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COSC 4364 Numerical Methods - Lecture09_Richardson_Extrapolation_Integration
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00:07 So that's why I am confident Okay I'm going to so I won't
00:29 the original extrapolation and invest for dealing derivatives today. Maybe you can start
00:34 in the next chapter in american Maybe there's something here. What is
00:43 doing our own? What's what's usually to? Okay so today, so
01:01 is where mentioned was the last time last lecture in terms of approximately approximating
01:09 evidence number out quick and what was one sided and they didn't call forward
01:17 passage as opposed to the minus its lost their backwards ah approximation of the
01:25 and then the censored version and showed the centered person and higher accuracy.
01:32 terms of part it scales with h distance between the point X where you
01:40 the derivative and mhm. Other It's plus or -8. So there's
01:47 moral in some ways. Was So the differences tend to be the
01:52 received. Yes. So just make that it is right this game of
02:01 . And then I showed this in example, I think it will sign
02:06 trying to find the derivative. And that for the center of different versions
02:14 single position accuracy for a much larger because tickets squares smaller, squaring your
02:21 a lot smaller. So that's so means. And also then because you
02:27 with that in american values then the transfer lots of significance is not quite
02:34 , still around just probably close to other. So you still do
02:39 But sometimes it's also say to a bit in terms of both the sympathy
02:44 there was a number of examples last and I think this is more or
02:50 when I stopped in terms of what the last time than looking at ah
02:59 approximation and the error terms and getting there there's formula that they're producing take
03:07 serious expansion for therefore the extra stage therefore the X. Therefore X minus
03:14 and just working it out. And I find that the but the center
03:21 , see if this kind of It works ok fine. So you
03:26 this please. Um in terms of errors, the terms is basically um
03:37 of a square to start with a . The next one is which 4:38
03:45 twice and then the next one is to the 6th. So they are
03:49 basically is to some even powers in of the area. So then this
03:59 richer than recognize this particular pattern and , okay if they won't find your
04:11 , let's figure out if you can rid of these eight square terms.
04:15 that case it would be something that . There's four approximately you seem in
04:23 page to get the desired level of . So through and show what is
04:33 and didn't whatever it's just on the that works. So for that as
04:43 look at this center difference that they if I am not going to play
04:48 on the fight to um come up something that is hire experts.
04:58 So now, so there was basically they have you Okay, okay,
05:03 look at the previous time so they fine and then this was the approximation
05:13 they have basically the error terms that . Find those things. Mhm.
05:21 information that was sent a difference. it was a good approximation of the
05:28 and you're kind of the aerostructures moving around that's been left on the inside
05:32 everything else. And then you can an expression supposed to half h make
05:38 half as large and just plug things . And obviously we have a square
05:45 and we have a square here. huh. Nothing else. Not changing
05:50 professions because it's yes you sir, of it's just plugging this thing
05:57 So now we can see so here have done ah coefficient these two Divided
06:06 four effectively. So basically you can if you multiply this guy by four
06:13 subtracted from this one, said Then get rid of the H two and
06:19 simply some part of But the registered done so and said they took the
06:27 one and multiply the edge of the versions before that, the eight square
06:37 term in terms of the better expansion . And now basically have Um you
06:45 divide up three here. I have where the leading every term is,
06:51 the before. So by this fairly manipulation. Now we have An approximation
06:59 the derivative that is actually accurate to 4th ordinance of the second quarter.
07:09 it's really simple observation but they're very . So I think on the next
07:16 of the show actually happened somehow given this fire and this fire expression is
07:26 it started out to do in the of difference using at this point and
07:30 point. And by doing this manipulation plugging in age of the two,
07:39 got actually two new points. So we also have evaluating the function.
07:44 this well and yeah, he's independent and difference Plus H of the
07:51 So yes we need two more functional to to the approximation of the
07:58 So you didn't come to Oakland for But they also got something to disorder
08:05 four instead of H two. Get with something larger. Thousands of
08:13 And yes, there are some arithmetic doing this evaluation but typically what's costly
08:22 the evaluation of the function unless it's very simple function. So um So
08:31 that's that's reduced the example earlier with . Mhm. Uh When my program
08:42 , you may not think much about because the standard function most definition of
08:49 is called this function but evaluation sign access a lot more cost than
08:56 So that's why I'm saying so even such simple and common function the same
09:02 with large and exponential. Ah there quite a computational expensive compared to her
09:11 subjects. And of course, most are more complex than such moves.
09:19 one aspect is again to do consensus it from what to do approximation.
09:25 derivative is the cost of actually about . Yeah. All right.
09:35 as I said, there's nothing but site says there's nothing magic about getting
09:40 of. So the second power once gets sort of the hang of
09:47 then okay, let's go for Just go for the next permanent.
09:50 you can get something that says to six and if you're not satisfied with
09:54 , you can do uh the same over and over again to get recognizable
10:02 . So it's a very nice and observation and it's quite wonderful.
