57 | , I'll give it to you About of them, Right? That's what |
|
| 05:06 | know, I would also believe that not what's going to happen. So |
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| 05:14 | does matter eventually do? That's one . And then the next time I |
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| 05:21 | to show you. Yes. So tests Each of their conditions, |
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| 05:30 | That compares whether X is greater than or not. And it also tests |
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| 05:37 | X is smaller than 10 or And if either one of them is |
|
| 05:42 | , that's an order, then it with that cost. So in this |
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| 05:49 | X -1, It's not smaller than , but it's smaller than 10. |
|
| 05:55 | in that case because it's an or , then it proceeds to do Why |
|
| 06:02 | 10 seconds and you can go through same logic happens with all the other |
|
| 06:07 | and that's fine. All of this up during the first statement after the |
|
| 06:13 | . Of course. So that's something , to me, one needs to |
|
| 06:19 | very aware of because our normal friends my normal stance would be the same |
|
| 06:23 | yours that they would fail the first and move on For the next week |
|
| 06:29 | to the past. But that's not happened. So when you have the |
|
| 06:35 | and to get what I think the one may have intended is that |
|
| 06:42 | need to force, it says that left and right sides of the street |
|
| 06:46 | the comparison as to the tool together the war. So to me that's |
|
| 06:55 | of a things are easier. You results in books because you think of |
|
| 07:02 | as an air conditioned utilities, but not the way for me. Mm |
|
| 07:12 | . So if you do that with then it comes out and that's, |
|
| 07:17 | think pretty much the comments have heads this place. Something here. And |
|
| 07:23 | there's somewhat references here when we find said there's many more slides that covers |
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| 07:31 | text of the exercises that three. you're just going through last time as |
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| 07:40 | , the ones who do not. , mm hmm. Just one. |
|
| 07:58 | . Mm hmm. Yes. So exercises of the whole, I would |
|
| 08:10 | it's mostly posters. Um, we're used to using um, what I |
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| 08:16 | the reputation for indexing or colon location , and the functions of manipulated it |
|
| 08:29 | . That's right. Can I go right. Um, there's a few |
|
| 08:40 | that were left last time and this be useful for you guys and some |
|
| 08:45 | the later assignments actually. So let's . Um So first thing here is |
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| 08:54 | using the symbolic package, so you'll to install that in the matter of |
|
| 08:58 | can simply do that. Um So math lab you can basically get a |
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| 09:04 | representation of different expressions if you want look at them in that we're not |
|
| 09:10 | not just solve them directly by putting values in them for that. You |
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| 09:15 | simply use the symbolic package. And first thing I'll show you how you |
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| 09:19 | solve uh linear equations or quadratic equations even higher order equations. Um So |
|
| 09:26 | the first thing you need to do define a symbolic variable. So here |
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| 09:31 | this case i they find X as symbolically able and then using the function |
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| 09:39 | , you can actually solve a linear quadratic equations. In this case I'm |
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| 09:45 | here for X. So that X equals two. That's a linear |
|
| 09:49 | And remember you need to provide double for the for getting the right |
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| 09:56 | Um So if you learn that it's straightforward, you get uh the solution |
|
| 10:01 | X S seven in that case the of a quadratic equation at x square |
|
| 10:08 | two, X plus one equals zero you solve it for X. Um |
|
| 10:13 | don't really need to give X specifically by default, X is the first |
|
| 10:18 | that meth lab looks for. Um still you can, you can say |
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| 10:24 | . Um So yeah, the solution a practical radic equation is the roots |
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| 10:29 | the quadratic equation. So in this why do contains minus one and minus |
|
| 10:33 | , which are the two routes um this particular equation? If there are |
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| 10:38 | roots, then you also mad lab gives you the imaginary uh results and |
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| 10:43 | imaginary roots for that equation as So it doesn't give you an |
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| 10:47 | There are no uh no roots. there are two different ways you can |
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| 10:55 | ah sort of functions or expressions. one is if you are not already |
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| 11:01 | of, is called a function And this is basically very useful when |
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| 11:07 | just want to evaluate an expression for numeric values. So it's like a |
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| 11:12 | version of defining a small small And in this case, what I've |
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| 11:17 | is I've defined F1 as a function and this is the syntax that you |
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| 11:23 | to provide it. So at the uh and then inside parenthesis you need |
|
| 11:27 | provide what the variables are in in that functions in this case it's |
|
| 11:33 | X, that's the variable. So what I've done here is I've defined |
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| 11:38 | one as a function handle to a that defines whose definition is basically X |
|
| 11:46 | and if you want to evaluate that just simply call that Function handled just |
|
| 11:52 | any other function and like here I've F12 and that gives you the answer |
|
| 11:58 | that and see that this is the of F. One here is actually |