10:09 um So in this case it's to like that's just 4-16. So they
10:17 one here at 16 years, you've 15, 16. It's very
10:26 Two. So that stuff well So attractive. Take it out what's
10:35 . Yeah, they have the from previous version their exports and now they
10:39 two more points against essentially there are more function of our relations. A
10:44 bit of very traditional help with but not much. And And you
10:50 increase the order of the approximation by square from student. It's four ft
10:58 . So it's Yeah, very The most commonly used together by your
11:08 approximation of. Okay, let's Yes. So this is 27.
11:17 I guess a little bit formal derivation the spoke through it quickly. I'm
11:21 going to not too much on but it's um kind of assaults in
11:31 military, simple schematic for how to and how to be a generation.
11:40 in this case it says that what want to approximate, what they choose
11:44 do in the book is best college . So that's and then we have
11:49 this case the approximation area for the ordinary that if you wanted to and
11:58 it defines this entity that has a and in its kind of how many
12:07 by this time. But the thanks in fact, so we started with
12:15 and then we have an expression for of you to and that may have
12:18 before then between another one too. effect of that before that change yourself
12:27 the 8 6. In terms of approximation. This is just and now
12:32 going to play with this. I did completely on the previous slides and
12:37 little bit more in general. So is the tough expression was what I
12:41 of the previous life. Nothing All right. And then it's just
12:46 little bit change of notation here that be used kind of makes it somewhat
12:51 enough to prove that this in So instead of having the miners for
12:56 lower case A to the K without man's question Acceptable depending on two
13:03 This is summation. Okay, um then we have the end that is
13:09 coming here. So no, the is to see that um uh that
13:22 , you know, it's sufficiently small or eight here. That's what the
13:27 for the five functions that basically things . So the evaluation points gets very
13:34 to the actual value where they wanted well approximated, it goes to
13:41 So the best for saying that this goes to zero very quickly. Mm
13:47 . And now any points of postulates general formula for how this, which
13:55 an extrapolation kind of works and it's not seeing all of that intuitive.
14:05 we have this first step from concrete , I never had a job and
14:11 of the two and it turned oh, The first step done in
14:17 to the age of the four was do this combination of The expression for
14:25 22 H and sort of, you imagine that it is funding M equals
14:33 up here, you get 4/3 and you have something which comes from that's
14:40 hubby defined fine. And then we another one that is done one over
14:45 Moments 1. So there's three. that's kind of some of the analogous
14:51 somewhat intuitive, but psychologically the general , that's the function of how many
15:01 and and how many times have The teachers and extrapolation beyond. So
15:12 this is just copy from the previous . And then now I want to
15:16 to convince you that this is in general the jewel. And then show
15:21 this mm hmm approximation scheme. And we're making something still table that shows
15:28 you can actually use this constructively. as I said, his number inductions
15:35 it's true. And then for Hey Mom is one. And then
15:40 try to prove it's true. 10 of this Richardson extrapolation formula. So
15:49 have this expression here. And so the formula on top, right?
15:57 soon it's two for that. This this formula holds for their minus one
16:06 then they want to see if it hosts when And so on the right
16:10 side, we're going to use this for M equals M -1. So
16:16 means that they were gone. And summation is um And the is one
16:24 it's the same thing to be on formula stand there. So this is
16:30 Presuming is true for this N Coming in a certain gift in that
16:35 right there and then the next step collect the things for hell. So
16:41 therefore to the far end And then have the -1 here. And so
16:48 the same thing as the denominator. that's fine There comes and then you
16:54 to be that these two parts and corresponding back steps. So in that
17:05 what you have is this one that this multiplier, go to the end
17:15 this guy finds this uh this one have also you're saying um capital a
17:27 if you want. But then this and this expression difference in terms of
17:34 this was there was an N -1 the argument for the second expressions with
17:38 while we have the N -1 So now if you manipulate this a
17:42 bit and make it to today and the denominator here that means they get
17:50 on top. So you have taken of them on this one. And
17:55 you have The two here on the of the K. There. And
18:00 the thing that ends up. Ah too so the two candidates also the
18:05 as for the case. But this of basically from this one is one
18:10 normalize it to this. So we're done yet. But soon. So
18:18 I'm gonna have something to show her this one, it's nice to have
18:22 control that eventually someone. So have . So this was from the previous
18:31 . And then we just defined that thing that is in front of age
18:35 the to Divided by 2 to the to the to the cave power is
18:40 next kind of version of a F. Sort of have this relationship
18:46 by definition and then we're going to out what this kind of behavior.
18:51 one thing that I can see immediately that if he happens to be
18:57 it's just that's somewhere here. Um Oh sorry, yes. And this
19:04 the one but the predictor is K equal for them. Which I meant
19:08 say. Then basically this becomes So in order a mm Somewhere here
19:17 is zero. So that means when summon do the summation here. Um
19:27 the two arguments are the same then zero. So if you don't plug
19:34 in, Give us the story from plus one and then you have the
19:41 . Um and that basically shows and and these guys are then bounded because
19:49 comes from this expression. So the is that the error than yes,
19:56 very quickly because you're it's you both down with them increase the number of
20:05 into which the extrapolation. So so getting too soon here. So so
20:13 is now if you look at it behavior as a function and that m
20:20 smaller. You don't get more steps you have divided age with a bunch
20:25 times. So that's the list of um that it's a very small
20:31 Generally particularly not the giant age to . And then it's um a part
20:41 a very small numbers of decreases. it's quick, quick unravel it and
20:49 see them how quickly, basically. . So, so here is a
21:03 . Look at these things, the column that essentially to evaluate the functions
21:15 for Like we started at age and was a judge of the two and
21:20 of the four. So on this kind of using agents The # two
21:26 before except us to put down and so that gives you Yeah, I
21:34 these guys uh and what this does it takes and then I said they
21:40 the columns here actually to start with one. So there's been zero second
21:49 zero. So that's the major function . And I'm up here and then
21:59 take place for the previous problem and and combined two adjacent 30 is one
22:07 say eight and 8 32. That's it is. Yeah. So that's
22:12 this kind of little scheme myself. use these dudes to get the first
22:17 in the next column. That's basically function of H&H 22. And then
22:23 too and it's before it's up to this and more right. And you
22:29 them successively better approximation of that. just Yeah. So it's uh at
22:38 conceptually quite simple. There's a different here, you know how this racism
22:47 works. And again, the first is the only thing that involves function
22:54 . Everything else is combination always functions in the book, I think things
23:06 this written down, they changed computation and and then I N J I'm
23:13 about that coming from results and there more or less the same once again
23:20 this historic the scent of difference, it and then do the original
23:33 This is just more of the Um yeah, this is what's in
23:43 box. Very simple code. Um we need to do and of course
23:51 need to do the function evaluations little to do that for some necks.
23:58 hmm multiple but then stuff. Um that was what I had to
24:13 It's just the next operation surprisingly simple in Singapore Malaysia. So yes,
24:24 just kind of somewhere a little bit we talked about and I had no
24:32 . The method of discourse um plugging . And so who was this fella
24:39 I think to me, in addition the visual and extrapolation in many Germany
24:46 probably better known as this for the forecast factory, the weather prediction.
24:59 , so the idea is that this prediction thing that to basically compartmentalized and
25:07 space around there are into small cells then there's busy people doing things and
25:16 one of themselves and then they will merge it all together and the weather
25:21 before you have barrels computers that was computing. I think a bunch of
25:25 students. Things for each one of little cells around the world. So
25:31 something weather modeling or climate modeling, went today you will come across somebody
25:41 this the distance got started and adopt little bit perspective in terms everybody knows
25:50 Gaza come wow. Alright. So different way of looking at it instead
26:07 it basing it on functional values, directly. Clinton also think of it
26:17 first time approximate and polynomial talked about and then instead of working with the
26:24 polynomial, once you have done so what's going to happen in the next
26:31 sites. Um but then that's less before. Well the normal says it's
26:43 problems if you have a large matter notes and are selected but nevertheless and
26:51 doctor while you're doing it and see it relates to get started. So
27:00 . But for this discussion of derivatives first time using the neutral form
27:10 the general form is on the and one can start to yeah, first
27:20 approximation approximation or function F. And neuter form, it looks like it's
27:26 function by the appointment of points and there was the soak between The two
27:32 you have like zero next one but you to be the first order approximation
27:38 It's very fine between the two points so we should do the fitting.
27:47 it may not be in the memory of us paper. Right. But
27:52 and the factor was that was best the soap as an approximation kind of
27:58 the director but If you wanted an then this would be a performance difference
28:05 . Not the best before. Um sorry about that. So that's what
28:18 said that you on to the next X zero as X X one or
28:25 plus H. This is nothing but out to be the former. So
28:32 you did the first order approximation of function, do you start polynomial and
28:39 the derivative of the polynomial? The is that the same thing as before
28:46 difference and we can also use um as X minus six and minus X
28:59 . And then so in this case consistent and that becomes the center point
29:08 to that approximation points are equally distant . X for you department.
29:22 That was the same thing I think one can answer what kind of
29:26 Oh, second order approximations and I three points to play with Thanks to
29:32 next one. And then we have expression for that. And then we
29:37 the derivative of the second is the . There's just a constance of the
29:44 so what survives is what they have their First of the polling on the
29:50 between ext two and ext one. then we have this expression and if
29:56 remember we talked about the divided It was the best player in the
30:03 . The slope of the two segments between next they're on X. One
30:08 X. One and X. And the distance between the two.
30:13 you're taking the derivative of This thing obviously have experts and that's the two
30:20 . and the other one is expands . one and That's zero times
30:24 or this is simple what you got the drill the feel of the product
30:28 the 3rd start. So we can a little bit tough how this actually
30:36 out. So I think 31 Monster essentially what they had before. So
30:45 can and see what happens if no it was at the bottom of the
30:53 slide. So uh if you have X. In the mid point that
31:03 Then this term is zero. So access the midpoint zero next 1.