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| 12:05 | function handle. And that will be from what I what I'll show you |
|
| 12:09 | is symbolic variables and symbolic functions. yeah, you can directly evaluate the |
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| 12:15 | handle by providing a numeric value and output of whatever you get by calling |
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| 12:21 | function. Again, just calling class on F12 tells you that it's a |
|
| 12:27 | dive of variable there. Uh Similarly can have a function that has multiple |
|
| 12:35 | in that. So here F two a function of X and Y. |
|
| 12:38 | it's X squared plus Y. Is definition of that function. And again |
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| 12:41 | same thing it's a function handle. then you can evaluate it by calling |
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| 12:47 | like any other function as F comma three. And then again the |
|
| 12:52 | is a double bag of variable. , so that's function handle it. |
|
| 12:59 | the symbolic functions are a little bit and they are useful when you want |
|
| 13:04 | um see sort of an expression type um um view of that function so |
|
| 13:12 | say I'll show you what that Um So here we have defined a |
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| 13:17 | t. You don't necessarily used to to use X. Or you can |
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| 13:22 | any different any other symbol. So have different T. Which is a |
|
| 13:27 | valuable. And then F. Is a symbolic function of T. |
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| 13:34 | F is a symbolic function of T definition is two times sine Please. |
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| 13:40 | mm hmm. And if I and print it out right. Okay. |
|
| 13:47 | just done it again. Yeah. . So now if I just simply |
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| 14:02 | F. D. Then you get same symbolic representation of that of that |
|
| 14:07 | . So it doesn't get evaluated for yet because obviously we haven't passed any |
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| 14:12 | to it. Um before I show how to evaluate it, we can |
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| 14:17 | can also perform differentiation. So by just calling the function on a on |
|
| 14:22 | symbolic function gives you a symbolic function it has returned. But whatever you |
|
| 14:28 | is a derivative of that function. if you perform differentiation of to sign |
|
| 14:33 | square, The differentiation of that is 4D costly square obviously. But still |
|
| 14:42 | what what you get is already done still a symbolic function. So you |
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| 14:46 | go and directly evaluated yet. So first step in evaluating it is |
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| 14:55 | I like to go by these you can just technically call the whole |
|
| 14:58 | result in just one statement burden. say I define a variable equals |
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| 15:06 | And now if I call this derivative we got uh and substitute the variable |
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| 15:14 | there was only one variable. I just provide a s the input to |
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| 15:18 | derivative. So what model of does it substitutes the valley of tea in |
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| 15:25 | derivative with the value of the So here it will substitute through with |
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| 15:32 | value of lee. And if you to what you get is eight times |
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| 15:38 | forward. So again, still it's a symbolic representation of what whatever that |
|
| 15:44 | was. It's still not uh still been evaluated. And the type of |
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| 15:52 | this substitution Um substitutions result by It's now it's a symbol. Now |
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| 15:59 | not a symbolic function. It's now symbol. And now you can evaluate |
|
| 16:05 | once you have a symbol. So evaluate it simply call the data type |
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| 16:12 | that you want to evaluate it in pass the uh the substitute of derivative |
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| 16:17 | it. And that will finally give um the american answer for that for |
|
| 16:25 | expression. There are quite a few involved but it's helpful if you just |
|
| 16:29 | to look at it in a symbolic . If you have a function, |
|
| 16:33 | want to see what it's delegated looks . And you can also get derivatives |
|
| 16:40 | higher order. So if you don't anything that's just the first order |
|
| 16:45 | uh you can just provide a comma then the order of the derivative to |
|
| 16:50 | uh let's say a second order derivative this case. If you pass |
|
| 16:54 | Mm hmm. Ah yes. If have a function right uh that they've |
|
| 17:05 | more number of variables in it. can also derive performed delegation for it |
|
| 17:13 | respect to a particular variable and there one difference between the previous example and |
|
| 17:18 | one is that there I first defined expression as a symbolic function here. |
|
| 17:26 | directly calling the same. The difference uh function that performs differentiation directly on |
|
| 17:32 | expression. So what I get as , as a result, it's directly |
|
| 17:38 | symbolic expression, not a symbolic It's a little bit confusing, very |
|
| 17:43 | if you spend a few minutes on that it will make sense. Ah |
|
| 17:49 | yeah here the function or the expression should say that I wanted to perform |
|
| 17:55 | for both sine X times the square I wanted to perform differentiation with respect |
|
| 18:01 | T. And so you just provide commodity as the second parameter to differentiation |
|
| 18:08 | . And differentiation would be performed with to T. And X will be |
|
| 18:15 | a constant in that case. So you get here is the differentiation of |
|
| 18:19 | expression with respect to T. And you check the class of whatever you |
|
| 18:26 | as a result is a symbol in you can directly evaluate. Um but |
|
| 18:31 | you have multiple variables inside your function is a slightly different way to get |
|
| 18:38 | that the symbolic representation for your So for that you can use the |
|
| 18:44 | subs stands for substitute um uh the name a list of the variable in |
|
| 18:53 | . Um It can mean whatever already but and then the values of the |
|
| 18:59 | those variables but the values of the should be same as this. The |
|
| 19:04 | of those values should be same as order of the parameters there. |
|
| 19:10 | And then you call double on that thing and that gives you the numeric |
|
| 19:16 | for that whole expression. So there two different ways. You can first |
|
| 19:23 | symbolic function followed by expression and then it or just directly evaluate a symbolic |
|
| 19:31 | . Yes. So using different can differentiation. Similar things you can |
|
| 19:37 | Um For integration it's just the difference that the function is called? It's |
|
| 19:44 | doesn't stand for typecasting for interior as programming language starts for stands for |
|
| 19:51 | And it's pretty much the same thing can perform integration of a symbolic expression |
|
| 19:55 | gives you the integration of the Xs uh integration is X square, |
|
| 20:01 | get a symbol in return and then can evaluate it for whatever value you |
|
| 20:06 | . So here I valued it for Yeah I didn't evaluate this one. |
|
| 20:14 | this is actually an indefinite integral Did did not provide any limit for |
|
| 20:19 | integration. You can also perform uh integral integral of the same function for |
|
| 20:27 | . You just need to provide the and upper bounds for that integration. |
|
| 20:31 | here what I've done is perform integration uh two times X in the range |
|
| 20:37 | , comma two. And that gives that is not four. The numerical |
|
| 20:45 | for that, that expression. So these things you will need it because |
|
| 20:52 | some point we'll get to differentiation and in the uh lecture. So the |
|
| 20:58 | will show you will obviously have some error guns. And then you can |
|
| 21:04 | uh the results that you get from processes that these direct functions and then |
|
| 21:10 | and see what the error looks So I take a look, it |
|
| 21:16 | be a little bit confusing in the the beginning but spend some time on |
|
| 21:20 | at all become clear. And just last thing if you want to time |
|
| 21:27 | in in meth labs, it's a simple process. You just need to |
|
| 21:33 | the whole section between pick and dog basically times the section that's that's between |
|
| 21:42 | . And I think we'll ask due time and see what the performance of |
|
| 21:46 | quotes looks like during in the Any questions on that? Yeah. |
|
| 22:03 | yeah yeah. This one, as said, it's a function handle. |
|
| 22:14 | it doesn't have any symbolic representation. just it's just like a shorthand function |
|
| 22:19 | you would define Or function in any programming language where you provide parameters and |
|
| 22:23 | gives you some results back the one line 22. It's a symbolic |
|
| 22:30 | So you can see it as a expression when you evaluate it in Medlab |
|
| 22:35 | then you can provide some numeric value then evaluate it on that value to |
|
| 22:40 | final numeric results for that. So just said your symbolic expressions are generally |
|
| 22:45 | when you want to see a symbolic for your whole function. Like you |
|
| 22:50 | to check what's the derivative or what the integral of a particular function or |
|
| 22:54 | equation. Thank you any other That's that's what I think on the |
|
| 23:08 | . Sure. Yes, variable. It just sort of registers variable name |
|
| 23:23 | a symbolic variable in math labs. , registry or memory. So mad |
|
| 23:27 | knows that it's a symbolic variable. . Um so if you if you |
|
| 23:32 | in directly just define a function like f. d. equals to 70 |
|
| 23:37 | metal will complain that it doesn't know he is. You need to tell |
|
| 23:41 | that it's a symbolic valuable. So nine functions that certain online line. |
|
| 23:52 | . It was like the second parameter the very black. Oh yeah. |
|
| 23:57 | I said so my club has sort I don't remember what it's called that |
|
| 24:04 | of it as an environment variable list some kind of thing in that it |
|
| 24:10 | defined default names for the variables that will consider first when evaluating a symbolic |
|
| 24:16 | . So let's say if I if don't give that comma X, it |
|
| 24:21 | uh in its environment variable list it considers X as the first video. |
|
| 24:27 | that it should look for while evaluating symbolic expression if you want. If |
|
| 24:32 | expression is defined, let's say using . Then you need to go and |
|
| 24:37 | tell it that they solve it for not X. So it it already |
|
| 24:48 | knows that it needs to look for first. That's that's defined in its |
|
| 24:54 | radio analyst. I don't know what exactly called. Yes. Yes. |
|
| 25:04 | will look for X in your in equation. Yeah, I think it's |
|
| 25:10 | it's called Semberas as who I Uh So I think this is what |
|
| 25:16 | fault. Look for. Look for keyword for on uh google and look |
|
| 25:22 | for what it means in Medlab. think it's a list that defines what |
|
| 25:28 | it looks for. All right. If you guys want to do some |
|
| 25:35 | , did you guys get a chance do the Maybe 5, 6 and |
|
| 25:41 | that we didn't get to do last you guys get that? Okay. |
|
| 25:47 | Yeah, maybe try to do these exercise them eight and 9. So |
|
| 25:52 | is basically just trying to solve for equations. So you can do that |
|
| 25:59 | using the function for her function Ah We're doing it as a |
|
| 26:06 | Thank you guys want to try now maybe. Okay, american right. |
|
| 26:17 | the function solved to define these quadratic and find out what their roots |
|
| 26:25 | And once you are done ah see you can get the derivative of these |
|
| 26:31 | functions and exercise mine. That should you started with some of the syntax |
|
| 26:43 | question. Um so on 9 17 introduce the variable Y um but I |
|
| 26:50 | it would be like a sims ahead that. Yes. Uh I think |
|
| 27:02 | I don't think you need to define . Ah I might be wrong. |
|
| 27:08 | know, these symbolic expressions are Well, no, this this is |
|
| 27:13 | a function handles. So in function . You don't specifically need to define |
|
| 27:19 | stems symbolic variables. Those are needed for the symbolic functions and symbolic |
|
| 27:28 | Sorry, say that again. I not post them yet, but you |
|
| 27:33 | , just take a look here. hmm. See if you can get |
|
| 27:49 | syntax is right for solve function and dysfunction. Remember for derivative you need |
|