31:10 that means re gaps effectively. What difference from the other science and china
31:18 view this as a correction chart making approximation better. But if you pick
31:23 that's the center point there so that doesn't get better without having high road
31:34 . Okay? So then I said before right so now this was but
31:47 looked at the error term for and thought of the normal approximation of the
31:53 to show that this is in the term looks like this. So now
31:58 we look at what's the consequence if are not interested in paranormal itself,
32:06 they're interested in the director of development they're going to use the then the
32:11 of the fallen normal too as an of the derivative. And then we're
32:18 to figure out what's the error, the derivative all the way to and
32:26 doing the formal way. So take derivative with respect to each one of
32:30 arguments that one with respect to his of the factor and then another one
32:35 respected. And that was your Some of them as they confirmed the
32:40 with respect to public effects. And is a very effective C because C
32:48 on may change a little bit depending where access in the among all the
32:55 are to use for calculations being So it's not necessarily the case of
33:01 . The six. So now I'm with, take a look at
33:08 Mhm. Um To understand the derivatives don't. So that's the one thing
33:21 , as I said here, so we know, Yeah. W
33:25 Extra was the product of X to one. Oh no points you used
33:33 for that separation uh electrical. Um what it means um w of access
33:44 minus X zero X minus X Except so if X is one of
33:50 notable points W X zero like So then they only air rises from
33:58 second term. That's what I think your it's been a different stuff.
34:10 Yeah so so and then if in case if you take just two points
34:22 assume that we're looking at to deliver one of the points then what we
34:27 is the W. X. is product just of X. 3 to
34:34 points. That's it. So that's it's come from and then that's
34:42 Uh huh. Okay. Okay. happened or something? Yeah. Um
35:02 take the derivative story which was not worked out but if you work it
35:09 what to do, question what we is ah this expression that I don't
35:20 it happening because this this will be thing we had before. Right?
35:26 two x minus that. Zero minus one. Um So right so someone
35:38 . Right. So this is what have essentially. So plugging exterior what
35:43 was then it's basically that's zero X zero minus X zero becomes
35:51 So that's what so this is what conservative thing is we evaluated for fc
36:01 . And then yes, certainly. this with the forward difference in terms
36:16 the approximation because it looks like x on us X one but that X
36:21 the X plus H. Then essentially what they had in terms of the
36:27 defense finished Syed was instead to take midpoint and then that is that you
36:41 the midpoint the since. Yeah. And this product expression things are and
36:59 selected the parabola 0, 0 and one. And that means basically somewhere
37:07 that paint the wall The derivative to . And then Since it's symmetric the
37:13 is zero at the big point. this is what this is. Um
37:19 then let's plug it in and this then. And so if that second
37:26 zero then you have to worry about first term. Mm hmm. So
37:34 this case it was just 21. we have x virus X zero and
37:45 26 one. And then in order get the second jump to zero,
37:50 used the big point. So access midpoint and if you put it into
37:56 expression then what you get is this is the coefficient different or the derivative
38:07 . This first point that makes error the end of time. So Now
38:15 that X0, next one. The -H. The before which is the
38:24 of difference loaded. It's worked. I mean it's kind of last year
38:32 , which is the next separation that talked about. But you can see
38:37 even if you look at it as thinking of doing the approximation of the
38:42 with a polynomial. And the other that polynomial um if you evaluate it
38:49 of one of the endpoints or the together the same. Absolutely. That
38:58 it's directed more construction And then of one can continue to do. I
39:09 to call a normal approximation and the approximation of preservatives. So what do
39:19 say? Ah Third order polynomial They needed four points. So if
39:29 take them kind of equal space in case age and double it. What
39:36 him? Changes to H. And . Of the two from wants to
39:39 it's basically using four points in order get um the third order polynomial equal
39:47 and have fun follows from this polynomial and plug in all the right things
39:58 what to eventually we'll find out yes this is the expression big. That's
40:07 now a derivative that is um basically so and if you go back and
40:18 at the Richardson formula, it's exactly the research and formulas it's using now
40:32 looks a little bit different than some the conditions because that's right and that's
40:37 they did. And then what happens use the largest point has been to
40:41 away instead of age. So that's that has to normalize it from
40:47 Yeah. You know take the riches formula At the maximum points playing
40:53 H. Instead of H. Or this one. The place age from
40:58 age of the tool Should be replaced each of the two. Can they
41:02 4/3 and remember that was also the and believing extrapolation. There was 4
41:12 the Power N divided by four. then there's one that is outfits
41:19 And about the best 4 -1 that see. So you can see there's
41:24 the same thing happens. There's two ways. Okay. Two the
41:31 That's the approximation. So so this so this and they are determined specifically
41:43 get some from the approach. wow. Oh okay. Um Any
41:57 so far there's two different ways but can see the better the same
42:02 They have the same waiting I So the same function value is the
42:09 waiting and the same order of accuracy . Use it as playing around with
42:16 polynomial Zor taking the functions directly and the cultural expressions. Well this was
42:26 the first order derivative. So the slide was to say something. What
42:31 you do? 2nd or whatever relatives awesome common issues. So again looking
42:43 the table series expansion again for the for X. Plus mine was
42:52 And then we can see if it just two. They have these two
42:59 . First a negative term disappears and the next term ends up having the
43:07 derivative. So if they want an of the approximation of the second
43:12 one can start up, I just we're having these two days and then
43:17 have F. Of X in there may or may not want but um
43:24 you don't have the first derivative. then they have the second derivative.