| 28:01 | define the symbolic variables and then the expression, maybe I can put them |
|
| 28:12 | by side for reference. All that be simply giving you the answers. |
|
| 28:19 | the first time you guys are doing probably. Mhm, mm hmm. |
|
| 28:54 | . Oh wait, you guys can't that right now. Nobody told me |
|
| 29:01 | didn't pay attention back there. I like, I don't know. I |
|
| 29:07 | it was on my screen but okay you guys should be able to see |
|
| 29:16 | . I didn't realize it wasn't showing there. Mhm. Mm hmm. |
|
| 29:26 | . Maybe you can raid five ISH . Yeah. What? Mhm. |
|
| 30:54 | . Thank you. Mm hmm. . Mhm. All right. Uh |
|
| 31:39 | . All right. Baby. For today. Mm hmm. No, |
|
| 32:09 | it's just a simple variable. If yeah, if you if you |
|
| 32:16 | a symbolic expression or a symbolic function some radial, then you need to |
|
| 32:21 | as a symbolic variable. Using Uh well, presently, uh this |
|
| 32:30 | function. Absolutely. You don't have ask the parable. What did you |
|
| 32:35 | have to? Right, right. , mm hmm. Yeah, I |
|
| 32:56 | I should have been clear with Um What the presidents of those symbolic |
|
| 33:03 | and Mad levels If you it only into play when you have multiple variables |
|
| 33:08 | your equation, let's say a function defined using T and X. Let's |
|
| 33:14 | then by default Madala will always perform or integration for X. Because that's |
|
| 33:20 | one that's uh first in its president's for the for the variables. If |
|
| 33:26 | want to perform differentiation and integration, , then you need to specify it |
|
| 33:30 | you need to do it for Yeah. Um I think I said |
|
| 33:37 | it will give error. I don't it will give It will perform differentiation |
|
| 33:42 | integration for X 1st and everything else be considered as a constant. |
|
| 33:58 | Yeah. Hm. Thank you. right. I think we can To |
|
| 34:15 | solutions. Um Alright. For Exercise . I think it was quite |
|
| 34:23 | Uh Let's say again. I specified . Um and has to be able |
|
| 34:33 | solve for the ready for the Children solve for that. Uh So yeah |
|
| 34:39 | remember for solving equations you need to uh double equals symbol in between uh |
|
| 34:49 | left hand side and right hand side the creation, that's all you need |
|
| 34:53 | take care about. And then as can see the last equation had imaginary |
|
| 34:58 | so they provide you with um imaginary at the output. Okay? Exercise |
|
| 35:06 | you have to perform differentiation for law over X. Power four. Yeah |
|
| 35:15 | yeah I just I didn't define a using uh using log one over |
|
| 35:20 | I just provided it as an expression the dysfunction and again explicitly mentioned experts |
|
| 35:27 | me to do that. Yeah. if one obviously gives you a symbolic |
|
| 35:35 | , that's the differentiation of one over . Powerful that's -4 over X. |
|
| 35:41 | then you can substitute uh that um expression, read whatever value you will |
|
| 35:49 | for so far asked for X equals , you can use the subs function |
|
| 35:53 | to get the result But I forgot step here and what you got from |
|
| 35:58 | is again a symbolic expression. So -2. Although it looks like a |
|
| 36:02 | but it's actually a symbolic radio voluntary its class. So to actually evaluate |
|
| 36:07 | whole thing you need to add double whatever data type you want and then |
|
| 36:13 | will give you, I think everything well. Yeah, now it's it's |
|
| 36:18 | double precision number Ah same thing that 2nd 1 ah here I asked you |
|
| 36:27 | Performed the differentiation of course explicitly. four with respect to plea and here |
|
| 36:33 | the way you can do it provided expression and the function and the parameter |
|
| 36:38 | origin. It should solve for substitute to substitute the values in that symbolic |
|
| 36:44 | for the other two variables. X . And then if you add double |
|
| 36:49 | in front it's solves actually for those and give you a double precision |
|
| 36:58 | Any questions on that? Okay. unfortunate. Oh jeez ! Mhm. |
|
| 37:21 | , yeah, next. Any additional from that time? Yeah. Mm |
|
| 37:48 | , mm hmm. Good. okay. A number of the exercises |
|
| 38:03 | talking about that's what's on the web I encourage you. Yeah. Um |
|
| 38:15 | your own let us know. Um I want to switch everything from the |
|
| 38:25 | . So the first thing I've thought this what's known as a very funny |
|
| 38:31 | analysis and give a couple of examples it's simply the purpose is to try |
|
| 38:36 | figure out um but it's known that is that given the results that they |
|
| 38:44 | tried to get some sense for what of input sizes could result in that |
|
| 38:54 | up. That's the backwash that tried protect you know what potential inputs could |
|
| 39:00 | their stuff. Mm hmm. And it is a couple of simple |
|
| 39:08 | Uh, I think three or four them just simply going through this notion |
|
| 39:13 | what happens. And um, so first thing is simply that feeling and |
|
| 39:22 | sort of relative errors when numbers are exactly represent a ble in the |
|
| 39:30 | So you know, the limited position . Well, depending on the data |
|
| 39:35 | that you have and you're lucky, projection can be represented exactly by most |
|
| 39:43 | the time. It's not. And it comes to floating point and that's |
|
| 39:51 | in front of this apprentices investment calling notion that most of the time in |
|
| 39:58 | course we didn't want members. So that one they have the relative error |
|
| 40:07 | representing the variable Z. And there's magic about that. Just five minutes |
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| 40:14 | the store and forward zone that depending in this case, actually, collective |
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| 40:22 | may depend not only on the but then also on the browning road |
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| 40:27 | we talked about. That's something I with the standard stuff. The |
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| 40:32 | something yeah, approximated in the representation there's your particular in that case. |
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| 40:38 | it can be exact. And the of the era of the maximum error |
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| 40:46 | this case is depending upon some of things you have going for you the |
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| 40:54 | bit, but they don't have what . And that's what the machine epsilon |
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| 40:59 | into play that was mentioned before. there is nothing difficult. So one |
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| 41:06 | is the first thing in best representing number in the computer. And the |
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| 41:11 | thing is what happens when you do of a couple of machine numbers like |
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| 41:18 | and Y on the second floor that outcome of that meeting of the and |
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| 41:26 | machines um potential translation around that happens upon the scheme you're using. So |
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| 41:36 | instance the application, the number of industry gaps those and so then you |
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| 41:46 | represent all indigenous to translate. All . So that's just an example what |
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| 41:53 | operations and then we get another And so yes, I guess an |
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| 42:02 | that did. So in this case first thing ah just looking at what |
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| 42:10 | when you do operations on the So assuming that whatever X, Y |
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| 42:15 | Z in this case or what they be. So there is no other |
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| 42:21 | representing the variables. Let's start what are. And things may errors may |
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| 42:27 | introduced because of the operation to do and that the outcome of the operation |
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| 42:32 | be represented from this case. In what to do first on X and |
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| 42:37 | . And then they're a modification of outcome of that physique. So this |
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| 42:44 | kind of the first step then that outcome of the addition results in the |
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| 42:50 | that are not necessarily accept we represent . So that's apartments are being the |
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| 42:56 | the true value of plus some That's terrible. And it's a pretty |
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| 43:05 | and you know their possession and single and that's actually so it is bounded |
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| 43:19 | the emotional next. Yeah. You putting around the next thing is you |
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| 43:25 | the outcome of that which is what is in the machine of the true |
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| 43:31 | . And you multiply with the true and then again that may not be |
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| 43:37 | of the machine so that the product them and the potential secondary. So |
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| 43:45 | and one can basically plug all these and get messy thing and it should |
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| 43:51 | carefully then you get what the outcome the drone number but a couple of |
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| 43:59 | and actually multiplication also in the So this was for the results of |
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| 44:03 | addition of potentially insulting the modification has another potential area. But since um |
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| 44:13 | machines absolute tends to get kind of . The product is very small. |
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| 44:17 | that small. It's roughly approximately two . There's some of the errors in |
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| 44:23 | tube operations. They had the So it's just but this sign is |
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| 44:30 | trying to come into that. There a chance that there will accumulate the |
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| 44:36 | operations to do every operation introduced. of course I'm a you're lucky and |
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| 44:43 | kind of cancel each other but if want to put the band on |
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| 44:48 | I have the worst places for each and then another. Now uh huh |
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| 45:01 | include that the starting point Z and . And Y may actually be some |
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| 45:07 | that come from outside and it's not in the machine. Not exactly. |
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| 45:12 | then you have the approximation areas and checks required. Has gotten the disease |
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| 45:17 | in here. Um and then you the operations on the outcome of and |
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| 45:24 | X into the computer as well. why. Okay it was just some |
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| 45:30 | that he gets also then and um is present. So plug it all |
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| 45:35 | . Um you got a similar to but you have also done really. |
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| 45:44 | the negative effects of from design Sorry if they haven't seen this implied |
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| 46:03 | there is a multiplication that they had adding the complexity of effects and life |
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| 46:09 | our machine exact numbers. So then if you then start to look at |
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| 46:17 | relative error in the some representation The point is that you do have |
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| 46:29 | fact that developing era ends up being one term that's three that came from |
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| 46:35 | previous time and than the relative And the product of these two guys |
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| 46:45 | bargain to some other X plus And that's kind of where things are |
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| 46:52 | because it looks like this is you a safe operations and additional operations but |