43:27 then you can also see that the term here in fact happens to have
43:33 opposite sign once it's nice because when have them, this disappears. So
43:37 we got something that is the best to the fore so on are
43:45 Do this some and so I say next but they move that sort of
43:51 the left hand side. Then what have this expression that combines these two
43:58 these two. This disappears me that one is saved and this disappears over
44:04 there's been an approximation of second If we sort of multiplied by well
44:12 don't have to multiply by two basically by the square. It's delicious,
44:17 half and a half years of So your best to divide by eight
44:23 . And then what is left is from here Divided by eight square.
44:30 that means the leading term is also it's spread but it says that this
44:35 expression, second order accurate. That's approximation of the 2nd division. And
44:45 also noticed that it's a sentence You have points where they want to
44:51 and then one left and one So that people accept to do everything
44:57 expressions and you already than get the order accurate. Also the secondary.
45:06 and that's a very common in this diphtheria but the good approximation or
45:16 But so many others you can have kinds of combinations of function values.
45:20 just playing on trying to eliminate but don't want it. And I think
45:30 certainly if you look at the books several of the exercises they will cancel
45:34 combination of function values. Wow, is just more or less for the
45:46 . You can do it based on serious suspension and special Minister.
45:59 yes. Some concrete examples question on to get rid of the approximation of
46:08 second. Mm hmm. Not for . So this was again the error
46:20 british slide. Right. So it this is orders all in order to
46:27 out the error. One needs to a bound for the fourth derivative in
46:30 case in the interval. Um probably interested in the the second derivative.
46:39 then what happened? Yeah. Step the distance between the United Nations points
46:48 the function. Yeah. So, this is just coming from the previous
46:54 and the error. So we have clean out the balance for right.
47:00 hmm, forced derivative in the interval talking about. Mhm. So the
47:07 X equals one side and again, a sine function. So it's easy
47:13 we know that any derivative is no than one regardless. So it cannot
47:19 the best estimate. But it's certainly an upper bound for this derivative.
47:24 they can replace this good one. then we have um basically, mhm
47:32 age of two. And now what expression is that related to what this
47:39 told example everywhere. And then it from and this formula and the way
47:44 did this example that says that for iteration of this loop to try to
47:50 the better, better approximation and Divided a factor of four instead of
47:58 So That's that's the 1/4 ah to second power for every iteration steps in
48:11 particular code it gets H goes from two H four The age of
48:20 wow. So this is to from too for is 25. Okay.
48:29 this gives you that trying to figure if I want a single position ah
48:36 and that you don't need and it . He said, okay, fine
48:40 a boss. So five iterations of particular told we'll get you second at
48:49 um single order position together. This not an exact evaluation. It's about
48:59 you may actually get as he did this case two gaps and but that's
49:06 higher order accuracy. Yeah. Oh, required 69. Okay.