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| 47:01 | sort of why it may also may very close to the negative of accident |
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| 47:05 | that's a plus but it may be numbers of this number can be very |
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| 47:10 | . So that means when you divide this number relative american factor very large |
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| 47:17 | X plus one will be very close zero. So that's the one thing |
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| 47:21 | be aware that things can blow up you are relatively speaking, they get |
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| 47:28 | of the cancelation of some of the . And I think the next fight |
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| 47:35 | just a simple in America example wow the sensor when it's again we talked |
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| 47:46 | it and the loss of significance in first lecture. So it's again, |
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| 47:52 | almost canceled each other out than relative can definitely large and that's pretty much |
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| 48:00 | I wanted to say about. Back when their analysis is just to |
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| 48:07 | it where there is shareholders seem to pretty much in every kinetic operations. |
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| 48:15 | addition to the earth of my When the first representing numbers expressions. |
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| 48:26 | the next thing was to talk about Hornet's ruler. Horner schema. That |
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| 48:32 | a commonly used and it's commonly used terms of dealing with for normals that |
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| 48:38 | talked about. Um Soon maybe I a couple of lectures on their own |
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| 48:46 | polynomial approximations are very common. It's about went to the university like |
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| 48:55 | You make observations recordings of something and get the table of stuff and sometimes |
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| 49:02 | convenient to represent collection of numbers by kind of a function. That means |
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| 49:12 | gives you some way of estimating or what the function or the observation might |
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| 49:22 | if you had looked at some other values and figure out what value would |
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| 49:27 | they want the truck. So if have a kind of a function you |
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| 49:32 | for him to say why would probably something like this, whether that's true |
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| 49:36 | not. That's a different story. the polynomial function actually represents whatever it |
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| 49:46 | , the deficit or something behind that's a different thing. But it |
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| 49:52 | it's often used as they don't use substance, degree 40% the table of |
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| 49:58 | or for that kind of I think surface that I have on the next |
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| 50:03 | so. And one of those computer expenses. It's amazing to use. |
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| 50:10 | is some kind of follow normal expression look at the surface there on some |
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| 50:14 | point. Not for the whole surface to the surface and down describe it |
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| 50:20 | some follow normal function and some of is just you can do it in |
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| 50:26 | case or another example of our almost sensitive, commonly used in a |
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| 50:32 | of senses. If you have some complex function like this expression here that |
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| 50:40 | not so easy to evaluate. Um maybe if one does some camera plug |
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| 50:46 | a few values in this direction perhaps try to be recursive into those values |
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| 50:53 | doctor might be described so much easter than they were. Yeah. Well |
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| 51:01 | taylor right. The data is one the that we talked about better pull |
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| 51:10 | as well as other polynomial. That's good point. So this is just |
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| 51:20 | that working polynomial is a common And this whole thing is something that |
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| 51:28 | a useful and trying to evaluate the foreign office. And then we'll talk |
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| 51:34 | different types of point. All So here is just mm hmm straightforward |
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| 51:43 | in normal and any degree polynomial with . And they're pretty much standard way |
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| 51:50 | recall it all now. Um That's proper politicians on the very first question |
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| 52:04 | . You know, collective value of A one and 9 If they want |
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| 52:09 | X amount of bias They want with and zero and keep calm down doing |
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| 52:16 | operations as their friends. That's The one operation or or well extra then |
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| 52:29 | you have to multiply unless the For yourself. And Okay. And |
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| 52:35 | president there will be. So if want to do it as this thing |
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| 52:39 | written Uh you would have I guess -1 for him. Whatever. Um |