49:29 . So investors for this got started 0.25 Being invested divided by four every
49:43 . So this is a racial form successes. It's just and then now
49:48 function value and the error. So this space and then see that this
49:56 actually, it ended up being enough do for installations in this code to
50:05 single position efforts square report for it gives the second for for that
50:17 , like only square members. So this fun. It's um so
50:27 h that's produced essentially the air that's in proportion to the reduction of the
50:37 square from the production, right? is what this sets. Obviously,
50:43 age is half, then the error be roughly four times smaller and not
50:49 have a sort of something. Now code, they didn't they do this
50:58 , right. two but it is by four in the steps of the
51:03 should get Yeah, kind of it more or less in the past 16
51:10 it's it's squared. Thanks. So didn't look at the numbers but this
51:20 you look so this was the starting for h and then The age of
51:24 448 0 because it's changing age ah and then once you look at the
51:34 in this space that goes down um awesome. 2 1⁄2 times 10 to
51:42 -3. Um and the next one 10 to the -4. So that
51:49 itself Will be a factor of There is actually more than a factor
51:53 10 is from 2.5 to 1.5. it may not be quite 16 but
51:59 the first is the factor of and this case again um more or less
52:11 . That's right. The first for first and the third, It's 10
52:18 the -5 actually. So it's More a factor of a 100 and 16
52:25 to 56 and maybe more or not quite ah 2 56. But
52:32 more than, as I said, that is about 200. The convergence
52:42 all this is again the estimate using balances for us, mm hmm More
52:50 four times to get to the single . And that's because there's an upper
52:56 . So within this number of divisions they should get single position in
53:03 what is it that it's even more . Yes, when you're estimating is
53:09 and then I can also look at at some point. Again get the
53:14 of significant significant numbers are yeah, technique the same. So, and
53:22 some point began pushes to, for , using single position and then we
53:28 get there. So lots of significance a reporter construct. Keep working on
53:41 . It was not just a And in the beginning of the 14th
53:50 . Yes. So, and I even more than in this case,
53:56 there's nothing um, that prevents it using other no points is that we
54:02 need to have equal space points that simple, straightforward way, but you
54:07 try to be clever and they can better approximation. Mhm, mm
54:18 One more thing, Yes, that strengthened. Mhm. I mentioned last
54:24 in terms of derivatives being very error and some lost the significance of just
54:40 joseph very evident things can go bad these 11 effective relatives because you end
54:48 taking difference between values that may be similar. So I think the next
54:54 is using the sense of difference. see what happens because some Airbnb evaluation
55:02 this case um this is the magnitude absolute terror evaluation. So these two
55:16 um and if you're unlucky the error these two functional organizations adds up,
55:25 don't cancel out that day. So that sense the Everett This expression the
55:33 here is two times the heroin, one of them. So that means
55:41 in fact if you take the difference the error then just coming from an
55:46 decision not the order of approximation and being rested The over age of three
55:52 from the top to the bottom. the one thing or the next slide
55:59 assumed that that's Uh huh. Look the translation editor that comes from the
56:09 approximation. Not because of America the if things are then exactly Arithmetic then
56:24 got another term ascension -4 but now I said they don't have this part
56:30 it is a plantation area that is the same. Mhm. What the
56:37 is about the same magnitude as your then you get a very large area
56:44 terms of evaluation of the difference that's whether lost 15 different. So it
56:52 shows when you work we're taking the . It can easily happen that because
56:59 lack of the american position he's got significant. So in this example like
57:08 some point this is kind of worse when you start. Mm hmm.
57:22 too much think about differentiation which It just reminds for the difference and
57:31 difference is order approximation, first order order to be um which is an
57:42 And it turns over and then how approximate for one step, two steps
57:48 riches and six and then there was center of difference. You want to
57:52 one step. You can also do some extrapolation on that so that can
57:57 used in many contexts. What's interesting So any questions on differentiation before I
58:09 a little bit without inflation. So gonna take home message. This was
58:18 last time um purpose I wanted to something and actually get to experience that
58:25 Higgs, I don't have to do working like no and more so than
58:33 any other scenarios trying to get used software package that does a good job
58:38 trying to guard against. It's also and other issues in terms of the
58:45 And if you have an analytic expression the function. Trying to Yes and
58:50 olympic expression for derivatives and use those of America. This can change integration
59:01 much less integration texture. Um They a nice thing in terms of download
59:09 favorites and other things so well we'll about your integration methods. Um everyone
59:23 , you know it's about american Have done it to have some metrics
59:27 you already used. Yeah. But , that's true. But that's kind
59:36 advanced for me. But yes. huh. I thought they had,
59:44 know, something known as that. is all you, yep. So
59:59 first some basic things that they don't around with a little bit mostly focused
60:06 what some definite integral. But you have a sort of unspecified. Mm
60:16 . Interpol and integration. Right, . For me, I think the
60:21 off looking at integration especially the son all the onions speaking the up and
60:31 and that's what decides the area under . But sometimes this is in the
60:39 stay just on dr calculus one have so called indefinitely. And so they
60:47 have any particular exactly. I just starting in the points of the integration
60:55 but if you have like this analytic um appointment, you know, but
61:01 easy to do that and it didn't away to come up with a new
61:05 that represents the integral in the sense um if you take the derivative of
61:11 integrated function they should get back to integrated. So in this case it
61:17 the derivative of the right hand side comes down and becomes X two.
61:21 then take the derivative of the right side. That gets exactly. But
61:25 function is that is being integrated and think it comes on another side.
61:32 sometimes but this part right inside this called the anti derivative in the sense
61:39 if you take the derivative or they derivatives of the function back That they
61:45 derivative of a constant as zero. that means also indefinite thing to
61:53 It's undetermined to some degree because we know constantly. Um and I remember
62:05 the definite integral basically you have you do this an olympics version and Daniels
62:12 in the balance and then what to is plug in the upper bounds and
62:18 of the integration. And then you the value for we have the derivative
62:24 the right hand side. But the down so nor wandering zero. So
62:30 is disappears after all this time. to to the parfait which is like
62:38 . And then we have plus And then lower Bonneville's 10s of subjective
62:42 objective c. so it just becomes that's the mechanic. So integration.
62:53 So this is all over this. through using the different slides and number
63:03 flights coming up then they would use if as the anti derivative or the
63:12 here. Or if you have the expression then this will be integrated version
63:18 the analytic expression. And as a take the value of the definite
63:23 Is that value of the securities act by Mr. Wonderful. And then
63:32 concrete examples dimensions this part. So . So this is kind of,
63:50 will use this a little bit make it it is in principle definitely
63:58 because of bounds. But now it's stress it. Sometimes it's of interest
64:05 have a kind of a general So they got effects so they comply
64:10 playing with the expression and see what when you change the X value are
64:14 to having just a fixed number that to be the case family new symbols
64:18 and B. They're supposed to bury . So if they have this form
64:24 context, the derivative of it the this is there. And the derivative
64:30 the function boost derivative is ah have or the lower things is that they
64:37 previous gives us back the the instagram being integrated and we're going to Houston
64:46 around. And this point this slide turning there's the chance that's the point
65:01 it many times. In fact, you need to be an american for
65:12 particular function E. To the X . There is no simple analytic function
65:19 derivative is X squared. So the sometimes it's fruitless to try to find
65:27 good analytic expression whose derivative people states it to the X squared is almost
65:34 voter find at least On this interval 0 to 1 for any given
65:39 And this this is you know, number it's not particularly that so it's
65:44 well defined but it doesn't have a function. And the derivative that you
65:51 use the in that case you need different procedure to figure out what the
65:56 is this into. Now. This pretty much what this site says.
66:07 no no. Do the thing that talked to mike and used at some
66:16 . It's not the simplest but the simplest inspiration. Mm hmm. It's
66:23 quite a good one. Ah There's one that is often talked about courses
66:35 what we'll talk about fine. It's as simple and more powerful.
66:44 So uh huh. Um that respect toxic. So I dropped the sword
66:57 let's see here. Um It's a right? The house. And then
67:04 has that's the theory function values. not a rectangle but it's best for
67:13 sides to it where it is being in the chapters or rules. This
67:18 looks like Function values at the two of an interval which in this case
67:25 . and experts one. And that's said to me the integration for integral
67:33 the sum of all the Function one in the commodity in the um So
67:42 true value of the integral is essentially area on the the blue curve and
67:51 chapters. So it doesn't quite capture but not too bad. So basically
67:56 captures the area between the two endpoints instead of the car between the two
68:03 value and points, we use the line between the sector. So that's
68:10 chapter. So once a year then if you try to integrate the
68:15 you generate a bunch of chapters avoids and figure the area for each one
68:23 the chapters awards and then you'll have call out and that's your approximation for
68:28 integral of the function between now. each type of sword is pretty simple
68:38 figure out what the area is because the heights in this case of the
68:46 . So companies think of replacing the over the rectangle average straddle between
68:55 So they have to sort of the . So that means that the success
69:02 this is sexy, average heights on chapter story to here and then some
69:10 them and then the width of the . This is not the area of
69:16 chapter sword. And then you add up for all the different chapters which
69:22 sort of the composite chapters on So to me that's a pretty simple
69:32 method response and then manipulate this expression little bit. So this is um
69:43 approximation. So that's the average height the width. Um Yes, it's
69:50 spaced points that is on this then all the with the chapter sources
69:59 the multiplier instead of this expression becomes will say to the two times and
70:04 we sum it up. So this what this says now to me that's
70:13 expression is at least very easy to for me says well the two endpoints
70:20 is awaited 5.5 and all the internal between the two endpoints, they're just
70:29 because every internal, the point is of left and right chapters. Also
70:37 has come to the point and so why all the function values except for
70:41 to and from values. I understand . So that's pretty much the jacket
70:49 on y'all. Ah mm hmm. instead of me to the classic square
70:56 this to the minus X squared that remembered statistics and medals and first and
71:10 simple code for the chapters on gold to one points and the rest of
71:14 is just turning up on the Yeah. So this place, what
71:22 says now, this particular um function E to the minus X squared.
71:31 Yes. In fact it's not a function in this case we can use
71:37 as a member function. They call for any argument in this case.
71:41 interval will get the number of And if you use the simple
71:48 in fact, it's really accurate. don't see any difference friends today.
71:52 , definitely. Thank you. So worked pretty well and there was 60
71:57 in this case and we'll get to your figure outs. What age do
72:02 need to use in order to Oh sweet. What? All
72:16 awesome. This episode Who was? the claim is essentially that this is
72:30 true instagram. This was what the or rule generates. And so this
72:36 now the editor. So this was this very simple room is in fact
72:43 the edge square efforts. So that that your shrink aah the area of
72:49 proportional to the square. So that's trick. So it's quite good despite
72:56 on a simplistic and I think the boat breaking bread. But so all
73:06 water in. So we have the and the next thing is basically
73:11 that's the claim. And then the is to do, there's some manipulation
73:15 to show that do that. Try convince you that that's actually true.