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| 52:50 | end additions. Right. But if evaluate the soul, here's one multiply |
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| 52:56 | this would impact the fact three Right? Because you have to square |
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| 53:01 | . That is one. And then have to find the results A |
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| 53:05 | So that's another one. So each of these terms then would have more |
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| 53:10 | more modifications. So if that's um be quite um, a large number |
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| 53:21 | operations. So he's in a better of doing it. Yeah. For |
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| 53:28 | , decline one of them to describe better way of evaluating Yes. |
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| 53:52 | in some ways right. Um, you're done say X square Next one |
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| 54:00 | X. You if he somehow already her X squared value then you only |
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| 54:08 | one more to the excuse. But the computed expect so yes, |
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| 54:18 | But another way of doing it is super hornish corner schema. That is |
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| 54:29 | that's implicitly, yeah says you. hmm. Like many powers of accepting |
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| 54:40 | . So I'm looking at this expression everything and the nesting level historically in |
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| 54:45 | most seriousness. one month 2.1 Ad you go to the next 111 by |
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| 54:51 | as basically one month supply and one all the time until you know, |
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| 54:56 | scenario and and activists, you So yes, computers are faster. |
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| 55:10 | mean all the characters but you know ? It doesn't take much to single |
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| 55:15 | will start the interest today. um, it does affect performance. |
|
| 55:24 | ? So you have to be basically squared versus to end operations. That |
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| 55:30 | in the end. It's large in of fire that before. And |
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| 55:36 | these days, computer society LTD and computer designed dying. How are you |
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| 55:44 | make it consumption. So it also extra operations also needs not necessarily |
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| 55:53 | Mm hmm. So trying to be about they're not doing more obvious than |
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| 56:02 | takes about their expressions. It's good and their power bill for now in |
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| 56:12 | of the performance. I would do or something. You want to know |
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| 56:20 | for most computations and difficult computers The thing that limits performances and created |
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| 56:26 | operation counts to the member group. , But never let us being prudent |
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| 56:34 | highly right two expressions. There's still . And this Thank you. A |
|
| 56:44 | of examples. Just taking more completely and just writing it down. That's |
|
| 56:49 | next question. Yeah. Obviously it means, you know this simple |
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| 57:00 | That is very simple. It's just long line of american inefficient. So |
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| 57:13 | questions on that. Okay. So something that there's no expectation. |
|
| 57:26 | manipulating polynomial that that would be used later as well. Many times |
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| 57:33 | And that's basically kind of factoring a . Zor finding roots of polynomial. |
|
| 57:39 | can use this constipation and I want came out for the schema for actually |
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| 57:48 | this factory an expression however normal. in this number are factor X miners |
|
| 57:59 | . And then you get the new all about you and some other kind |
|
| 58:07 | expression that turns out to be a value for for the nobel path, |
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| 58:13 | value of our And we'll tell you those kind of things work. |
|
| 58:18 | And then it looks funny. The is it's our happens to be in |
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| 58:24 | . this polynomial then this is the , which is the zero In fact |
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| 58:31 | it's so that's concerning is kind of procedure you can we also use some |
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| 58:38 | rules to follow. Okay, so do you have to and settled down |
|
| 58:45 | the polynomial too? Is that the motor one order less than your written |
|
| 58:51 | the normal by X. So some in Q. There the senate basically |
|
| 59:00 | . To the power and but at highest degree of Cube is and -1 |
|
| 59:06 | of reduce the water for a normal time they try to sell. |
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| 59:14 | Yeah. So that's what I So here's kind of how everyone is |
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| 59:20 | it. Princeton and papers already seem steam up. It was kind of |
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| 59:24 | down all the conditions of the original starting the coefficient for the highest total |
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| 59:30 | of X. And then there now highest part in the is one over |
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| 59:38 | left. So it kind of goes the column for thanks. Okay, |
|
| 59:42 | the power of land line is one you keep breaking it down of course |
|
| 59:46 | don't know to devalue us and the one's supposed to figure out what they |
|
| 59:52 | . But it turns out that the for the highest order um X. |
|
| 59:57 | the two polynomial. It's the same in the original polynomial and then you're |
|
| 60:07 | . Um the other coefficients in the of the lower of the polynomial by |
|
| 60:15 | the coefficients in the queue polynomial with and you take the monster previously |
|
| 60:24 | So this one, you know, now we know but the computer this |
|
| 60:27 | that we can find the And -2 then let go successfully down. They're |
|
| 60:34 | and then just to find all the for the coupon but it's very simple |
|
| 60:41 | , this is under the manger term if it's zero, that that happens |
|
| 60:45 | be an exception. Um So this against the schema. It's a service |
|
| 60:54 | . And then so then so that's way of it. So the fact |
|
| 61:03 | impala normal, it would be the system distension. So you know, |
|
| 61:15 | Britain or in the comments on that I guess a concrete example. Um |
|
| 61:20 | will start with the coefficients. So b now written starting with the highest |
|
| 61:27 | . So first so it's a one minus 47 minus five minus two. |
|
| 61:33 | We want to top it off x three. Um Some of the three |
|
| 61:39 | they are so Copy that this coefficient , multiplied by three And add them |
|
| 61:46 | to -1. Take Finance one comes -3. Had adopted four down |
|
| 61:53 | No until they got 19. So are now the coefficients in the polynomial |
|
| 61:59 | degree three. Um So hopefully This factor of X -3. First coefficient |
|
| 62:11 | one. Next Professionals -1. Mayhem. So I want to graduate |
|
| 62:23 | . Queen then what we have to might see is zero. So this |
|
| 62:28 | called. So I think especially is out what I just said, whether |
|
| 62:43 | going and if one doesn't believe it we can plug in three. And |
|
| 62:54 | expression directly obviously. So three to power of four is 81 ah 32 |
|
| 63:03 | part three is 27. Right? four, that's 108 And seven times |
|
| 63:12 | 63. And he has pocketed. . It's interesting this one double check |
|
| 63:21 | to that's the proceedings Francis straightforward. yeah, let's see what else is |
|
| 63:37 | ? No, I guess the same . Right. So in this case |
|
| 63:43 | do now is that so same Do you want to follow the things |
|
| 63:49 | then turns out now they're reminded of . So that turns to is a |
|
| 63:57 | to the question. So that means . Ah its support actually constitute. |
|
| 64:05 | there's a question about the zero and expression of zero. So yes, |
|
| 64:12 | again. So um Right, so was like that then let's see what |
|
| 64:28 | . Um Yeah. Um So one one cannot tell you this and we'll |
|
| 64:38 | more about that in a later election differentiation. But these foreigners rule is |
|
| 64:46 | and derivatives are falling over. So we have the polynomial deflated the fact |
|
| 64:56 | our experiments are and now they want take the derivative. Um It can |
|
| 65:00 | the chain rule, right? But derivative of the phenomenal then is ah |
|
| 65:07 | derivative of the first Factor X Which is one times two. And |
|
| 65:13 | then expands our transit derivative of the . And and this is a constant |
|
| 65:20 | to exit the derivative of that So now we can go a lot |
|
| 65:26 | then continue if I want to do . Or someone can evaluate this may |
|
| 65:31 | uh the reputation perhaps about your arm that evaluates to A. And |
|
| 65:38 | And to do that you can go and factor too. And the remainder |
|
| 65:44 | what we want to do in terms so I can keep doing this to |
|
| 65:50 | higher order derivatives. That particular values being exploited by using mm hmm. |
|
| 66:01 | in this case seems normal. So can also be on population probably using |
|
| 66:08 | other functions other than the point in marriage. And I tell you |
|
| 66:15 | remember talking about differentiation later on differentiation in America very tricky, very unforgiven |
|
| 66:26 | integration that smooth himself. So whenever want to do things, symbolic things |
|
| 66:32 | the differentiation is proposed. Mhm. That's suspicious. Yes, showing do |
|
| 66:41 | exercise. Mhm. The first to the queue and then yeah. Ah |
|
| 66:52 | next step for one more around all off externals are in order to ah |
|
| 67:03 | get the cube of water. Let's my example here. Um Oh then |
|
| 67:13 | question of design before and the steps front of Q. And A. |
|
| 67:23 | have this one and then the next was basically to do uh the organization |
|
| 67:32 | the derivative by doing one more In fact long time I got this |
|
| 67:41 | the sermon and then come on. can, you can, who doesn't |
|
| 67:47 | it's just about things incorrectly and consider first time in my life. What |
|
| 68:00 | this? It's simply cold again. every time you basically Get one More |
|
| 68:07 | . Look it's just 1014. Uh . For this one it's been much |
|
| 68:18 | I have for today actually. So time we'll talk about taylor series that |
|
| 68:23 | mentioned here as one of the type taylor series. It's very good. |
|
| 68:30 | already familiar with it and it cannot sure you do get it into being |
|
| 68:37 | intuitive and so on top of your because it will be used extensively and |
|
| 68:44 | up properties province of approximation states. often falls back to using tendency this |
|
| 68:52 | try to listen to. I got approximation. So it's kind of that |
|
| 68:59 | used to all the chapters in the . Yeah, the best thing. |
|
| 69:05 | that's part of being familiar with it be comfortable that taylor series is important |
|
| 69:13 | then this is called an elimination of it was quickly. Hopefully everybody remember |
|
| 69:20 | song central system of equations using and elimination of steam was invented by, |
|
| 69:30 | know, recovered. Thank you. . How many do you remember that |
|
| 69:43 | ? Some systems of the Persians. . But so talk a little bit |
|
| 69:55 | athlete control errors in elimination. So for the next lecture and stay from |
|
| 70:07 | on for complex. Besides for lectures years old. Don't That's enough. |
|
| 70:27 | . Song as well. Yeah. doable of our convenience for students and |
|
| 70:38 | be uncomfortable for whatever reason. That provide some access recordings and uploading. |
|
| 70:47 | I will be here for you. you would prefer. Okay. |
|
| 70:54 | And we use the same. I we need to think about. I |
|
| 71:08 | remember the first scheduled somewhere tonight and just a little confused for sure. |
|
| 71:33 | hmm. I get my authority. hmm. I lost my question. |
|
| 71:49 | , Thank you. That's good to . Okay. Can you find my |
|
| 71:57 | and maybe I'll talk to you. , ah, thanks. Oh, |
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| 72:11 | the curse. Anonymous Spinoza. Well, that's the story |
|