73:20 if you just look at kind of of these uh trapezoid from 8 to
73:27 plus H, then there's no need , especially the two endpoints. So
73:32 one chapter. So that thing is the true interval is the chapter is
73:38 value and it's all lovely minus the B minus eight B is a plus
73:46 . So that becomes engaged to the parliament and the next question is really
73:54 . Um so now having an So now we're going to play around
73:57 this notion that you status years expression you're sure of this things like this
74:05 . So in this case now we this consider relative induced the arbitrary africom
74:18 . And so that means this is anti derivative of the integration argument.
74:24 steps. So I want to take derivative about this. You almost forget
74:29 we used to take a serious expansion anti derivative and look at the argument
74:35 A plus H and take a series pensions. So the first thing based
74:40 prime and no, because this is basis of the integration of This expression
74:54 from 8-8, basically we don't go rest or or Capital F.
74:59 A. is zero. And causes the one never after the lower bound
75:04 the same. Uh Then we have next time the derivative of capital
75:11 Lowercase sir. And then we can in the best place for them what
75:18 are. So then they investigate the here for that. This one
75:29 This is a crime that sort of case F. So we have facebook
75:34 plugging in the proper um about missing of power. Yeah. So that's
75:44 they have The one on the Now we can also do that say
75:49 certainly question of the lowercase function D. In the garden itself camera
75:55 a straightforward to this area expression and they can do says we want to
76:04 chapter soviet rule for this ah integral top from eight A plus H.
76:12 the two endpoints divided by two. to get there to this, we
76:17 therefore they In both sides and divide two. But this is what
76:25 Yeah. So then no. So the best figure ah 2.5 away
76:37 adding things to both sides, we to evaporate and then with a viable
76:42 multiplied by H. So uh the of recent question straightforward. Yeah.
76:50 more. So now only that's protected value. It's defined that it's gonna
77:00 I. And subtract this. It's . So now they have iron minus
77:06 . And look at perhaps on the hand side. So these are the
77:10 . These are also the same. um The only difference is happens here
77:16 age 23. This is three factorial the upper and this is number
77:20 So they are now. So let's there. It's like the difference that
77:26 get this stuff. So this is manipulation to show that simple Simpson's formula
77:37 what is it? A small interval proportional today. A third. But
77:44 was a small intervention to do it A and B. He added up
77:48 a bunch of these chapters toys. then we need to assume that in
77:54 worst case the errors will add So in that case age is in
78:03 um the man is a divided by of points are best because you
78:09 Mhm. Demon to add the A bunch of time. So
78:18 this tells you What's left is placed eight square. Um Thanks. Uh
78:27 number of times you have things that minus a divided by H. And
78:35 you had it that many times. that's why you get the -8.
78:40 to do is pick ourselves the divisions denominated. That's something that for the
78:47 the interval between A and B. guests something else. Second order of
78:53 for X square. You know, think the next thing is example
79:02 It's all the stuff but it's just this thing again in this table contract
79:07 there. Ah they want to this an upper bound. So ever heard
79:14 was the expression for the area We need to find the bound for
79:17 second derivative of E. To the six squared two E. to the
79:23 takes the first derivative take the second . Try to put the ball on
79:27 expression For the inter fall between zero 1. So this one is always
79:38 less than one is one divided by to do something on top here.
79:45 at one point it is zero. that there's this minus students divided by
79:50 . And at the other end It's . So it's two divided by
79:55 So who is always divided by And then after about this too.
80:05 that's an army know that had a on this and then they can use
80:09 money. Say that was once it 1/12 and then they can figure out
80:14 we need to get the desired there is on this side. So I
80:20 want to have in terms of the , what it needs to be in
80:27 to get In this case the accuracy the four digits. The same procedure
80:34 the time. Trying to find the tried to find in africa and I
80:39 that is one more example same Um, look at derivatives, the
80:46 has the same formal thing. It's the second derivative we need to
80:51 And in this case There's some things I found in 19. Oh
81:03 Let's take this one more time So we should think about this.
81:10 forward. Just do the same thing or not. Mm hmm. So
81:29 form of the work. Right and the instagram or the function that being
81:35 the successful points The left 1 to right point. It doesn't matter that
81:43 a zero on the yes. They tried to bump up in and
81:49 ask your software to do something sine zero divided by zero. It's not
81:57 . So all of them follow up time and tell you how would you
82:08 or in order to figure it out it's already been figured out.
82:14 So maybe I just, yep. that's what you can use the table
82:24 function of, say next. You necessarily need to use it all the
82:31 . But at least if you use paper series function is clear then when
82:35 is zero, Then it's recently assumed cynics of the X is one.
82:42 you can just use it for the endpoint and then depending upon your hmm
82:48 and for some small value observation, you can switch some potential to science
82:54 . So it's again, sorry, have to be created in some of
82:57 situations to figure out what's a good of approaching it from the country
83:05 Okay. Thank you. Mhm. . You
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